This is a multi-part message in MIME format. ---------------0107090858597 Content-Type: text/plain; name="01-251.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-251.keywords" Liouvillean, ergodic theory, spectral theory, weak mixing, quantum statistical mechanics ---------------0107090858597 Content-Type: application/postscript; name="erwe_us.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="erwe_us.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: erwe.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman Times-Bold Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -t letter -o erwe_us.ps erwe %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.06.29:1420 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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Fl(e)1712 927 y Fc(i)p Fe(t)p Fg(L)1838 968 y Fl(=)28 b(e)1985 927 y Fc(i)p Fe(t)p Fg(L)2083 968 y Fm(J)n(;)1522 1194 y Fl(e)1565 1153 y Fc(i)p Fe(tL)1662 1194 y Fl(e)1705 1153 y Fc(i)p Fe(s)p Fg(L)1838 1194 y Fl(=)g(e)1985 1153 y Fc(i)p Fe(s)p Fg(L)2090 1194 y Fl(e)2133 1153 y Fc(i)p Fe(tL)2231 1194 y Fm(:)3589 968 y Fn(\(2.1\))0 1412 y(By)d(T)-8 b(omita-T)g(ak)o(esaki)24 b(theorem,)g Fm(J)9 b Fk(M)p Fm(J)37 b Fl(=)27 b Fk(M)1673 1375 y Fg(0)1696 1412 y Fn(.)0 1582 y(The)e(natural)g(cone)g Fi(P)34 b Fn(is)24 b(the)h(closure)g(of)g(the)g(set)g Fi(f)p Fm(AJ)9 b(AJ)g Fl(\012)17 b Fi(j)g Fm(A)28 b Fi(2)g Fk(M)p Fi(g)g(\032)g(H)q Fn(.)k(The)25 b(cone)g Fi(P)33 b Fn(is)25 b(self-dual,)0 1702 y(namely)i Fi(P)42 b Fl(=)34 b Fi(f)p Fl(\011)f Fi(2)h(H)18 b(j)f Fl(\(\011)p Fm(;)g Fl(\010\))32 b Fi(\025)i Fl(0)28 b(for)c(all)i(\010)34 b Fi(2)g(H)q(g)p Fn(.)41 b(F)o(or)27 b(e)n(v)o(ery)h(state)f Fm(\021)38 b Fi(2)c Fk(M)3022 1717 y Fg(\003)3061 1702 y Fn(,)28 b(there)h(is)e(a)h(unique) 0 1822 y(v)o(ector)f Fl(\012)345 1837 y Fe(\021)420 1822 y Fi(2)33 b(P)j Fn(such)27 b(that)g Fm(\021)t Fl(\()p Fm(A)p Fl(\))32 b(=)h(\(\012)1461 1837 y Fe(\021)1503 1822 y Fm(;)17 b(A)p Fl(\012)1690 1837 y Fe(\021)1732 1822 y Fl(\))p Fn(.)39 b(Moreo)o(v)o(er)l(,)27 b(the)g(state)g Fm(\021)k Fn(is)c Fm(\034)11 b Fn(-in)l(v)n(ariant)27 b(if)n(f)g Fm(L)p Fl(\012)3522 1837 y Fe(\021)3597 1822 y Fl(=)33 b(0)p Fn(.)0 1943 y(In)25 b(particular)l(,)f(if)h(zero)h(is)e (a)h(simple)f(eigen)l(v)n(alue)g(of)h Fm(L)p Fn(,)g(then)f Fm(!)29 b Fn(is)24 b(the)h(unique)f Fm(\034)11 b Fn(-in)l(v)n(ariant)24 b(state)g(in)h Fk(M)3637 1958 y Fg(\003)3676 1943 y Fn(.)0 2213 y Fo(Pr)n(oof)33 b(of)f(Theor)n(em)j(1.1.)53 