***********This paper was created in Microsoft word 4.0 on the Macintosh, and is saved in RTF format. If you open a file containing the body of the paper Word will convert it. Presumably this will work on PC-type computer as well. The figures are embedded in the text as postscipt. Delete everything above this line.********** {\rtf1\mac\deff2 {\fonttbl{\f0\fswiss Chicago;}{\f2\froman New York;}{\f3\fswiss Geneva;}{\f4\fmodern Monaco;}{\f13\fnil Zapf Dingbats;}{\f14\fnil Bookman;}{\f16\fnil Palatino;}{\f18\fnil Zapf Chancery;}{\f20\froman Times;}{\f21\fswiss Helvetica;} {\f22\fmodern Courier;}{\f23\ftech Symbol;}{\f34\fnil New Century Schlbk;}{\f1024\fnil Princeton;}{\f1162\fnil KochRoman;}}{\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\green255 \blue255;\red0\green255\blue0;\red255\green0\blue255; \red255\green0\blue0;\red255\green255\blue0;\red255\green255\blue255;}{\styl esheet{\s231\li1440\dxfrtext180\tldot\tx8280 \f20 \sbasedon0\snext0 toc 3;}{\s232\li440\dxfrtext180\tldot\tx8280 \f20 \sbasedon0\snext0 toc 2;}{\s233\dxfrtext180\tldot\tx8280 \f20 \sbasedon0\snext0 toc 1;}{\s242\dxfrtext180 \f20 \sbasedon0\snext0 page number;}{\s244\dxfrtext180\tqc\tx4320\tqr\tx8640 \f20 \sbasedon0\snext244 header;}{\s245\dxfrtext180 \f20\fs18\up6 \sbasedon0\snext0 footnote reference;}{\s246\dxfrtext180 \f20\fs20 \sbasedon0\snext246 footnote text;}{\s250\li1440\dxfrtext180 \f20\fs20\ul \sbasedon0\snext0 heading 6;}{\s252\li800\dxfrtext180 \f21 \sbasedon0\snext0 heading 4;}{\s253\li720\sb200\dxfrtext180 \b\f21 \sbasedon0\snext0 heading 3;}{ \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 \sbasedon0\snext0 heading 2;}{\s255\sb240\sa240\dxfrtext180 \b\scaps\f21\fs28 \sbasedon0\snext0 heading 1;}{\dxfrtext180 \f20 \sbasedon222\snext0 Normal;}{\s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \sbasedon11\snext1 Equation;}{\s2\sl-360\dxfrtext180\tx720\tx1440\tx7920 \f21 \sbasedon0\snext2 Fixed;}{\s4\qj\li720\ri720\sa300\keep\dxfrtext180 \f20\fs20 \sbasedon0\snext4 Caption;}{\s6\fi-540\li540\dxfrtext180\tx540\tx1440\tx7920 \f20 \sbasedon0\snext6 References;}{\s7\qc\keep\keepn\dxfrtext180\tqc\tx4760\tx8820 \f20 \sbasedon1\snext7 Figure;}{\s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 \sbasedon0\snext8 theorem;}{\s9\qc\keep\keepn\posxc\dxfrtext180 \f20 \sbasedon0\snext9 table;}{ \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 \sbasedon0\snext11 Text;}}\margl1440\margr1440\widowctrl\ftnbj \sectd \pgnrestart\pgnx720\pgny720\linemod0\linex0\cols1\endnhere \pard\plain \s255\qc\sb240\sa240\dxfrtext180 \b\scaps\f21\fs28 Exit Times and Transport for Symplectic Twist Maps\par \pard\plain \qc\dxfrtext180 \f20 {\fs28 R.W. Easton, J.D. Meiss and S. Carver\par Program in Applied Mathematics\par University of Colorado\par Boulder, CO 80309-0526\par \par November 3, 1992\par \par }\pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 Abstract\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 The exit time decomposition of a set yields a description of t he transport through the set as well as a visualization of the invariant structures inside it. We construct several sets, based on the ordering properties for orbits of twist maps, that are computationally easier to deal with than the construction of reson ances. Furthermore these sets can be constructed for four and higher dimensional twist mappings. For the four dimensional case\'d1using the example of Froeshl\'8e\'d1we find \'d2practically\'d3 invariant volumes surrounding elliptic fixed points. The boundaries of these regions are remarkably sharp, and apart from some \'d2holes\'d3, the regions appear to be connected.\par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 1. Introduction\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 An understanding of transport phenomena in symplectic mappings is of importance in many applications including plasma confinement, particle accelerators, chemical reaction dynamics, and fluid mixing {\fs20\up10 1-3}. Indeed, the Poincar\'8e mapping for any Hamiltonian system is symplectic, and Hamiltonian descriptions are fundamental for most physical systems.\par \tab The primary goal of a transport theory is to characterize the rates at which phase space volume moves from one region to another. Often the regions of interest are dictated by the application\'d1 for example in a particle accelerator the phase space domain corresponding to a well collimated, low temperature beam\'d1or one may choose the regions to study the transition between qualitatively different kinds of motion.\par \tab Understanding transport is a first step in the statistical analysis of dynamical systems. If a system has simple statistical properties, then it could be treated using ergodic theory. Such systems include Anosov systems, Bernoulli systems, and the like. For example, a measure preserving transformation f of a finite measure space with measure {\f23 m} is said to be {\i mixing} if given any two measurable sets A and B, {\f23 m}(f{\fs20\up6 n}(A){\f23 \'c7}B{\f23 ) \'ae m(}A{\f23 )m}(B) as n{\f23 \'ae\'a5}. Unfortunately such ideal properties are rarely found in physically motivated maps.\par \tab For the case of two dimensional maps, the description of transport in terms of turnstiles {\fs20\up10 4}, resonances {\fs20\up10 5,6} and lobes {\fs20\up10 7} is well developed an d efficient. The turnstile picture gives good predictions as to the scaling of transport near its onset, and describes the most effective barriers to transport (the cantori). The resonances give a complete partition of phase space into dynamically distinct behaviors {\fs20\up10 5,8} . A complete description of transport can be given in terms of the iterates of the lobes of the turnstiles of these resonances; unfortunately, the lobe dynamics approach does not provide insight into transport rates, nor into the fundamental question of the existence of algebraic decays. Simple approximate descriptions of this transport process, in terms of a Markovian description, while yielding easily computable predictions, give only a qualitative and difficult to justify description of the transport (including an explanation for algebraic decay {\fs20\up10 9}) except in special cases where they are quantitatively accurate {\fs20\up10 10}.\par \tab For symplectic maps of four or more dimensions, it is difficult to obtain a description of transport in terms of escape through turnstiles, except in the special case of a two dimensional map with a turnstile that is weakly coupled to an integrable two dimensional map {\fs20\up10 11} . Otherwise, for fully coupled 4d mappings, there has not been a construction of turnstiles or resonances.\par \tab Nonetheless, as we will show below, it is quite clear computationally that structures analogous to resonances exist in these systems. In fact one would like to develop a notion of \'d2practical stability\'d3\'d1 orbits are trapped in a region of phase space for all practical purposes even if, as one believes, many will eventually escape by the Arnol\'d5d mechanism (Arnol\'d5d diffusion). Practical stability would extend the theory of Nekhoroshev {\fs20\up10 12} , which guarantees that the actions of an integrable Hamiltonian perturbed by {\f18 O}({\f23 e)} will drift only by {\f18 O}({\f23 e) }in exponentially long times (these ideas are exploited in the work of {\fs20\up10 13} ). Our concern is not this case of weak perturbation, but that of fully coupled systems.\par \tab In this paper we develop several new statistical measures describing transport. These are based on the exit time decomposition of a set {\fs20\up10 6}, which we review in \'a4 2. The sets we consider are tailored to probing the orbits trapped in the neighborhood of an elliptic periodic orbit. \par \tab For the case of twist maps, the Lagrangian description of the dynamics leads to the notion that orbits trapped near an elliptic (minimax) orbit must live in the orbit of the gaps formed by its hyperbolic (minimizing) partner. We discuss the exit time deco mposition the set based on these gaps in \'a45 and prove that the set of orbits trapped forever in the strips is identical to those orbits trapped in the resonance. We also introduce the concept of the width of an orbit in \'a4 6, which is easier to compute than the escape from the orbit of gaps since the hyperbolic orbit need not be computed. The set of orbits with width less than one is also identical to the invariant set of the resonance.\par \tab Generalizations of these concepts to higher dimensions are discussed in \'a47.\par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 2. Exit Times\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 A typical question one woul d like to answer is illustrated by the case of a periodically driven frictionless pendulum. Suppose the pendulum is initially swinging back and forth. What is the probability that n periods later it will be rotating? There is a region in phase space that c orresponds to rocking motion. The fraction of initial states that exit this region in exactly n periods is the desired probability. In order to discuss questions like these, we introduce several measures of escape times.\par \tab Let f be a mapping on some phase space, and A and B be sets in this space. As a basis for our statistical measures we focus attention on the minimum time for points of A to reach B. Define a {\i crossing time functio}n t: A{\f23 \'ae}[0,{\f23 \'a5} ] by \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 {\dn32 \tab }{\dn32 {\pict\macpict\picw247\pich38 077e00000000002600f7110101000afc18fc18040e04df0700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb7787072000000100014000c0000a0008c a15745000603ee00090003a15745000a70ee00140554696d6573a15745000a71ee001400c000 000174a15745000a71ee001400 c000000128a15745000a71ee001400c000000161a15745000a71ee001400c000000129a15745 000a71ee001400c000000120a15745000a71ee001400c00000013da15745000a71ee001400c0 00000120a15745000a71ee001400c000000120a0008ca15745000602ee120100006100000022 00140028010e00526200000023 00140027010e005a610000001d00140023005a0052620000001e00140022005a005a61001400 1d0028002300080052620014001e002800220000005a61001400220028002800bc0052620014 00230028002700b4005aa0008ca15745000603ee000100032c000800140554696d6573030014 0d000c2e0004000000002a1608 74286129203d2020a0008ca15745000605ee02010100a0008ca15745000603ee001e0003a157 45000b70ee00170653796d626f6ca15745000a71ee001700c0000001a5a15745000a71ee0014 00c0000001202c000900170653796d626f6c03001728000d002501a5a15745000a71ee001400 c000000169a15745000a71ee00 1400c000000166a15745000a71ee001400c000000120a15745000a71ee001400c000000120a0 008ca15745000602ee50030000a0008ca15745000603ee000100030300142909052069662020 a15745000a71ee001400c00000016629100166a0008da0008ca15745000603ee00000003a000 8da0008ca15745000603ee0001 0003a15745000a71ee001400900000016e0d00092800090042016ea0008da0008da15745000a 71ee001400c000000120a15745000a71ee001400c000000128a15745000a71ee001400c00000 0161a15745000a71ee001400c000000129a15745000a71ee001400c000000120a15745000a71 ee001700c0000001cf0d000c2b 0504052028612920a15745000a71ee001400c000000120030017291301cfa15745000a71ee00 1400c000000142a15745000a71ee001400c000000120a15745000a71ee001400c000000120a15745000a71ee001400c000000166a15745000a71ee001400c00000016fa15745000a71ee001400c000000172a15745000a71ee 001400c000000120a15745000a71ee001400c000000161a15745000a71ee001400c00000016c a15745000a71ee001400c00000016ca15745000a71ee001400c000000120a15745000a71ee00 1400c00000016ea15745000a71ee001400c000000120a15745000a71ee001400c00000013ea1 5745000a71ee001400c0000001 20a15745000a71ee001400c00000013003001429091120422020666f7220616c6c206e203e20 30a0008da0008ca15745000603ee002b0003a15745000a71ee001400c000000174a15745000a 71ee001400c000000168a15745000a71ee001400c000000165a15745000a71ee001400c00000 0120a15745000a71ee001400c0 0000016ca15745000a71ee001400c000000165a15745000a71ee001400c000000161a1574500 0a71ee001400c000000173a15745000a71ee001400c000000174a15745000a71ee001400c000 000120a15745000a71ee001400c00000016ea15745000a71ee001400c000000120a15745000a 71ee001400c000000173a15745 000a71ee001400c000000175a15745000a71ee001400c000000163a15745000a71ee001400c0 00000168a15745000a71ee001400c000000120a15745000a71ee001400c000000174a1574500 0a71ee001400c000000168a15745000a71ee001400c000000161a15745000a71ee001400c000 000174a15745000a71ee001400 c000000120a15745000a71ee001400c000000120a0008ca15745000602ee50030000a0008ca1 5745000603ee00010003280022002517746865206c65617374206e2073756368207468617420 20a15745000a71ee001400c00000016629620166a0008da0008ca15745000603ee00000003a0 008da0008ca15745000603ee00 010003a15745000a71ee001400900000016e0d000928001e008b016ea0008da0008da1574500 0a71ee001400c000000120a15745000a71ee001400c000000128a15745000a71ee001400c000 000161a15745000a71ee001400c000000129a15745000a71ee001400c000000120a15745000a 71ee001700c0000001ce0d000c 2b0504052028612920a15745000a71ee001400c000000120030017291301cea15745000a71ee 001400c000000142a15745000a71ee001400c000000120a15745000a71ee001400c000000120 a15745000a71ee001400c00000016fa15745000a71ee001400c000000174a15745000a71ee00 1400c000000168a15745000a71 ee001400c000000165a15745000a71ee001400c000000172a15745000a71ee001400c0000001 77a15745000a71ee001400c000000169a15745000a71ee001400c000000173a15745000a71ee 001400c00000016503001429090d204220206f7468657277697365a0008da0008da0008da000 8da0008da100b6000400010001 a100b6000400010001a00083ff}}\tab (1)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 If B is the complement of A then the crossing time function is called the {\i exit time} and denoted t{\fs20\up6 +}. \tab If the map f is invertible, we define the {\i backward exit time} for a point a{\f23 \'ce}A as\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\dn32 {\pict\macpict\picw252\pich38 07e200000000002600fc110101000afc18fc18040e04e40700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb7787072000000100014000c0000a0008c a15745000603ee00090003a0008ca15745000602ee50030000a0008ca15745000603ee000100 03a15745000a70ee0014055469 6d6573a15745000a71ee001400c0000001742c000800140554696d65730300140d000c2e0004 000000002a160174a0008da0008ca15745000603ee00000003a0008da0008ca15745000603ee 00010003a15745000a71ee00140090000001d00d0009280012000301d0a0008da0008da15745 000a71ee001400c000000128a1 5745000a71ee001400c000000161a15745000a71ee001400c000000129a15745000a71ee0014 00c000000120a15745000a71ee001400c00000013da15745000a71ee001400c000000120a157 45000a71ee001400c000000120a0008ca15745000602ee1201000061000000270014002d010e 005262000000280014002c010e 005a610000002200140028005a0052620000002300140027005a005a61001400220028002800 0800526200140023002800270000005a61001400270028002d00bc005262001400280028002c 00b4005aa0008ca15745000603ee000100030d000c2b050407286129203d2020a0008ca15745 000605ee02010100a0008ca157 45000603ee001d0003a15745000b70ee00170653796d626f6ca15745000a71ee001700c00000 01a5a15745000a71ee001400c0000001202c000900170653796d626f6c03001728000d002a01 a5a15745000a71ee001400c000000169a15745000a71ee001400c000000166a15745000a71ee 001400c000000120a15745000a 71ee001400c000000120a0008ca15745000602ee50030000a0008ca15745000603ee00010003 0300142909052069662020a15745000a71ee001400c00000016629100166a0008da0008ca157 45000603ee00000003a0008da0008ca15745000603ee00020003a15745000a71ee0014009000 00012da15745000a71ee001400 900000016e0d00092800090047022d6ea0008da0008da15745000a71ee001400c000000128a1 5745000a71ee001400c000000161a15745000a71ee001400c000000129a15745000a71ee0014 00c000000120a15745000a71ee001700c0000001ce0d000c2b08040428612920a15745000a71 ee001400c00000012003001729 1001cea15745000a71ee001400c000000141a15745000a71ee001400c000000120a15745000a 71ee001400c000000120a15745000a71ee001400c000000166a15745000a71ee001400c00000 016fa15745000a71ee001400c000000172a15745000a71ee001400c000000120a15745000a71 ee001400c000000161a1574500 0a71ee001400c00000016ca15745000a71ee001400c00000016ca15745000a71ee001400c000 000120a15745000a71ee001400c00000016ea15745000a71ee001400c000000120a15745000a 71ee001400c00000013ea15745000a71ee001400c000000120a15745000a71ee001400c00000 01300300142909112041202066 6f7220616c6c206e203e2030a0008da0008ca15745000603ee002a0003a15745000a71ee0014 00c000000174a15745000a71ee001400c000000168a15745000a71ee001400c000000165a157 45000a71ee001400c000000120a15745000a71ee001400c00000016ca15745000a71ee001400 c000000165a15745000a71ee00 1400c000000161a15745000a71ee001400c000000173a15745000a71ee001400c000000174a1 5745000a71ee001400c000000120a15745000a71ee001400c00000016ea15745000a71ee0014 00c000000120a15745000a71ee001400c000000173a15745000a71ee001400c000000175a157 45000a71ee001400c000000163 a15745000a71ee001400c000000168a15745000a71ee001400c000000120a15745000a71ee00 1400c000000174a15745000a71ee001400c000000168a15745000a71ee001400c000000161a1 5745000a71ee001400c000000174a15745000a71ee001400c000000120a15745000a71ee0014 00c000000120a0008ca1574500 0602ee50030000a0008ca15745000603ee00010003280022002a17746865206c65617374206e 207375636820746861742020a15745000a71ee001400c00000016629620166a0008da0008ca1 5745000603ee00000003a0008da0008ca15745000603ee00020003a15745000a71ee00140090 0000012da15745000a71ee0014 00900000016e0d000928001e0090022d6ea0008da0008da15745000a71ee001400c000000128 a15745000a71ee001400c000000161a15745000a71ee001400c000000129a15745000a71ee00 1400c000000120a15745000a71ee001700c0000001cf0d000c2b08040428612920a15745000a 71ee001400c000000120030017 291001cfa15745000a71ee001400c000000141a15745000a71ee001400c000000120a1574500 0a71ee001400c000000120a15745000a71ee001400c00000016fa15745000a71ee001400c000 000174a15745000a71ee001400c000000168a15745000a71ee001400c000000165a15745000a 71ee001400c000000172a15745 000a71ee001400c000000177a15745000a71ee001400c000000169a15745000a71ee001400c0 00000173a15745000a71ee001400c00000016503001429090d204120206f7468657277697365 a0008da0008da0008da0008da0008da100b6000400010001a100b6000400010001a00083ff}} \tab (2)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 The {\i transit time }for a point in A is the sum of the forward and backward exit times for that point.\par \tab The subset of points in A that exit in time j is denoted {\f18 E}{\fs20\dn4 j}, those that have backward exit time j are denoted {\f18 I}{\f18\fs20\dn4 j} (for {\i incoming)}, and those with transit time j are denoted {\f18 T}{\fs20\dn4 j}. The {\i exit set }for A is the set {\f18 E} {\f23 \'ba} {\f18 E}{\f18\fs20\dn4 1}, and the {\i entry set} is {\f18 I} = {\f18 I}{\fs20\dn4 1}{\f18 .} If the set A is measurable then so are various sets defined by the exit time functions. \par \tab The {\i survival sequence} associated with A is the sequence \{s{\fs20\dn4 j}\} where s{\fs20\dn4 j }is the measure of the set of points in A that exit in time greater than or equal to j. The {\i escape time distribution}, e{\fs20\dn4 j }{\f23 \'ba} s{\fs20\dn4 j+1}\'d0 s{\fs20\dn4 j}, is the measure of particles that escape at time j.\par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 3. Twist Maps\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 For our purposes a {\i twist map} is an exact symplectic map{\fs18\up6 \'a0{\footnote \pard\plain \s246\dxfrtext180 \f20\fs20 {\fs18\up6 \'a0} A mapping is exact symplectic if the integral of y dx over an arbitrary closed loop is invariant upon iteration.}} \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab (x{\f23 \'a2,}y{\f23 \'a2}) = f(x,y)\tab (3){\fs20\up6 \par }\pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 on T{\fs20\up6 d}{\f23 \'b4}R{\fs20\up6 d}, such that the map {\f23 j: }(x,y) {\f23 \'ae }(x,x{\f23 \'a2}(x,y)){\f23 }is a diffeomorphism {\fs20\up10 14,15} It is sufficient that the matrix \'b6x{ \f23 \'a2}/\'b6y be nonsingular, and we will require this to be the case. We denote an orbit by the sequence (x{\fs20\dn4 t}, y{\fs20\dn4 t}) = f(x{\fs20\dn4 t-1},y{\fs20\dn4 t-1}).\par \tab A twist map can written in a Lagrangian form {\f23 L}: (x{\fs20\dn4 t-1},x{\fs20\dn4 t}) {\f23 \'ae} (x{\fs20\dn4 t},x{\fs20\dn4 t+1}), and an orbit of such a map is determined by a sequence \{...x{\fs20\dn4 t}, x{\fs20\dn4 t+1},.....\} . Often we will consider a lift of the map in order to distinguish between orbits that rotate and those that do not, so that the configuration x{\f23 \'ce}R{\fs20\up6 d}, rather than T{\fs20\up6 d}.\par \tab An important property of orbits of a twist map is the {\i frequency vector }, {\f23 w\'ce}R{\fs20\up6 d}, defined as the limit\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {{\pict\macpict\picw52\pich24 02d80000000000180034110101000afc18fc180400041c0700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000b0017000c0000a0008c a15745000603ee00050003a15745000b70ee00170653796d626f6ca15745000a71ee001700c0 000001772c000900170653796d 626f6c0300170d000c2e0004000000002a0d0177a15745000a70ee00140554696d6573a15745 000a71ee001400c0000001202c000800140554696d657303001429080120a15745000a71ee00 1400c00000013d2903013da15745000a71ee001400c00000012029070120a0008ca157450006 02ee50e40000a0008ca1574500 0603ee00010003a0008ca15745000602ee00120000a100b6000400010001a100b60004000100 04070001000122000a002a0a00a0008ca15745000603ee00010003a0008ca15745000602ee50 030000a0008ca15745000603ee00010003a15745000a71ee001400c000000178280007002a01 78a0008da0008ca15745000603 ee00010003a15745000a71ee00140090000001740d00092b06020174a0008da0008ca1574500 0603ee00000003a0008da0008da0008da0008ca15745000603ee00010003a15745000a71ee00 1400c0000001740d000c280015002d0174a0008da0008da0008da0008ca15745000603ee0003 0003a15745000e70ee00150948 656c766574696361a15745000a71ee001500c00000016c2c000c00150948656c766574696361 03001528000d0016016ca15745000a71ee001500c00000016929030169a15745000a71ee0015 00c00000016d2903016da0008da0008ca15745000603ee00030003a15745000a71ee00140090 000001740300140d0009280015 00150174a15745000a71ee00170090000001ae030017290301aea15745000a71ee0017009000 0001a5290901a5a0008da0008ca15745000603ee00000003a0008da0008da0008da100b60004 00040001a100b6000400010001a00083ff}}\tab (4)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 if it exists. In particular, a periodic orbit is defined by the condition x{\fs20\dn4 t+n }= x{\fs20\dn4 t}+m, so it has a rational frequency {\f23 w} = m/n, where m{\f23 \'ceZ}{\fs20\up6 d} , and n is the minimal period of the orbit.\par \tab The standard example of a twist mapping is\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {{\pict\macpict\picw108\pich35 0414000000000023006c110101000afc18fc18040b04540700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000d0014000c0000a0008c a15745000603ee00050003a15745000a70ee00140554696d6573a15745000a71ee001400c000 0001662c000800140554696d65 730300140d000c2e0004000000002a160166a15745000a71ee001400c00000013a2904013aa1 5745000a71ee001400c00000012029030120a15745000a71ee001400c00000012029030120a0 008ca15745000602ee12010000610000000f00120015010e0052620000001000120014010e00 5a610000000a00120010005a00 52620000000b0012000f005a005a610012000a0024001000080052620012000b0024000f0000 005a610012000f0024001500bc005262001200100024001400b4005aa0008ca15745000603ee 00010003a0008ca15745000604ee000200aea0008ca15745000603ee000d0003a15745000a71 ee001400c00000017928000d00 120179a15745000a71ee001400c00000012029060120a15745000b70ee00170653796d626f6c a15745000a71ee001700c0000001ae2c000900170653796d626f6c030017290301aea1574500 0a71ee001400c000000120030014290c0120a15745000a71ee001400c00000017929030179a1 5745000a71ee001400c0000001 2029060120a15745000a71ee001400c0000001d0290301d0a15745000a71ee001400c0000001 2029060120a15745000a71ee001700c0000001d1030017290301d1a15745000a71ee001400c0 0000017603001429090176a15745000a71ee001400c00000012829060128a15745000a71ee00 1400c00000017829040178a157 45000a71ee001400c00000012929060129a0008da0008ca15745000603ee00110003a1574500 0a71ee001400c00000017828001f00120178a15745000a71ee001400c00000012029060120a1 5745000a71ee001700c0000001ae030017290301aea15745000a71ee001400c0000001200300 14290c0120a15745000a71ee00 1400c00000017829030178a15745000a71ee001400c00000012029060120a15745000a71ee00 1400c00000012b2903012ba15745000a71ee001400c00000012029070120a15745000a71ee00 1400c00000017929030179a15745000a71ee001400c00000012029060120a15745000a71ee00 1400c0000001d0290301d0a157 