b Fn(Let)33 b Fm(E)38 b Fn(be)33 b(an)g(eigen)l(v)n(alue)e(of)i Fm(L)g Fn(and)g Fl(\012)2455 2228 y Fe(E)2547 2213 y Fn(a)g(\(normalized\))f (eigen)l(v)o(ector)g(as-)0 2333 y(sociated)d(to)g Fm(E)6 b Fn(.)44 b(W)-8 b(e)30 b(sho)n(w)e(\002rst)h(that)g Fl(\012)1440 2348 y Fe(E)1529 2333 y Fn(is)g(a)h(c)o(yclic)f(and)g (separating)g(v)o(ector)g(for)g Fk(M)p Fn(.)44 b(Note)29 b(that)g(since)0 2453 y Fm(J)9 b(L)28 b Fl(=)g Fi(\000)p Fm(LJ)9 b Fn(,)26 b Fl(\012)588 2468 y Fg(\000)p Fe(E)730 2453 y Fl(:=)i Fm(J)9 b Fl(\012)994 2468 y Fe(E)1079 2453 y Fn(is)24 b(an)h(eigen)l(v)o(ector)f(of)h Fm(L)h Fn(associated)e(to)g(the)h(eigen)l(v)n(alue)f Fi(\000)p Fm(E)6 b Fn(.)0 2624 y(The)25 b(states)f Fm(!)487 2639 y Fg(\006)p Fe(E)601 2624 y Fl(\()p Fm(A)p Fl(\))k(=)g(\(\012)990 2639 y Fg(\006)p Fe(E)1104 2624 y Fm(;)17 b(A)p Fl(\012)1291 2639 y Fg(\006)p Fe(E)1406 2624 y Fl(\))25 b Fn(are)h Fm(\034)11 b Fn(-in)l(v)n(ariant,)24 b(and)g(hence)i(for)f(all)f Fm(A)k Fi(2)g Fk(M)p Fn(,)1363 2844 y Fl(\(\012)p Fm(;)17 b(A)p Fl(\012\))28 b(=)f(\(\012)1935 2859 y Fg(\006)p Fe(E)2050 2844 y Fm(;)17 b(A)p Fl(\012)2237 2859 y Fg(\006)p Fe(E)2352 2844 y Fl(\))p Fm(:)1172 b Fn(\(2.2\))0 3064 y(Thus,)29 b(if)g Fm(A)p Fl(\012)487 3079 y Fe(E)582 3064 y Fl(=)35 b(0)p Fn(,)30 b(then)f Fm(A)p Fl(\012)35 b(=)g(0)29 b Fn(and)g Fm(A)35 b Fl(=)g(0)29 b Fn(\(since)g Fl(\012)g Fn(is)g(separating\).)42 b(Hence)30 b Fl(\012)3161 3079 y Fe(E)3250 3064 y Fn(is)e(separating.)0 3184 y(T)-8 b(o)23 b(pro)o(v)o(e)g(that)g Fl(\012)617 3199 y Fe(E)701 3184 y Fn(is)g(c)o(yclic,)g(let)h Fm(P)1277 3148 y Fg(0)1323 3184 y Fn(be)g(the)f(orthogonal)g(projection)g(on)p 2593 3103 235 4 v 23 w Fk(M)p Fl(\012)2768 3199 y Fe(E)2851 3184 y Fn(and)h Fm(Q)3096 3148 y Fg(0)3147 3184 y Fl(=)k Ff(1)17 b Fi(\000)i Fm(P)3497 3148 y Fg(0)3519 3184 y Fn(.)31 b(Then)0 3304 y Fm(Q)77 3268 y Fg(0)132 3304 y Fi(2)h Fk(M)335 3268 y Fg(0)385 3304 y Fn(and)27 b Fl(\(\012)664 3319 y Fe(E)724 3304 y Fm(;)17 b(Q)845 3268 y Fg(0)868 3304 y Fl(\012)938 3319 y Fe(E)998 3304 y Fl(\))32 b(=)f(0)p Fn(.)37 b(Let)27 b Fm(Q)k Fl(:=)h Fm(J)9 b(Q)1829 3268 y Fg(0)1853 3304 y Fm(J)g Fn(.)37 b(Then)26 b Fm(Q)i Fn(is)e(an)h(orthogonal)f(projection,)g Fm(Q)32 b Fi(2)g Fk(M)p Fn(,)0 3425 y(and)1408 3618 y Fl(0)c(=)f(\(\012)1696 3633 y Fg(\000)p Fe(E)1811 3618 y Fm(;)17 b(Q)p Fl(\012)2002 3633 y Fg(\000)p Fe(E)2117 3618 y Fl(\))1485 3834 y(=)27 b(\(\012)p Fm(;)17 b(Q)p Fl(\012\))29 b(=)e Fi(k)p Fm(Q)p Fl(\012)p