45000a71ee001400c00000012029060120a15745000a71ee001700c0000001d1030017290301 d1a15745000a71ee001400c00000017603001429090176a15745000a71ee001400c000000128 29060128a15745000a71ee001400c00000017829040178a15745000a71ee001400c000000129 29060129a0008da0008da0008d a0008da0008da100b6000400010001a100b6000400010001a00083ff}}\tab (5)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 where the potential v(x) is periodic, v(x+m) = v(x) for m{\f23 \'ce}Z{\fs20\up6 d}. The map can be expressed in Lagrangian form as\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {{\pict\macpict\picw126\pich35 03f0000000000023007e110101000afc18fc18040b04660700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000d0017000c0000a0008c a15745000603ee00050003a15745000b70ee00170653796d626f6ca15745000a71ee001700c0 0000014c2c000900170653796d 626f6c0300170d000c2e0004000000002a16014ca15745000a70ee00140554696d6573a15745 000a71ee001400c00000013a2c000800140554696d65730300142908013aa15745000a71ee00 1400c00000012029030120a15745000a71ee001400c00000012029030120a0008ca157450006 02ee1201000061000000130012 0019010e0052620000001400120018010e005a610000000e00120014005a0052620000000f00 120013005a005a610012000e0024001400080052620012000f002400130000005a6100120013 0024001900bc005262001200140024001800b4005aa0008ca15745000603ee00010003a0008c a15745000604ee000200aea000 8ca15745000603ee00060003a15745000a71ee001400c00000017828000d00190178a1574500 0a71ee001400c00000012029060120a15745000a71ee001700c0000001ae030017290301aea1 5745000a71ee001400c000000120030014290c0120a15745000a71ee001400c0000001782903 0178a15745000a71ee001700c0 000001a2030017290601a2a0008da0008ca15745000603ee00150003a15745000a71ee001400 c00000017803001428001f00160178a15745000a71ee001700c0000001a2030017290601a2a1 5745000a71ee001400c00000012003001429030120a15745000a71ee001700c0000001ae0300 17290301aea15745000a71ee00 1400c000000120030014290c0120a15745000a71ee001400c00000013229030132a15745000a 71ee001400c00000017829060178a15745000a71ee001700c0000001a2030017290601a2a157 45000a71ee001400c00000012003001429030120a15745000a71ee001400c0000001d0290301 d0a15745000a71ee001400c000 00012029060120a15745000a71ee001400c00000017829030178a15745000a71ee001400c000 00012029060120a15745000a71ee001400c0000001d0290301d0a15745000a71ee001400c000 00012029060120a15745000a71ee001700c0000001d1030017290301d1a15745000a71ee0014 00c00000017603001429090176 a15745000a71ee001400c00000012829060128a15745000a71ee001400c00000017829040178 a15745000a71ee001700c0000001a2030017290601a2a15745000a71ee001400c00000012903 001429030129a0008da0008da0008da0008da0008da100b6000400010001a100b60004000100 01a00083ff}}\tab (6)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 For the two dimensional case, (5) becomes the \'d2standard map\'d3 upon the choice\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {{\pict\macpict\picw107\pich27 02e000000000001b006b110101000afc18fc18040304530700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000e0014000c0000a0008c a15745000603ee00130003a15745000a70ee00140554696d6573a15745000a71ee001400c000 0001762c000800140554696d65 730300140d000c2e0004000000002a0d0176a15745000a71ee001400c00000012829060128a1 5745000a71ee001400c00000017829040178a15745000a71ee001400c00000012929060129a1 5745000a71ee001400c00000012029040120a15745000a71ee001400c00000013d2903013da15745000a71ee001400c0000001 2029070120a15745000a71ee001400c0000001d0290301d0a15745000a71ee001400c0000001 2029060120a0008ca15745000602ee00120000a100b6000400010001a100b600040001000407 0001000122000a002a1300a0008ca15745000603ee00010003a15745000a71ee001400c00000 016b2800090030016ba0008da0 008ca15745000603ee00020003a15745000a71ee001400c000000134280018002a0134a0008c a15745000602ee50030000a0008ca15745000603ee00010003a15745000a71ee001400c00000 01b9290601b9a0008da0008ca15745000603ee00000003a0008da0008ca15745000603ee0001 0003a15745000a71ee00140090 000001320d000928001400370132a0008da0008da0008da0008da15745000a71ee001400c000 0001200d000c28000d003d0120a15745000a71ee001400c00000016329030163a15745000a71 ee001400c00000016f2905016fa15745000a71ee001400c00000017329060173a15745000a71 ee001400c00000012829050128 a15745000a71ee001400c00000013229040132a15745000a71ee001400c0000001b9290601b9 a15745000a71ee001400c00000017829070178a15745000a71ee001400c00000012929060129 a0008da100b6000400040001a100b6000400010001a00083ff}}\tab (7)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 For all k>0 this map has hyperbolic fixed points {\i p} and {\i q} located at (-.5,0 ) and (.5, 0 ). These points are connected by heteroclinic orbits going from the first to the second and from the second back to the first forming a heteroclinic cycle. There is an elliptic fixed point located at (0,0) for 00, f{\fs20\up6 j}({\i z}) belongs to the half open segment W{\fs20\up6 s}[{\i a,} f({\i a})) of stable manifold between {\i a} and f({ \i a}). This segment is outside the closure of f(T){\f23 \'c7}S and hence f{\fs20\up6 j}({\i z}) does not belong to this closed set. This is a contradiction because f{\fs20\up6 j}({\i z}) does belong to f(T){\f23 \'c7}S . Hence the orbit of {\i z} must exit from S.\par \tab By twist, every point on the vertical half lines above {\i b} and {\i q} maps outside the closure of S. The vertical segment {\f23 g} between {\i a} and {\i b} maps into T. Thus every point in the closure of T that does not belong to W{\fs20\up6 s}[{ \i a},{\i q}] exits to the right from the closure of S.{\f13 o\par }\tab In the proof of Prop. 2 we did not assume that the segment of stable manifold W{\fs20\up6 s}(f({\i a}),{\i q}) is contained in the strip between v{\fs20\dn4 q} and v{\fs20\dn4 f(a)} ; however, Prop. 2 actually implies that this is true and thus that {\i a}{\fs20\dn4 1} = {\i a}. This follows since every point in T{\fs20\dn4 1} must exit S, so T{\fs20\dn4 1} cannot intersect W{\fs20\up6 s}[{\i a},{\i q} ]. Applying the same conclusion to the original strip S shows that the preimage of W{\fs20\up6 s}[{\i a},{\i q}] cannot intersect T.\par \tab Proof of theorem 1: The complement of the resonance zone can be broken into four pieces. Each piece is the part of a vertical strip above or below a segment of stable or unstable manifold forming the boundary of the resonance zone. For example, one piece is the part of the strip between the vertical lines v{\fs20\dn4 a} and v{\fs20\dn4 q} that is above the stable segment W{\fs20\up6 s}[{\i a},{\i q} ]. By Prop. 2 the forward orbits of all points in this piece exit this strip to the right. A second piece of the complement is the part of the vertical strip between the lines v{\fs20\dn4 p} and v{\fs20\dn4 a} that is above the unstable segment W{ \fs20\up6 u}[p,a]. Proposition 2 applied to f{\fs20\up6 -1}, which is a left-twisting map, shows that the backward orbit of any point in this piece exits this strip to the l eft. The two pieces of the complement that are below the resonance zone are treated similarly.{\f13 o\par }\tab For an (m,n) resonance, the strip V is replaced by a set of n strips using the gaps in the minimizing periodic orbit. Let V{\fs20\dn4 t} be the vertical strip defined by the closure of the gap g{\fs20\dn4 t } associated with a minimizing (m.n) orbit: V{\fs20\dn4 t} = \{z : {\f23 p(}z) {\f23 \'ce }\|{\f23 O(}{\f23\fs20\up20 _}{\f23 ,}g){\fs20\dn4 t}\}. Then an orbit is said to be in the set of strips V{\fs20\dn4 t} if there is a k{\f23 \'ceZ} such\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\f23 p(}z{\fs20\dn4 t}) {\f23 \'ce }\|{\f23 O(}{\f23\fs20\up20 _}{\f23 ,}g){\fs20\dn4 t+k} for all t{\f23 \'ceZ} \tab (13)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Since the stable and unstable manifolds of the minimizing orbit and its right neighbor intersect to form a heteroclinic cycle, the resonance construction can be generalized to this case {\fs20\up10 5 } and is easy to generalize Th. 1:\par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Theorem 3}: For a twist map satisfying hypothesis A for the (m,n) resonance, the set of points whose orbits are contained in the sequence of strips V{\fs20\dn4 t} associated with the (m,n) resonance is identical to the maximal invariant set of the (m,n) resonance.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 \tab Thus the invariant sets contained in the strips are identical to those in the resonanc e zones. Furthermore a result of Easton implies that the asymptotic escape rates for these sets are the same, since they can be made into nested isolating blocks {\fs20\up10 22} . It is numerically simpler to construct the exit time decomposition for the strip than for the resonance itself. However, our prime motivation for this construction is its generalization to four dimensions, which we defer until \'a47.\par \tab One deficiency of the strip is that it uses vertical lines as boundaries. These appear artificially since they have no invariant definition\'d1 though of course the vertical is privileged due to the definition of twist. We discuss an alternate definition next.\par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 6. The Width Function\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 For a symplectic twist map, f, of the plane, the {\i width }of a point z=(x,y) relative to frequency {\f23 w} is defined as \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\dn26 {\pict\macpict\picw215\pich27 116000000000001b00d71101a10064000eb77870720000000d000000000000a101f200164578 b972000d00000000001b00d74772897068af626a01000afff5ffc6002601112c000900170653 796d626f6c0300170d00092e000400ff01002b0a0a01202c000800140554696d65730300140d 000c2902012007000000002300 00a000bea100c0000d63757272656e74706f696e7420a000bf22000100010000a000bea100c0 03fe7472616e736c6174652063757272656e74706f696e74207363616c65200d31207365746c 696e656361702030207365746c696e656a6f696e202032207365746d697465726c696d69740d 36206172726179206375727265 6e746d617472697820616c6f616420706f7020706f7020706f70206162732065786368206162 7320616464206578636820616273206164642065786368206162732061646420322065786368 20646976202f6f6e6570782065786368206465660d5b20302e3233383830362030203020302e 32333838303620302030205d20 636f6e6361740d2f54207b2f6673697a65206578636820646566202f66756e64657220657863 682030206e6520646566202f666974616c2065786368206673697a65206d756c203420646976 20646566202f66626f6c6420657863682030206e65206465660d2f666d6174205b206673697a 65203020666974616c20667369 7a65206e656720302030205d206465662066696e64666f6e7420666d6174206d616b65666f6e 7420736574666f6e74206d6f7665746f7d2062696e64206465660d2f74207b66756e64657220 7b63757272656e74706f696e742030206673697a652031322064697620726d6f7665746f2032 20696e64657820737472696e67 776964746820726c696e65746f206673697a6520323420646976207365746c696e6577696474 68207374726f6b65206d6f7665746f7d2069660d66626f6c64207b63757272656e74706f696e 74206673697a6520323420646976203020726d6f7665746f203220696e6465782073686f7720 6d6f7665746f7d206966207368 6f777d2062696e64206465660d30203539202f54696d65732d526f6d616e2030203020302035 3020540d28772920740d3335203731202f53796d626f6c20302030203020333820540d287729 20740d3835203539202f54696d65732d526f6d616e20302030203020353020540d287a292074 0d2f48207b6e65777061746820 6d6f7665746f203220636f7079206375727665746f203220636f7079206375727665746f2066 696c6c7d2062696e64206465660d373320323120373320343620373620323120383420323120 373620343620480d373320373120373320343620373620373120383420373120373620343620 480d3131392032312031313920 343620313136203231203130382032312031313620343620480d313139203731203131392034 3620313136203731203130382037312031313620343620480d313434203539202f54696d6573 2d526f6d616e20302030203020353020540d283d2920740d323738203539202f53796d626f6c 20302030203020353020540d28 702920740d333139203539202f54696d65732d526f6d616e20302030203020353020540d2866 2920740d33333820a100c004003339202f54696d65732d526f6d616e20302030203020353020 540d28742920740d333531203539202f54696d65732d526f6d616e2030203020302035302054 0d285c282920740d3336372035 39202f54696d65732d526f6d616e20302030203020353020540d287a2920740d333839203539 202f54696d65732d526f6d616e20302030203020353020540d285c292920740d333036203120 3330362033362033303920312033313820312033303920333620480d33303620373120333036 20333620333039203731203331 382037312033303920333620480d343138203120343138203336203431352031203430362031 2034313520333620480d34313820373120343138203336203431352037312034303620373120 34313520333620480d343331203539202f54696d65732d526f6d616e20302030203020353020 540d282d2920740d3435392035 39202f53796d626f6c20302030203020353020540d28772920740d343933203539202f54696d 65732d526f6d616e20302030203020353020540d28742920740d2f4c77207b6f6e6570782073 7562207365746c696e6577696474687d2062696e64206465660d31204c77202f53202f737472 6f6b65206c6f6164206465660d 2f4d207b63757272656e746c696e657769647468202e35206d756c2061646420657863682063 757272656e746c696e657769647468202e35206d756c206164642065786368206d6f7665746f 7d2062696e64206465660d2f4c207b63757272656e746c696e657769647468202e35206d756c 20616464206578636820637572 72656e746c696e657769647468202e35206d756c206164642065786368206c696e65746f7d20 62696e64206465660d3237362031204d0d3236362031204c0d323636203733204d0d32373620 3733204c0d530d33204c77203236362031204d0d323636203731204c0d530d31204c77203530 372031204d0d3531372031204c 0d353137203733204d0d353037203733204c0d530d33204c77203531352031204d0d35313520 3731204c0d530d313834203539202f54696d65732d526f6d616e20302030203020353020540d 287375702920740d32313320313032202f54696d65732d526f6d616e20302030203020333820 540d28742920740d3534332035 39202f54696d65732d526f6d616e20302030203020353020540d285c3236312920740d363630 203539202f53796d626f6c20302030203020353020540d28702920740d373031203539202f54 696d65732d526f6d616e20302030203020353020540d28662920740d373139203335202f5469 6d65732d526f6d616e20302030 203020333820540d28742920740d373239203539202f54696d65732d526f6d616e2030203020 3020353020540d285c282920740d37343520353920a100c0020b2f54696d65732d526f6d616e 20302030203020353020540d287a2920740d373637203539202f54696d65732d526f6d616e20 302030203020353020540d285c 292920740d363838203720363838203339203639312037203730302037203639312033392048 0d3638382037312036383820333920363931203731203730302037312036393120333920480d 3739362037203739362033392037393320372037383420372037393320333920480d37393620 37312037393620333920373933 203731203738342037312037393320333920480d383039203539202f54696d65732d526f6d61 6e20302030203020353020540d282d2920740d383337203539202f53796d626f6c2030203020 3020353020540d28772920740d383731203539202f54696d65732d526f6d616e203020302030 20353020540d28742920740d31 204c77203635382037204d0d3634382037204c0d363438203733204d0d363538203733204c0d 530d33204c77203634382037204d0d363438203731204c0d530d31204c77203838352037204d 0d3839352037204c0d383935203733204d0d383835203733204c0d530d33204c772038393320 37204d0d383933203731204c0d 530d353830203539202f54696d65732d526f6d616e20302030203020353020540d28696e6629 20740d36303220313032202f54696d65732d526f6d616e20302030203020333820540d287429 20740da000bfa000be01000a00000000001b00d728000e000001770300170d00092b08030177 0300140d000c28000e0014017a 070001000160000500120015001a010e005a6800b4005a60000500140015001c0000005a6800 5a005a290f013d03001729200170030014290a0166280009005101742b030501282904017a29 050129600000004a00100052010e005a6800b4005a600000005c001000640000005a68005a00 5a290a012d0300172907017703 0014290801740800092200000043fe0022001100410200220000004100112200000079020022 0011007bfe00220000007b001128000e002c037375700d00092b070a01740d000c28000e0082 01d0030017291c0170030014290a01660d000928000800ad01740d000c2b020601282904017a 2905012908000860000200a600 1200ae010e005a6800b4005a60000200b6001200be0000005a68005a005a290a012d03001729 07017703001429080174080009220002009efe00220012009c0200220002009c001022000200 d3020022001200d5fe0022000200d5001028000e008b03696e660d00092b050a0174a000bfa1 006400f64578b97200015d7c45 7870727c5b233e60625f5f5f7d292c232062283c245e222054696d65737e3a203b2c2620775e 222153796d626f6c5e3a213b29775f3a203b2c203c632124315e3b6250387a7d3b2c20202c5d 203c6325234428223c63213d5128273a21703c632124312824245e3a20665f5e742c487a2c49 7d7d202c4d203a21773a20747d 7d207d28217375707d5e3b29745f7d3b2c202f30203c632523445e3c63213d5128273a213b62 5038703c632124312824245e3a20665f5e742c487a2c497d7d3b2c202c4d203a21773a20747d 7d2821696e667d5e3b29745f7d7d2320622044206221282062214c2157577d5d7c5ba1574500 0677ee00030000a15745000603 ee000c0003a15745000602ee50030000a15745000603ee00010003a15745000a70ee00140554 696d6573a15745000a71ee001400c000000177a15745000603ee00010003a15745000b70ee00 170653796d626f6ca15745000a71ee0017009000000177a15745000603ee00000003a1574500 0a71ee001400c000000120a157 45000602ee11110000a15745000603ee00010003a15745000a71ee001400c00000017aa15745 000a71ee001400c000000120a15745000a71ee001400c000000120a15745000a71ee001400c0 0000013da15745000a71ee001400c000000120a15745000602ee50e40000a15745000603ee00 020003a15745000602ee177100 00a15745000603ee00070003a15745000a71ee001700c000000170a15745000602ee11110000 a15745000603ee00040003a15745000602ee50030000a15745000603ee00010003a15745000a 71ee001400c000000166a15745000603ee00000003a15745000603ee00010003a15745000a71 ee001400c000000174a1574500 0a71ee001400c000000128a15745000a71ee001400c00000017aa15745000a71ee001400c000 000129a15745000a71ee001400c000000120a15745000a71ee001400c00000012da15745000a 71ee001400c000000120a15745000a71ee001700c000000177a15745000a71ee001400c00000 0174a15745000a71ee001400c0 00000120a15745000603ee00030003a15745000a71ee001400c000000173a15745000a71ee00 1400c000000175a15745000a71ee001400c000000170a15745000603ee00010003a15745000a 71ee0014009000000174a15745000603ee00000003a15745000a71ee001400c000000120a157 45000a71ee001400c0000001d0 a15745000a71ee001400c000000120a15745000602ee50e40000a15745000603ee00010003a1 5745000602ee17710000a15745000603ee00070003a15745000a71ee001700c000000170a157 45000602ee11110000a15745000603ee00040003a15745000602ee50030000a15745000603ee 00010003a15745000a71ee0014 00c000000166a15745000603ee00000003a15745000603ee00010003a15745000a71ee001400 9000000174a15745000a71ee001400c000000128a15745000a71ee001400c00000017aa15745 000a71ee001400c000000129a15745000a71ee001400c000000120a15745000a71ee001400c0 0000012da15745000a71ee0014 00c000000120a15745000a71ee001700c000000177a15745000a71ee001400c000000174a157 45000603ee00030003a15745000a71ee001400c000000169a15745000a71ee001400c0000001 6ea15745000a71ee001400c000000166a15745000603ee00010003a15745000a71ee00140090 00000174a15745000603ee0000 0003ff}} \tab (14)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 where t is any integer. The set of points with w{\f23\fs20\dn4 w} {\f23 \'a3 1} will be denoted {\f18 W}{\f23\fs20\dn4 w}. This set is invariant, since according to the definition w{\f23\fs20\dn4 w} (z) = w{\f23\fs20\dn4 w}(f(z)). If the width of an orbit is bounded, then the orbit \{z{\fs20\dn4 t}\} has rotation number {\f23 w}, according to the definition (4). \par \tab A similar set is {\f18 L}{\f23\fs20\dn4 w}, which denotes the set of points whose orbits remain close to the linear orbit\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\dn6 {\pict\macpict\picw145\pich13 0d4e00000000000d00911101a10064000eb778707200000003000000000000a101f200164578 b972000300000000000d00914772897068af626a01000afff8ffd7001500ba2c000800140554 696d65730300140d000c2e000400ff01002b0a0a01202c000900170653796d626f6c0300170d 0009290301202c001000120d5a 617066204368616e636572790300120d000c290201200700000000230000a000bea100c0000d 63757272656e74706f696e7420a000bf22000100010000a000bea100c003f97472616e736c61 74652063757272656e74706f696e74207363616c65200d31207365746c696e65636170203020 7365746c696e656a6f696e2020 32207365746d697465726c696d69740d362061727261792063757272656e746d617472697820 616c6f616420706f7020706f7020706f70206162732065786368206162732061646420657863 682061627320616464206578636820616273206164642032206578636820646976202f6f6e65 70782065786368206465660d5b 20302e3233383830362030203020302e32333838303620302030205d20636f6e6361740d2f54 207b2f6673697a65206578636820646566202f66756e64657220657863682030206e65206465 66202f666974616c2065786368206673697a65206d756c20342064697620646566202f66626f 6c6420657863682030206e6520 6465660d2f666d6174205b206673697a65203020666974616c206673697a65206e6567203020 30205d206465662066696e64666f6e7420666d6174206d616b65666f6e7420736574666f6e74 206d6f7665746f7d2062696e64206465660d2f74207b66756e646572207b63757272656e7470 6f696e742030206673697a6520 31322064697620726d6f7665746f203220696e64657820737472696e67776964746820726c69 6e65746f206673697a6520323420646976207365746c696e657769647468207374726f6b6520 6d6f7665746f7d2069660d66626f6c64207b63757272656e74706f696e74206673697a652032 3420646976203020726d6f7665 746f203220696e6465782073686f77206d6f7665746f7d2069662073686f777d2062696e6420 6465660d30203432202f5a6170664368616e636572792d4d656469756d4974616c6963203020 30203020353020540d284c2920740d3239203535202f53796d626f6c20302030203020333820 540d28772920740d3636203432 202f54696d65732d526f6d616e20302030203020353020540d283d2920740d31323620343220 2f54696d65732d526f6d616e20302030203020353020540d287a2920740d313438203432202f 54696d65732d526f6d616e20302030203020353020540d283a2920740d313838203432202f53 796d626f6c2030203020302035 3020540d28702920740d323238203432202f54696d65732d526f6d616e203020302030203530 20540d28662920740d323438203231202f54696d65732d526f6d616e20302030203020333820 540d28742920740d323538203432202f54696d65732d526f6d616e2030203020302035302054 0d285c282920740d3237352034 32202f54696d65732d526f6d616e20302030203020353020540d287a2920740d323937203432 202f54696d65732d526f6d616e20302030203020353020540d285c292920740da100c004002f 48207b6e657770617468206d6f7665746f203220636f7079206375727665746f203220636f70 79206375727665746f2066696c 6c7d2062696e64206465660d323136202d312032313620323620323139202d3120323237202d 312032313920323620480d323136203533203231362032362032313920353320323237203533 2032313920323620480d333236202d312033323620323620333233202d3120333135202d3120 33323320323620480d33323620 35332033323620323620333233203533203331352035332033323320323620480d3333392034 32202f54696d65732d526f6d616e20302030203020353020540d285c3236312920740d333736 203432202f54696d65732d526f6d616e20302030203020353020540d287a2920740d34313020 3432202f54696d65732d526f6d 616e20302030203020353020540d285c3236312920740d343437203432202f53796d626f6c20 302030203020353020540d28772920740d343831203432202f54696d65732d526f6d616e2030 2030203020353020540d28742920740d2f4c77207b6f6e65707820737562207365746c696e6577696474687d2062696e642064 65660d32204c77202f53202f7374726f6b65206c6f6164206465660d2f4d207b63757272656e 746c696e657769647468202e35206d756c2061646420657863682063757272656e746c696e65 7769647468202e35206d756c206164642065786368206d6f7665746f7d2062696e6420646566 0d2f4c207b63757272656e746c 696e657769647468202e35206d756c2061646420657863682063757272656e746c696e657769 647468202e35206d756c206164642065786368206c696e65746f7d2062696e64206465660d31 3735202d31204d0d313735203532204c0d530d353036202d31204d0d353036203532204c0d53 0d353231203432202f53796d62 6f6c20302030203020353020540d285c3234332920740d353630203432202f54696d65732d52 6f6d616e20302030203020353020540d28312920740d313134202d3120313134203133203131 37202d3120313234202d312031313720313320480d3131372032372031313720313320313134 20323720313037203237203131 3420313320480d31313720323720313137203431203131342032372031303720323720313134 20343120480d3131342035352031313420343120313137203535203132342035352031313720 343120480d353937202d312035393720313320353934202d3120353837202d31203539342031 3320480d353934203237203539 3420313320353937203237203630342032372035393720313320480d35393420323720353934 20343120353937203237203630342032372035393720343120480d3539372035352035393720 34312035393420a100c000133535203538372035352035393420343120480da000bfa000be01 000a00000000000d009128000a 0000014c0300170d00092b070301770300140d000c28000a0010013d290e017a2906013a0300 1729090170030014290a01660d0009280005003c01740d000c2b020501282904017a29050129 07000100016000000035000c003b010e005a6800b4005a6000000047000c004d0000005a6800 5a005a290a01d02909017a2908 01d00300172909017703001429080174080009220000002b000c2200000078000c030017290a 01a303001429090131080008600000001c00060022010e005a60000000170006001d005a005a 6000060017000c001d0000005a600006001c000c002200b4005a60000000880006008e000000 5a600000008d0006009300b400 5a600006008d000c0093010e005a6000060088000c008e005a005aa000bfa1006400c34578b9 7200015d7c457870727c5b233e60625f5f5f7d2925232062283c245e22205a61706620436861 6e636572797e3a203b625038266335352a4c5e222153796d626f6c5e3a21775f22222a7e3a22 202c5d203c6321284128287a2c 5a203c63213121282b3a212663302020703c632124312824245e3a22266335352a665f5e742c 487a2c497d7d202f30207a202f30203a21773a22747d7d203a2126633020202e433a22266335 352a20317d7d7d2320622044206221282062214c2157577d5d7c5ba15745000677ee00030000 a15745000603ee00050003a157 45000602ee50030000a15745000603ee00010003a15745001270ee00120d5a61706620436861 6e63657279a15745000a71ee001200c00000014ca15745000603ee00010003a15745000b70ee 00170653796d626f6ca15745000a71ee0017009000000177a15745000603ee00000003a15745 000a70ee00140554696d6573a1 5745000a71ee001400c000000120a15745000a71ee001400c00000013da15745000a71ee0014 00c000000120a15745000602ee12210000a15745000603ee00080003a15745000a71ee001400 c00000017aa15745000a71ee001400c00000013aa15745000a71ee001400c000000120a15745 000602ee14410000a157450006 03ee000b0003a15745000a71ee001700c000000170a15745000602ee11110000a15745000603 ee00040003a15745000602ee50030000a15745000603ee00010003a15745000a71ee001400c0 00000166a15745000603ee00000003a15745000603ee00010003a15745000a71ee0014009000 000174a15745000a71ee001400 c000000128a15745000a71ee001400c00000017aa15745000a71ee001400c000000129a15745 000a71ee001400c000000120a15745000a71ee001400c0000001d0a15745000a71ee001400c0 00000120a15745000a71ee001400c00000017aa15745000a71ee001400c000000120a1574500 0a71ee001400c0000001d0a157 45000a71ee001400c000000120a15745000a71ee001700c000000177a15745000a71ee001400 c000000174a15745000a71ee001400c000000120a15745000a71ee001700c0000001a3a15745 000a71ee001400c000000120a15745000a71ee001400c000000131ff}}.\tab (15)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 This is an interesting set, since there is an {\f23 w }such that each ordered orbit, (8), is in {\f18 L}{\f23\fs20\dn4 w }{\fs20\up10 3} . In particular the minimizing and minimax orbits of frequency {\f23 w} are in {\f18 L}{\f23\fs20\dn4 w}. We can show that {\f18 W}{\f23\fs20\dn4 w} is a subset of {\f18 L}{\f23\fs20\dn4 w}; moreover,\par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Lemma 4}: {\f18 W}{\f23\fs20\dn4 w} is the maximal invariant subset of {\f18 L}{\f23\fs20\dn4 w}.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Proof: First we show that {\f18 W}{\f23\fs20\dn4 w} {\f23 \'cc }{\f18 L}{\f23\fs20\dn4 w}; for if z {\f23 \'ce}{\f18 W}{\f23\fs20\dn4 w}, then, letting x{\fs20\dn4 t} = \'b9(f{\fs20\up6 t} (z)), we have\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab \'d01 {\f23 \'a3 [}x{\fs20\dn4 t+j} - {\f23 w}(t+j)] - [x{\fs20\dn4 t} -{\f23 w}t] {\f23 \'a3} 1\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 for any t and j, since otherwise the width would be too large. Rearranging this gives |x{\fs20\dn4 t+j} \'d0 x{\fs20\dn4 t} \'d0 {\f23 w}j| < 1, and thus z{\fs20\dn4 t} {\f23 \'ce }{\f18 L}{ \f23\fs20\dn4 w}. Furthermore, if there is a point z {\f23 \'ce }{\f18 L}{\f23\fs20\dn4 w} whose orbit is also in {\f18 L}{\f23\fs20\dn4 w}, then we have | x{\fs20\dn4 t+j} \'d0 x{\fs20\dn4 j} \'d0 {\f23 w}t | {\f23 \'a3} 1 for any t and j, so \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab | x{\fs20\dn4 t+j} \'d0 {\f23 w(}j+t) \-\'d0 (x{\fs20\dn4 j }\'d0 {\f23 w}j) | {\f23 \'a3} 1\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 which implies that w{\f23\fs20\dn4 w}(z{\fs20\dn4 j}) {\f23 \'a3} 1.{\f13 q\par }\tab We will now show that the set of orbits of width one is also contained within the resonance, and therefore computations of the width function can be used to describe exit times. We begin by considering the (0,1) resonance, setting {\f23 w} = 0/1, and dropping the subscript, {\f18 W}{\fs20\dn4 }= {\f18 W}{\fs20\dn4 0}.\par {\b Hypothesis B: }We assume in addition to hypothesis A, that the upper turnstile is entirely to the left of the vertical line v{\fs20\dn4 a}, and that its iterate is entirely to the right of v{\fs20\dn4 a}. Consequently the upper entry set {\f18 E } is entirely to the left of v{\fs20\dn4 a} and the upper exit set {\f18 I} is entirely to the right of v{\fs20\dn4 a}, recall Fig. 6. We make a similar assumption about the lower turnstile, relative to the line v{\fs20\dn4 b}.\par This hypothesis appears true for the standard map from numerical computations. For the case considered by Veerman and Tangerman {\fs20\up10 21} , the boundaries of the turnstile can be chosen to be Lipschitz graphs, and so the turnstile is contained in a vertical strip, and the exit set and the preimage of the entry set are separated by a vertical line. This, while not quite equivalent to hypothes is B, is also sufficient to prove the next theorem.\par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Theorem 5:} For a twist map satisfying hypothesis B, the set of orbits {\f18 W} is the maximal invariant set in the (0,1) resonance R (or more precisely the set of integer translates of R). \par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 We begin the proof with a simpler proposition. Denote the vertical line through {\i q} by v{\fs20\dn4 q}, and choose another vertical line {\f18 r }to the right of {\i q}. Let S{\f23 \'a2} denote the open strip between v{\fs20\dn4 q} and {\f18 r}. Let W{\fs20\up6 u} denote the right-going unstable manifold of {\i q}.{\b }The point of {\f18 r }{\f23 \'c7 }W{\fs20\up6 u} closest to {\i q} along W{\fs20\up6 u} is denoted {\i a}{\f23 \'a2} . Thus the open segment of unstable manifold W{\fs20\up6 u}({\i q},{\i a}{\f23 \'a2}) is contained in S{\f23 \'a2}. By the Jordon curve theorem, the set S{\f23 \'a2}-W{\fs20\up6 u}({\i q},{\i a}{\f23 \'a2} ) is the disjoint union of two open connected sets T{\f23 \'a2} and B{\f23 \'a2}, with T{\f23 \'a2} \'d2above\'d3 B{\f23 \'a2}. \par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Proposition 6:} Every point in the closure of T{\f23 \'a2}, except {\i q}, either exits to the right from the closure of S{\f23 \'a2} or enters B{\f23 \'a2}.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Proof: The inverse of f is also a twist map that twists to the left. Therefore the closure C of f{\fs20\up6 -1}(T{\f23 \'a2}){\f23 \'c7T\'a2}{\dn8 } is a compact set. By the twist condition, the boundary of the set T{\f23 \'a2}-C maps to the right of {\f18 r} or into B{\f23 \'a2}, thus each point of the closure of T{\f23 \'a2} not in C exits S{\f23 \'a2} to the right, or enters B{\f23 \'a2} . On the other hand, suppose {\i z} belongs to C. If the orbit of {\i z} does not enter B{\f23 \'a2} and does not exit from S{\f23 \'a2} then it must remain in C. Thus its omega limit set, {\f23 W}({\i z}), is a non-empty invariant set. Note that {\f23 W} ({\i z}) is not contained in W{\fs20\up6 u}[q,{\i a}{\f23 \'a2}] because if it were then {\f23 W}(z) would have to be the fixed point {\i q}. In this case some iterate of {\i z} would belong to the local stable manifold of {\i q} ; however, the local stable manifold of {\i q} does not intersect C, because, for a right twist map, the right going branc h of the local unstable manifold is above the right going branch of the local stable manifold (as is easy to demonstrate by looking at the matrix Df({\i q})). Therefore there exists a point {\i w} in {\f23 W}(z) that is not in W{\fs20\up6 u}[{\i q},{\i a} {\f23 \'a2}] . However in this case, according to Prop. 2 for f{\fs20\up6 -1 }applied to the strip S{\f23 \'a2}, the backward orbit of {\i w} must exit the strip to the left. 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203433322e3820a100c00002430da100c0000f3436342e30343634203433322e3820a100c000 0f3436352e33203433312e3534363420a100c0000a3436352e332034333020a100c00002430d a100c0000f3436352e33203432 382e3435333620a100c0000f3436342e30343634203432372e3220a100c0000c3436322e3520 3432372e3220a100c00002430da100c0000f3436302e39353336203432372e3220a100c0000f 3435392e37203432382e3435333620a100c0000a3435392e372034333020a100c00002430da1 00c00006312e3220770da100c0 0002620da100c00002550da100c00002550da100c00004766d720da100c00013656e64202520 4672656548616e64446963740da100c0000f656e642025204648494f446963740da000bfff}} \par \pard\plain \s4\qj\li720\ri720\sa300\keep\posxr\posyb\dxfrtext180 \f20\fs20 Fig. 8. Sketch for the proof of Proposition 6.\par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Proposition 7:} Consider the union of the two strips S and S{\f23 \'a2}. Let {\i z} belong to the closure of T{\f23 \'c8}T{\f23 \'a2} minus W{\fs20\up6 s}[{\i a},{\i q}] and W{\fs20\up6 u}[{\i q},{\i a}{\f23 \'a2}]. If the orbit of {\i z} does not intersect B or B{\f23 \'a2} then it must cross S{\f23 \'c8}S{\f23 \'a2} from left to right.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Proof: Suppose first that {\i z} belongs to the closure of T minus W{\fs20\up6 s}[{\i a},{\i q}]. By Prop. 2, the orbit of {\i z} exits the strip S to the right. Since by hypothesis it cannot enter B{\f23 \'a2} it must enter T{\f23 \'a2} or exit S{\f23 \'c8}S{\f23 \'a2} to the right as required. On the other hand, if it enters T{\f23 \'a2} or starts in the closure of T{\f23 \'a2 } then Prop 6 implies that it must also exit S{\f23 \'a2} to the right. Finally, to show that the backward orbit must exit to the left, apply the preceding argument to f{\fs20\up6 -1}.{\f13 o}\par Proof of Theorem 5: The upper boundary of the resonance R and its translates form a \'d2partial barrier\'d3 separating the plane into top and bottom components. If an orbit stays above R and its translates, then Prop. 7 implies that the orbit has width greater than 1. Orbits that cross the partial barrier must intersect the sets {\f18 E} or {\f18 I} or their translates. Consider an orbit that crosses from below through the set {\f18 E}. If it recrosses the barrier then some point on the orbit must be in {\f18 I\~\~}or one of its translates. By hypothesis B the distance between {\f18 E } and each of the translates of {\f18 I\~}is greater than one, so in this case the width would be greater than one. On the other hand, an orbit that intersects {\f18 E\~\~}cannot re-enter the resonance through {\f18 I} , because by hypothesis B, the image of {\f18 E} is to the right of the vertical line{\f18 }that forms the right boundary of T, and every point in T exits to the right by Prop. 2.\par A similar argument works for orbits that start below the chain of translates of R. Thus the only orbits of width less than or equal to 1 must be contained in one of the translates of R.{\f13 o\par \tab } The generalization of the theorem to an (m,n) resonance that satisfies hypothesis B is straightforward. It is based on the fact that such a resonance has a turnstile in only one of the n islands, and thus when a point is above the resonance, it is forced t o rotate around the cylinder at least m times before it could possibly enter the resonance. {\cf1 \par }\tab We show computations of the exit times using the width criterion in Figs. 9 and 10. One advantage of the width criterion, over the strip, is that there is no artificial discontinuity on vertical lines\'d1 the discontinuity appearing with the choice w = 1 is d ynamically significant since this corresponds to the orbit undergoing a complete rotation around the cylinder. Secondly, one does not have to know the position of the hyperbolic periodic points in order to compute the width. Otherwise Fig. 9 is quite si milar to Fig. 4. The discontinuities appear to lie along approximations of the stable and unstable manifolds. Another advantage of the width function is that it allows classification of the trapped orbits. In the figures we color those orbits with w < 1 fo r the first 100 iterates according to their widths\'d1 this shows the structure of the trapped invariant set. Figure 10 displays a computation for the case when the fixed point at x = 0 has become unstable to a period two orbit. The light blue curves outline t he stable and unstable manifolds in a striking way. \par \tab As k increases, the computations give a nice visualization of the formation of the horseshoe in the resonance zone R; this occurs slightly below k = 8.3. The exit time decomposition in the case of the horseshoe corresponds simply to the cantor set constru ction of the horseshoe itself. \par \pard\plain \s4\qj\li720\ri720\sa300\keep\dxfrtext180 \f20\fs20 Fig. 9. Transit times computed for the standard map at k = 2.0 using the width function; boundaries are the same as Fig. 3. The colors ranging from black through orange to yellow then white correspond to orbits whose width exceeds one during its first 100 iterates. Orbits with w<1 are colored green to magenta to blue in the direction of decreasing width. These correspond to the orbits trapped in the resonance.\par \pard \s4\qj\li720\ri720\sa300\keep\dxfrtext180 Fig. 10. Transit times for k = 4.7 using the width function; boundaries are the same as Fig. 3. Colors ranging from light to dark blue to yellow correspond to the time at which the width exceeds one, up to 100 iterates. Orbits with w<1 are colored shades of red.\par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 7. Four Dimensions\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Little is known about transport in high dimensional phase spaces. Numerical studies indicate that some structure analogous to resonances appears to exist {\fs20\up10 23} . Some results were obtained by Wiggins {\fs20\up10 11} , who studied a weakly coupled pair of area preserving maps, one of which is integrable and the other of which has a transverse homoclinic connection. In this case, stability of normally hyperbolic sets implies that the standard two dimensional transport r esults persist for weak coupling.\par \tab Here we consider an example introduced by Froeschl\'8e {\fs20\up10 24} corresponding to coupled standard maps The family on T{\fs20\up6 2}{\f23 \'b4}R{\fs20\up6 2} is given by (5) with the potential \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\dn28 {\pict\macpict\picw308\pich30 089200000000001e0134110101000afc18fc180406051c0700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000e0014000c0000a0008c a15745000603ee000b0003a15745000a70ee00140554696d6573a15745000a71ee001400c000 000176a15745000a71ee001400 c000000128a0008ca15745000602ee50030000a0008ca15745000603ee000100032c00080014 0554696d65730300140d000c2e0004000000002a10027628a15745000a71ee001400c0000001 78290a0178a0008da0008ca15745000603ee00010003a15745000a71ee00140090000001310d 00092b06020131a0008da0008c 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45000a71ee001400c000000120a15745000a71ee001400c000000163a15745000a71ee001400 c000000120a15745000a71ee001400c000000163a15745000a71ee001400c00000016fa15745 000a71ee001400c000000173a15745000a71ee001400c000000120a0008ca15745000602ee17 71000022000700fafe0023000b 230200220007012d020023000b23fe00a0008ca15745000603ee000600030d000c28001000cc 0a29202b206320636f7320a15745000a71ee001400c000000132a15745000a71ee001700c000 000170292f0132a0008ca15745000602ee50030000a0008ca15745000603ee00020003030017 29060170a15745000a71ee0014 00c000000128a15745000a71ee001400c0000001780300142907022878a0008da0008ca15745 000603ee00010003a15745000a71ee00140090000001310d00092b0a020131a0008da0008ca1 5745000603ee00000003a0008da0008da15745000a71ee001400c00000012ba0008ca1574500 0602ee50030000a0008ca15745 000603ee000100030d000c2800100117012ba15745000a71ee001400c00000017829070178a0 008da0008ca15745000603ee00010003a15745000a71ee00140090000001320d00092b060201 32a0008da0008ca15745000603ee00000003a0008da0008da15745000a71ee001400c0000001 290d000c28001001290129a000 8da0008da0008da0008da0008da100b6000400040001a100b6000400010001a00083ff}}\tab (16)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 There are (generically) four fixed points at (x,y) = ({\fs20\up6 1}/{\fs20\dn4 2 }(m,n), 0), where m,n {\f23 \'ce \{0,1\}}\'d1 these correspond to critical points of v. Typically one of these points is hyperbolic, one elliptic, and two are half elliptic and half hyperbolic (i.e. one eigenvalue pair of unit modulus and the other real). For small c and positive a and b, are four hy perbolic fixed points in C located at (-.5,-.5,0,0), (.5,-.5,0,0), (-.5,.5,0,0), and (.5,.5,0,0). The point with coordinates (0,0,0,0) is elliptic. \par \tab An analogue of the strip considered before is the \'d2cylinder\'d3 \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\fs20\dn8 {\pict\macpict\picw204\pich13 0da800000000000d00cc1101a10064000eb778707200000003000000000000a101f200164578 b972000300000000000d00cc4772897068af626a01000afff8ffc8001501042c000900170653 796d626f6c0300170d000c2e000400ff01002b0a0a01202c000800140554696d657303001429 0301200700000000230000a000 bea100c0000d63757272656e74706f696e7420a000bf22000100010000a000bea100c003fc74 72616e736c6174652063757272656e74706f696e74207363616c65200d31207365746c696e65 6361702030207365746c696e656a6f696e202032207365746d697465726c696d69740d362061 727261792063757272656e746d 617472697820616c6f616420706f7020706f7020706f70206162732065786368206162732061 6464206578636820616273206164642065786368206162732061646420322065786368206469 76202f6f6e6570782065786368206465660d5b20302e3233383830362030203020302e323338 38303620302030205d20636f6e 6361740d2f54207b2f6673697a65206578636820646566202f66756e64657220657863682030 206e6520646566202f666974616c2065786368206673697a65206d756c203420646976206465 66202f66626f6c6420657863682030206e65206465660d2f666d6174205b206673697a652030 20666974616c206673697a6520 6e656720302030205d206465662066696e64666f6e7420666d6174206d616b65666f6e742073 6574666f6e74206d6f7665746f7d2062696e64206465660d2f74207b66756e646572207b6375 7272656e74706f696e742030206673697a652031322064697620726d6f7665746f203220696e 64657820737472696e67776964 746820726c696e65746f206673697a6520323420646976207365746c696e6577696474682073 74726f6b65206d6f7665746f7d2069660d66626f6c64207b63757272656e74706f696e742066 73697a6520323420646976203020726d6f7665746f203220696e6465782073686f77206d6f76 65746f7d2069662073686f777d 2062696e64206465660d30203432202f54696d65732d526f6d616e2030203020302035302054 0d28432920740d3435203432202f54696d65732d526f6d616e20302030203020353020540d28 3d2920740d313035203432202f54696d65732d526f6d616e20302030203020353020540d287b 5c282920740d31343620343220 2f54696d65732d526f6d616e20302030203020353020540d28782920740d313731203432202f 54696d65732d526f6d616e20302030203020353020540d282c2920740d313834203432202f54 696d65732d526f6d616e20302030203020353020540d28792920740d323039203432202f5469 6d65732d526f6d616e20302030 203020353020540d285c292920740d323338203432202f53796d626f6c203020302030203530 20540d285c3331362920740d323836203432202f54696d65732d526f6d616e20302030203020 353020540d28522920740d333330203232202f54696d65732d526f6d616e2030203020302033 3820540d28342920740d333631 203432202f54696d65732d526f6d616e20302030203020353020540d283a2920740d34313320 343220a100c003b12f54696d65732d526f6d616e20302030203020353020540d28782920740d 343439203535202f54696d65732d526f6d616e20302030203020333820540d28312920740d2f 4c77207b6f6e65707820737562 207365746c696e6577696474687d2062696e64206465660d32204c77202f53202f7374726f6b 65206c6f6164206465660d2f4d207b63757272656e746c696e657769647468202e35206d756c 2061646420657863682063757272656e746c696e657769647468202e35206d756c2061646420 65786368206d6f7665746f7d20 62696e64206465660d2f4c207b63757272656e746c696e657769647468202e35206d756c2061 646420657863682063757272656e746c696e657769647468202e35206d756c20616464206578 6368206c696e65746f7d2062696e64206465660d3338382034204d0d333838203533204c0d53 0d3437392034204d0d34373920 3533204c0d530d343934203432202f53796d626f6c20302030203020353020540d285c323433 2920740d353333203432202f54696d65732d526f6d616e20302030203020353020540d28302e 352c2920740d363436203432202f54696d65732d526f6d616e20302030203020353020540d28 782920740d363832203535202f 54696d65732d526f6d616e20302030203020333820540d28322920740d3632312034204d0d36 3231203533204c0d530d3731322034204d0d373132203533204c0d530d373237203432202f53 796d626f6c20302030203020353020540d285c3234332920740d373636203432202f54696d65 732d526f6d616e203020302030 20353020540d28302e352920740d2f48207b6e657770617468206d6f7665746f203220636f70 79206375727665746f203220636f7079206375727665746f2066696c6c7d2062696e64206465 660d3933202d32203933203132203936202d3220313033202d3220393620313220480d393620 32362039362031322039332032 3620383620323620393320313220480d39362032362039362034302039332032362038362032 3620393320343020480d39332035342039332034302039362035342031303320353420393620 343020480d383431202d322038343120313220383338202d3220383331202d32203833382031 3220480d383338203236203833 3820313220383431203236203834382032362038343120313220480d38333820323620383338 20343020383431203236203834382032362038343120343020480d3834312035342038343120 343020383338203534203833312035342038333820343020480da000bfa000be01000a000000 00000d00cc28000a0000014329 0b013d290e027b28290a01782906012c2903017929060129030017290701ce030014290c0152 0d0009280005004f01340d000c2b0805013a290c01780d00092b090301310700010001080009 220001005e000c2200010072000c0300170d000c28000a007701a3030014290904302e352c29 1b01780d00092b090301322200 010096000c22000100aa000c0300170d000c28000a00ae01a3030014290a03302e3508000860 000000170006001d010e005a600000001200060018005a005a6000060012000c00180000005a 6000060017000c001d00b4005a60000000c3000600c90000005a60000000c8000600ce00b400 5a60000600c8000c00ce010e00 5a60000600c3000c00c9005a005aa000bfa1006400c24578b97200015d7c457870727c5b233e 60625f5f5f7d2925232062283c22202a7e3a203b625038266335352a43202c5d203c63212841 28372d5b2c48782c4c792c4920222153796d626f6c5e3a2126633020202f2e3a20266335352a 20245e525f282220347d202c5a 203c63213121282220245e78282220317d5f7d7d203a212e433a2020302c4e352c4c203c6321 3121282220245e78282220327d5f7d7d203a212e433a2020302c4e357d7d7d23206220442062 21282062214c2157577d5d7c5ba15745000677ee00030000a15745000603ee00050003a15745 000a70ee00140554696d6573a1 5745000a71ee001400c000000143a15745000a71ee001400c000000120a15745000a71ee0014 00c00000013da15745000a71ee001400c000000120a15745000602ee12210000a15745000603 ee001d0003a15745000a71ee001400c00000017ba15745000a71ee001400c000000128a15745 000a71ee001400c000000178a1 5745000a71ee001400c00000012ca15745000a71ee001400c000000179a15745000a71ee0014 00c000000129a15745000a71ee001400c000000120a15745000b70ee00170653796d626f6ca1 5745000a71ee001700c0000001cea15745000a71ee001400c000000120a15745000602ee5003 0000a15745000603ee00010003 a15745000a71ee001400c000000152a15745000603ee00000003a15745000603ee00020003a1 5745000a71ee0014009000000120a15745000a71ee0014009000000134a15745000a71ee0014 00c000000120a15745000a71ee001400c00000013aa15745000a71ee001400c000000120a157 45000602ee14410000a1574500 0603ee00020003a15745000a71ee001400c000000120a15745000602ee50030000a157450006 03ee00010003a15745000a71ee001400c000000178a15745000603ee00020003a15745000a71 ee0014009000000120a15745000a71ee0014009000000131a15745000603ee00000003a15745 000a71ee001400c000000120a1 5745000a71ee001700c0000001a3a15745000a71ee001400c000000120a15745000a71ee0014 00c000000130a15745000a71ee001400c00000012ea15745000a71ee001400c000000135a157 45000a71ee001400c00000012ca15745000a71ee001400c000000120a15745000602ee144100 00a15745000603ee00020003a1 5745000a71ee001400c000000120a15745000602ee50030000a15745000603ee00010003a157 45000a71ee001400c000000178a15745000603ee00020003a15745000a71ee00140090000001 20a15745000a71ee0014009000000132a15745000603ee00000003a15745000a71ee001400c0 00000120a15745000a71ee0017 00c0000001a3a15745000a71ee001400c000000120a15745000a71ee001400c000000130a157 45000a71ee001400c00000012ea15745000a71ee001400c000000135ff}}{\fs20\dn8 };\tab (17)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 this set is quite natural when c = 0. We study the maximal invariant set in C in terms of the exit time function.\par \tab The stable and unstable manifolds of the hyperbolic fixed points are two dimensional and therefore do not separate volumes in phase space. It is no longer possible to construct a resonance zone whose boundary consists of pieces of stable and unstable manifolds of hyperbolic fixed points. However, it does make sense to study the maximal invariant set contained in C. The geometry of this set may reveal how Arnold diffusion occurs in the neighborho od of the elliptic fixed point (0,0,0,0).\par \tab The invariant set contained in C is compact, a fact that follows from\par \pard\plain \s8\li720\ri720\sb120\sa120\dxfrtext180 \f20 {\b Proposition 8:} The set C {\f23 \'c7}f{\fs20\up6 -1}(C) is diffeomorphic to a four dimensional cube.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Proof: Let {\f23 j}: R{\fs20\up6 4 }{\f23 \'ae }R{\fs20\up6 4} be the smooth map defined by\par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\f23 j}(x,y) = (x,\'b9(f(x,y)) = (x,x{\f23 \'a2} ) .\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Since f is a twist map, {\f23 j}{\dn8 }restricted to the set \'b9{\fs20\up6 -1}(x) is one to one, and the set \{\'b9(f(x,y)) : y{\f23 \'ce}R{\fs20\up6 2}\}{\dn8 } is the entire plane. It follows that {\f23 j}{\dn6 }is one to one and onto. The Jacobian matrix of {\f23 j} has the form \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {{\pict\macpict\picw81\pich62 0aa500000000003e00511101a10064000eb778707200000020000000000000a101f200164578 b972002000000000003e00514772897068af626a01000affecffe70052006a2c000900170653 796d626f6c0300170d000c2e000400ff01002b0a0a01202c000800140554696d657303001429 0301200700000000230000a000 bea100c0000d63757272656e74706f696e7420a000bf22000100010000a000bea100c0040074 72616e736c6174652063757272656e74706f696e74207363616c65200d31207365746c696e65 6361702030207365746c696e656a6f696e202032207365746d697465726c696d69740d362061 727261792063757272656e746d 617472697820616c6f616420706f7020706f7020706f70206162732065786368206162732061 6464206578636820616273206164642065786368206162732061646420322065786368206469 76202f6f6e6570782065786368206465660d5b20302e3233383830362030203020302e323338 38303620302030205d20636f6e 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313920302031313920313330203132322030203134332030203132322031333020480d313139 2032363020313139203133302031323220323630203134332032363020313232203133302048 0d33333720302033333720313330203333342030203331332030203333342031333020480d33 33372032363020333337203133 3020333334203236302033313320323630203333342031333020480da000bfa000be01000a00 000000003e005128001e000001440300172909016a030014290a013d280013002b0149291301 3028002a002501b629060178030017290601a203001428003a002701b6290501780700010001 08000922002f00250e0028002a 003901b629060178030017290601a203001428003a003b01b62906017922002f00390e000800 08600000001d003e0029010e005a6800b4005a6000000044003e00500000005a68005a005aa0 00bfa10064009e4578b97200015d7c457870727c5b233e60625f5f5f7d2926232062283c2220 2a7e3a203b625038266325352a 44222153796d626f6c5e3a216a3a20202c5d203c632124315e5b2222205e495e305e3c322823 2e56783a2126202e427d28223a20266325352a2e56787d7d5e3c3228232e56783a2126202e42 7d28223a20266325352a2e56797d7d7d7d7d2320622044206221282062214c2157577d5d7c5b a15745000677ee00030000a157 45000603ee00060003a15745000a70ee00140554696d6573a15745000a71ee001400c0000001 44a15745000b70ee00170653796d626f6ca15745000a71ee001700c00000016aa15745000a71 ee001400c000000120a15745000a71ee001400c00000013da15745000a71ee001400c0000001 20a15745000602ee11110000a1 5745000603ee00010003a15745000605ee02020101a15745000603ee00010003a15745000a71 ee001400c000000149a15745000603ee00010003a15745000a71ee001400c000000130a15745 000603ee00010003a15745000602ee00120000a15745000603ee00030003a15745000a71ee00 1400c0000001b6a15745000a71 ee001400c000000178a15745000a71ee001700c0000001a2a15745000603ee00020003a15745 000a71ee001400c0000001b6a15745000a71ee001400c000000178a15745000603ee00010003 a15745000602ee00120000a15745000603ee00030003a15745000a71ee001400c0000001b6a1 5745000a71ee001400c0000001 78a15745000a71ee001700c0000001a2a15745000603ee00020003a15745000a71ee001400c0 000001b6a15745000a71ee001400c000000179ff}} \par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 By the twist condition, the 2 by 2 submatrix \'b6x{\f23 \'a2}/\'b6y is non-singular; therefore D{\f23 j} is non-singular. Hence by the inverse function theorem, {\f23 j} has a smooth inverse and is therefore a diffeomorphism. From the definition of {\f23 j} it follows that the set C {\f23 \'c7}f{\fs20\up6 -1}(C) is mapped to the four dimensional cube\tab \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 {\dn10 \tab }{\dn8 {\pict\macpict\picw247\pich17 06c600000000001100f7110101000afc18fc1803f904df0700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb7787072000000040014000c0000a0008c a15745000603ee00050003a15745000a70ee00140554696d6573a15745000a71ee001400c000 00014ba15745000a71ee001400 c000000120a15745000a71ee001400c00000013da15745000a71ee001400c000000120a0008c a15745000602ee1221000061000000170008001b010e005262000000180008001a010e005a61 0000001400080018005a0052620000001500080017005a005a61000800140010001800080052 6200080015001000170000005a 61000800170010001b00bc005262000800180010001a00b4005a61000000f2000800f6000800 5262000000f3000800f50000005a61000000f5000800f900bc005262000000f6000800f800b4 005a61000800f5001000f9010e005262000800f6001000f8010e005a61000800f2001000f600 5a005262000800f3001000f500 5a005aa0008ca15745000603ee002b00032c000800140554696d65730300140d000c2e000400 0000002a0d044b203d20a15745000a71ee001400c00000017ba15745000a71ee001400c00000 0128a15745000a71ee001400c000000178a15745000a71ee001400c00000012ca15745000a71 ee001400c000000178a1574500 0b70ee00170653796d626f6ca15745000a71ee001700c0000001a2291a057b28782c78a15745 000a71ee001400c0000001292c000900170653796d626f6c030017291a01a2a15745000a71ee 001400c000000120a15745000a71ee001700c0000001ce0300142902022920a15745000a71ee 001400c0000001200300172907 01cea0008ca15745000602ee50030000a0008ca15745000603ee0001000303001429080120a1 5745000a71ee001400c00000015229030152a0008da0008ca15745000603ee00000003a0008d a0008ca15745000603ee00020003a15745000a71ee0014009000000120a15745000a71ee0014 0090000001340d000928000900 50022034a0008da0008da15745000a71ee001400c000000120a15745000a71ee001400c00000 013aa15745000a71ee001400c000000120a0008ca15745000602ee14410000a100b600040001 0001a100b600040002000407000100012200040060000c220004006c000ca0008ca157450006 03ee000200030d000c2b070403 203a20a15745000a71ee001400c000000120a0008ca15745000602ee50030000a0008ca15745 000603ee00010003290a0120a15745000a71ee001400c00000017829030178a0008da0008ca1 5745000603ee00010003a15745000a71ee001400900000016a0d00092b0602016aa0008da000 8ca15745000603ee00000003a0 008da0008da0008da0008da15745000a71ee001400c000000120a15745000a71ee001700c000 0001a30d000c28000d006d0120a15745000a71ee001400c000000120030017290301a3a15745 000a71ee001400c000000130a15745000a71ee001400c00000012ea15745000a71ee001400c0 00000135a15745000a71ee0014 00c00000012ca15745000a71ee001400c000000120a0008ca15745000602ee14410000220000 00900010220000009e0010a0008ca15745000603ee0002000303001429060620302e352c20a1 5745000a71ee001400c000000120a0008ca15745000602ee50030000a0008ca15745000603ee 00020003291b0120a15745000a 71ee001400c000000178a15745000a71ee001700c0000001a229030178030017290601a2a000 8da0008ca15745000603ee00010003a15745000a71ee001400900000016a0300140d00092b02 02016aa0008da0008ca15745000603ee00000003a0008da0008da0008da0008da15745000a71 ee001400c000000120a1574500 0a71ee001700c0000001a30d000c28000d009f0120a15745000a71ee001400c0000001200300 17290301a3a15745000a71ee001400c000000130a15745000a71ee001400c00000012ea15745 000a71ee001400c000000135a15745000a71ee001400c00000012ca15745000a71ee001400c0 00000120a15745000a71ee0014 00c000000166a15745000a71ee001400c00000016fa15745000a71ee001400c000000172a157 45000a71ee001400c000000120a15745000a71ee001400c00000016aa15745000a71ee001400 c000000120a15745000a71ee001400c00000013da15745000a71ee001400c000000120a15745 000a71ee001400c000000131a1 5745000a71ee001400c00000012ca15745000a71ee001400c00000013203001429061120302e 352c20666f72206a203d20312c32a0008da0008da0008da100b6000400040002a100b6000400 010001a00083ff}}{\dn10 } .{\f13 o}\par \pard\plain \dxfrtext180 \f20 \par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 \tab It follows immediately from Proposition 8 that the maximal invariant set contained in C is compact. The geometry of this invariant set can be probed by taking two dimensional slices of C and restricting the exit time function to these slices. The interse ction of the invariant set with such a slice is the set of points with infinite forward and backward exit times. Pixels are colored as before according to exit time from C. The curves that separate regions of different colors are evide ntly generated by the intersections of preimages of the three dimensional boundary of C with the chosen two dimensional plane slice of C. This is reasonable because a three dimensional submanifold intersects a two dimensional submanifold of a four dimensio nal manifold generically in a one dimensional manifold or curve.{\cf3 \par }\tab For the four dimensional case, the width of an orbit can be computed in any direction in the configuration space. For lack of a compelling choice, we take the width to be the maximum of the widths computed for each component; in other words, for z = (x{\fs20\dn4 1},x{\fs20\dn4 2},y{\fs20\dn4 1},y{\fs20\dn4 2}) we compute the widths \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab {\fs20\dn12 {\pict\macpict\picw291\pich20 08800000000000140123110101000afc18fc1803fc050b0700000000230000a00082a101f200 04b7787072a15745000677ee00030000a10064000eb77870720000000b0014000c0000a0008c a15745000603ee00150003a0008ca15745000602ee50030000a0008ca15745000603ee000100 03a15745000a70ee0014055469 6d6573a15745000a71ee001400c0000001772c000800140554696d65730300140d000c2e0004 000000002a090177a0008da0008ca15745000603ee00010003a15745000a71ee001400900000 01310d00092b09020131a0008da0008ca15745000603ee00000003a0008da0008da15745000a 71ee001400c000000128a15745 000a71ee001400c00000017aa15745000a71ee001400c000000129a15745000a71ee001400c0 00000120a15745000a71ee001400c00000013da15745000a71ee001400c000000120a0008ca1 5745000602ee50e40000a0008ca15745000603ee000800030d000c280009000e06287a29203d 20a15745000a71ee001400c000 000128a0008ca15745000602ee50030000a0008ca15745000603ee00010003292e0128a15745 000a71ee001400c00000017829040178a0008da0008ca15745000603ee00020003a15745000a 71ee0014009000000131a15745000a71ee00140090000001740d00092b0602023174a0008da0 008ca15745000603ee00000003 a0008da0008da15745000a71ee001400c000000129a15745000a71ee001400c000000120a157 45000a71ee001400c0000001d0a15745000a71ee001400c000000120a0008ca15745000602ee 50e40000a0008ca15745000603ee000400030d000c280009004e042920d020a15745000a71ee 001400c000000128a0008ca157 45000602ee50030000a0008ca15745000603ee0001000329200128a15745000a71ee001400c0 0000017829040178a0008da0008ca15745000603ee00020003a15745000a71ee001400900000 0131a15745000a71ee00140090000001740d00092b0602023174a0008da0008ca15745000603 ee00000003a0008da0008da157 45000a71ee001400c000000129a15745000a71ee001400c0000001200d000c28000900800229 20a0008da0008ca15745000603ee00030003a15745000a71ee001400c000000169a15745000a 71ee001400c00000016ea15745000a71ee001400c000000166280009005e03696e66a0008da0 008ca15745000603ee00010003 a15745000a71ee00140090000001740d00092b05080174a0008da0008ca15745000603ee0000 0003a0008da0008da15745000a71ee001400c0000001200d000c28000900870120a0008da000 8ca15745000603ee00030003a15745000a71ee001400c000000173a15745000a71ee001400c0 00000175a15745000a71ee0014 00c000000170280009002803737570a0008da0008ca15745000603ee00010003a15745000a71 ee00140090000001740d00092b070b0174a0008da0008ca15745000603ee00000003a0008da0 008da15745000a71ee001400c00000013ba15745000a71ee001400c000000120a15745000a71 ee001400c000000120a1574500 0a71ee001400c000000120a15745000a71ee001400c000000120a0008ca15745000602ee5003 0000a0008ca15745000603ee000100030d000c280009008a053b20202020a15745000a71ee00 1400c000000177290f0177a0008da0008ca15745000603ee00010003a15745000a71ee001400 90000001320d00092b09020132 a0008da0008ca15745000603ee00000003a0008da0008da15745000a71ee001400c000000128 a15745000a71ee001400c00000017aa15745000a71ee001400c000000129a15745000a71ee00 1400c000000120a15745000a71ee001400c00000013da15745000a71ee001400c000000120a0 008ca15745000602ee50e40000 a0008ca15745000603ee000800030d000c28000900a706287a29203d20a15745000a71ee0014 00c000000128a0008ca15745000602ee50030000a0008ca15745000603ee00010003292e0128 a15745000a71ee001400c00000017829040178a0008da0008ca15745000603ee00020003a157 45000a71ee0014009000000132 a15745000a71ee00140090000001740d00092b0602023274a0008da0008ca15745000603ee00 000003a0008da0008da15745000a71ee001400c000000129a15745000a71ee001400c0000001 20a15745000a71ee001400c0000001d0a15745000a71ee001400c000000120a0008ca1574500 0602ee50e40000a0008ca15745 000603ee000400030d000c28000900e7042920d020a15745000a71ee001400c000000128a000 8ca15745000602ee50030000a0008ca15745000603ee0001000329200128a15745000a71ee00 1400c00000017829040178a0008da0008ca15745000603ee00020003a15745000a71ee001400 9000000132a15745000a71ee00 140090000001740d00092b0602023274a0008da0008ca15745000603ee00000003a0008da000 8da15745000a71ee001400c000000129a15745000a71ee001400c0000001200d000c28000901 19022920a0008da0008ca15745000603ee00030003a15745000a71ee001400c000000169a157 45000a71ee001400c00000016e a15745000a71ee001400c00000016628000900f703696e66a0008da0008ca15745000603ee00 010003a15745000a71ee00140090000001740d00092b05080174a0008da0008ca15745000603 ee00000003a0008da0008da15745000a71ee001400c0000001200d000c28000901200120a000 8da0008ca15745000603ee0003 0003a15745000a71ee001400c000000173a15745000a71ee001400c000000175a15745000a71 ee001400c00000017028000900c103737570a0008da0008ca15745000603ee00010003a15745 000a71ee00140090000001740d00092b070b0174a0008da0008ca15745000603ee00000003a0 008da0008da0008da100b60004 00010001a100b6000400010001a00083ff}}\tab (18)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 and define \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab w(z) = max(w{\fs20\dn4 1}(z),w{\fs20\dn4 2}(z)) .\tab (19)\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 Thus an orbit can acquire large width either by rotating in the x{\fs20\dn4 1} or the x{\fs20\dn4 2} direction. \par \tab As a corollary to Prop. 8, the set of orbits in T{\fs20\up6 2}{\f23 \'b4}R{\fs20\up6 2} for which w{\b\fs20\dn4 }{\f23 \'a3 1} is also compact, since it is contained in the set for which \par \pard\plain \s1\qj\sa200\sl280\keep\dxfrtext180\tqc\tx4760\tx8820 \f20 \tab max(|x{\fs20\dn4 l} \'d0 x{\fs20\dn4 1}{\f23 \'a2}|, |x{\fs20\dn4 2} \'d0 x{\fs20\dn4 2}{\f23 \'a2}|) {\f23 \'a3} 1.{\cf5 \par }\pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 {\cf5 \tab }There are a number of interesting two dimensional slices; in Fig 11 we show four. The top two panels a re the canonical planes and since the complementary canonical pair is set to zero, the exit time distribution appears similar to two dimensional case; however, in detail the shapes are unlike the structure of the standard map pictures\'d1 note particularly the red regions in the left panel. The bottom left panel shows the momentum plane for {\b x} = {\b 0}. It shows a connected trapped (black) region surrounding {\b y} = {\b 0} that is the analogue of the island surrounding the fixed point in the two dimensional case. Most orbits within this region lie on invariant two-tori. It appears that this region is a four-volume. Note however the appearance of escaping (white and red) regions within the connected black component\'d1 these can never occur in the two dimensional case. The last panel shows the configuration plane for {\b y }= {\b 0}; since the domain of the figure is (-1,1){\f23 \'b4} (-1,1) there are nine copies of the elliptic point shown. Note that the boundary between fundamental domains on T{\fs20\up6 2} is complex implying that we should use a different norm than that in (19). \par \tab The orbits trapped near the elliptic fixed point are outlined even with short time computations of the width function, as shown in Fig. 12 for the {\b x }= {\b 0} slice. Here the first 100 iterates are colored using the scale shown. Those orbits that escape in times 100 to 10{\fs20\up6 6} are shown in white, and those whose exit time is larger than 10{\fs20\up6 6\~} are black. The set of trapped orbits becomes increasingly well defined for long times and there are disconnected components (corresponding to islands around t he main island) and holes as before. Nevertheless, most orbits escape in the first 100 iterations, and by 10{\fs20\up6 6} iterations, as far at the computations are concerned, almost no orbits escape.\par \pard\plain \s4\qj\li720\ri720\sa300\keep\dxfrtext180 \f20\fs20 Fig. 11. Exit times using the width function for the Froeshl\'8e map with a = 1.0, b = 2.0, and c = 0.8. Each panel represents a different slice: the planes (x{\fs18\dn4 2},y{\fs18\dn4 2} ) = {\b 0}, (x{\fs18\dn4 1},y{\fs18\dn4 1}) = {\b 0},{\b }(x{\fs18\dn4 1},x{\fs18\dn4 2}) = {\b 0}, and (y{\fs18\dn4 1},y{\fs18\dn4 2}) = {\b 0}. The domain of each panel is (-1,1){\f23 \'b4(-1,1). } Colors range from blues for short times, to reds for intermediate times, to greys for longer times. Points that do not escape in 100 iterates are colored black.\par \pard \s4\qj\li720\ri720\sa300\keep\dxfrtext180 Fig. 12. Exit times using the width function for the slice {\b x} = {\b 0} for the Froeshl\'8e map with a = 1.0, b = 0.5 and c = 1.0. The domain of the figure is (y{\fs18\dn4 1},y{\fs18\dn4 2}) = (-0.5,0.5){ \f23 \'b4}(-0.5,0.5). The colors ranging from blue to white indicate exit times between 1 and 100 as shown in the color scale at the bottom. Points that have w<1 for 10{\fs18\up6 6} iterates are colored black.\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 \par \tab The escape time distribution computed using the width function for various parameter values is shown in Fig. 13. The distribution is power law in nature, e{\fs20\dn4 t} ~ t{\fs20\up6 -}{\f23\fs20\up6 a}, with an exponent that ranges from about {\f23 a}= -1.1 for the case c = 0.01, to {\f23 a} = 1.3 for c = 0.1. No exponents significantly deviating from these have been observed, but it is not known whether there is a universal exponent associated with this process. These exponents are considerably smaller than those observed for the two dimensi onal case, recall Fig. 5, though much longer computations would be required to confirm this smaller value for {\f23 a}.\par \pard\plain \s7\qc\keep\keepn\dxfrtext180\tqc\tx4760\tx8820 \f20 {{\pict\macpict\picw464\pich330 241200000010014a01e0001102ff0c00ffffffff001000000000000001e00000014a00000000 000000a0008200a000c4001e001a00000000d4000001000a00000010014a01e00009aaaaaaaa aaaaaaaa002000e7006d00e701a1002000b0006d00b001a100200079006d007901a100200042 006d004201a10009ff00ff00ff 00ff000020011e00d4004300d40020011e013b0043013b00a0008c001add6b08c206a2000100 0a0042006d011e01a20009ffffffffffffffff0022007d006d00000023121600230d0300230a 000023080f00230701002306ff00230509002305fd002304030023040400230406002303f800 230304002303040023030b0023 02fa00230300002302040023020e002302fb0023020300230200002302fb002302fd00230103 00230212002303f700230401002302fd00230309002304fe00230a07002308fe002307030023 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00a1009a0008000000000000000000a00098000d00120015bdd40028013a0107017400a00099 00a0009700a0008300a0008300ff}}\par \pard\plain \s4\qj\li720\ri720\sa300\keep\dxfrtext180 \f20\fs20 Fig 13. Normalized escape time distributions (time at which the width exceeds unity) for initial points in the {\b x} = {\b 0} slice for a=1.0 and b=0.5 with varying values of c. The vertical axis is the probability of escape per unit time for those particles that do eventually escape. The computations were done for a box of size 500{\f23 \'b4500 } pixels, each was iterated up to 10{\fs18\up6 6} steps. The statistics become noisy when e{\fs18\dn4 t }{\f23 <} 2(10){\fs18\up6 -8 }as only a few particles escape during each temporal bin of 5000 iterates. \par \pard \s4\qj\li720\ri720\sa300\keep\dxfrtext180 Fig. 14. Enlargement of a small region in the {\b x }= {\b 0} plane for a=1.0, b = 1.5, c = 0.5. The domain of the figure is (0.085,-0.24){\f23 \'b4} (0.145,-0.228). Exit times for the first 1000 iterates are shown using the width function. The color scale is shown at the bottom of the figure. Note the disconnected red regions, and the varying complexity of the border of the non-escaping black region. \par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 8. Conclusions\par \pard\plain \s11\qj\sa160\sl-280\dxfrtext180\tx440 \f20 The exit and transit time decompositions of a set yield statistical measures of transport w ith physical relevance. For the two dimensional case, resonance zones are the dynamically natural regions to use for these studies; however, we have shown that the set of orbits trapped a vertical strip, or set of strips, containing the resonance is ident ical to that of the resonance itself. Thus the exit time distribution of the resonance and the strip have the same asymptotic form. Furthermore, since the stable and unstable manifolds of the resonance need not be constructed to study escape from the strip , the computations are easier. A third measure of escape is provided by the width function, that measures the deviation of an orbit from a given frequency of rotation. For a given rational frequency, the set of orbits of width less than one is identical to those trapped in the resonance. An orbit whose width exceeds one has rotated around the cylinder at least once more or less then it should have. The calculation of the width function does not require that the hyperbolic periodic orbit be found, and thus i t is easily implemented.\par \tab For four and higher dimensional systems there is no simple construction of a resonance zone. Nevertheless, computations of escape from a four dimensional cylinder as well as those of orbits with width less than one show that there are four dimensional reg ions trapped around an elliptic fixed point analogous to those orbits trapped in a 2D resonance. These volumes include the families of invariant tori given by the KAM theorem, but undoubtedly also include orbits that are not strict ly stable, and that will eventually escape. However, the computations imply there is a zone of \'d2practical\'d3 stability, and it is this concept that the computations quantify. The boundary of this trapped set contains smooth regions as well as disconnected tu bes that appear to wend their way into the midst of the trapped regions. We do not have, as of yet, a dynamical explanation of these.\par \tab A number of other open questions remain. We do not know whether the set of orbits trapped in the cylinder C is identical to those of width less than one, as it is in the two dimensional case. Furthermore, though the choice of the strip in the two dimensional case is natural, since its boundaries are formed from vertical lines through the hyperbolic fixed points, for four dim ensions only the 2 dimensional edges of the cylinder C are similarly natural\'d1 there is no dynamical reason to use 3 dimensional planes to connect these edges. Similarly it is not clear which of the translates of the hyperbolic fixed points should form the p roper fundamental domain for the definition of the cylinder. A related issue arises with the proper choice of norm for the width function. In the norm we used, (19), there is an artificial use of the square in configuration space as the boundary. \par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 Acknowledgements\par \pard\plain \dxfrtext180 \f20 Partial support for this research was obtained from the National Science Foundation through grant DMS-9001103. S. 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Meiss, \'d2Periodic Orbits for Reversible, Symplectic Mappings,\'d3 Physica, {\b 35D}, 65-86 (1989).\par \par 24.\tab Froeschle, C., \'d2Numerical Study of a Four Dimensional Mapping,\'d3 Astron. and Astrophys, {\b 16}, 172-189 (1972).\par \par \par \pard\plain \s254\sb480\sa240\keepn\dxfrtext180 \b\f21 Color Figures\par \pard\plain \dxfrtext180 \f20 \par \pard \dxfrtext180 \tab We have not included copies of the color figures with this preprint, because of the cost. Should you wish to have high quality color xerox copies of these 7 figures, please send $10.00 to\par \pard \dxfrtext180 \par \tab \tab James Meiss\par \tab \tab Program in Applied Mathematics\par \tab \tab Box 526\par \tab \tab University of Colorado\par \tab \tab Boulder, CO 80304 \par }