Fi(k)2304 3793 y Fc(2)2344 3834 y Fm(:)0 4051 y Fn(Since)e Fl(\012)h Fn(is)e(separating,)g Fm(Q)k Fl(=)g(0)c Fn(and)h Fm(P)1420 4015 y Fg(0)1471 4051 y Fl(=)i Ff(1)p Fn(.)k(Hence)25 b Fl(\012)2035 4066 y Fe(E)2120 4051 y Fn(is)g(c)o(yclic.)0 4222 y(Let)i Fm(U)236 4185 y Fg(0)287 4222 y Fn(be)h(the)f(linear)g (map)g(de\002ned)g(on)g Fk(M)p Fl(\012)1633 4237 y Fe(E)1720 4222 y Fn(by)g Fm(U)1923 4185 y Fg(0)1947 4222 y Fm(A)p Fl(\012)2090 4237 y Fe(E)2182 4222 y Fl(=)32 b Fm(A)p Fl(\012)p Fn(.)39 b(Since)28 b Fl(\012)2816 4237 y Fe(E)2903 4222 y Fn(is)f(separating,)g(the)g(map)0 4342 y Fm(U)76 4306 y Fg(0)122 4342 y Fn(is)22 b(well-de\002ned.)30 b(Cyclicity)22 b(of)g Fl(\012)1327 4357 y Fe(E)1410 4342 y Fn(and)g(\(2.2\))h(yield)e(that)h Fm(U)2259 4306 y Fg(0)2305 4342 y Fn(e)o(xtends)g(to)g(a)g(unitary)g(map)g(on)g Fi(H)q Fn(.)30 b(Since)1251 4562 y Fm(U)1327 4521 y Fg(0)1351 4562 y Fm(AB)5 b Fl(\012)1573 4577 y Fe(E)1661 4562 y Fl(=)28 b Fm(AB)5 b Fl(\012)28 b(=)g Fm(AU)2268 4521 y Fg(0)2292 4562 y Fm(B)5 b Fl(\012)2441 4577 y Fe(E)2501 4562 y Fm(;)0 4782 y(U)76 4746 y Fg(0)128 4782 y Fi(2)28 b Fk(M)327 4746 y Fg(0)350 4782 y Fn(.)0 4952 y(Let)d Fm(L)224 4967 y Fe(E)309 4952 y Fn(be)g(the)f(Liouvillean)f(associated) i(to)f Fl(\012)1665 4967 y Fe(E)1725 4952 y Fn(.)31 b(Clearly)-6 b(,)25 b Fm(L)2184 4967 y Fe(E)2271 4952 y Fl(=)j Fm(L)22 b Fi(\000)h Fm(E)31 b Fn(and)1398 5172 y Fm(U)1474 5131 y Fg(0)1498 5172 y Fm(LU)1640 5131 y Fg(0\003)1727 5172 y Fl(=)d Fm(L)1897 5187 y Fe(E)1985 5172 y Fl(=)f Fm(L)c Fi(\000)f Fm(E)6 b(:)1208 b Fn(\(2.3\))p eop %%Page: 5 5 5 4 bop 3730 100 a Fn(5)0 407 y(Since)28 b(zero)g(is)f(a)g(simple)f (eigen)l(v)n(alue)h(of)g Fm(L)p Fn(,)i Fm(E)k Fn(is)27 b(also)g(a)h(simple)e(eigen)l(v)n(alue.)38 b(Hence)27 b(eigen)l(v)n(alues)g(of)g Fm(L)0 527 y Fn(are)f(simple.)j(Note)c(that) f(if)h Fm(U)38 b Fl(:=)28 b Fm(J)9 b(U)1321 491 y Fg(0)1345 527 y Fm(J)g Fn(,)25 b(then)f(\(2.3\))h(implies)1549 735 y Fm(U)10 b(LU)1767 694 y Fg(\003)1836 735 y Fl(=)27 b Fm(L)c Fl(+)f Fm(E)6 b(:)1358 b Fn(\(2.4\))0 989 y(Let)21 b(no)n(w)f Fm(E)416 1004 y Fe(i)445 989 y Fn(,)i Fm(i)28 b Fl(=)f(1)p Fm(;)17 b Fl(2)k Fn(be)g(tw)o(o)g(eigen)l(v)n(alues)f(of)h Fm(L)h Fn(and)f(let)g Fm(U)2143 953 y Fg(0)2133 1014 y Fe(i)2167 989 y Fn(,)g Fm(U)2279 1004 y Fe(i)2329 989 y Fn(be)g(as)h(in)e(\(2.3\),)i(\(2.4\).)30 b(Set)21 b Fm(W)42 b Fl(:=)27 b Fm(U)3619 953 y Fg(0)3609 1014 y Fc(2)3649 989 y Fm(U)3715 1004 y Fc(1)3755 989 y Fn(.)0 1109 y(Then)e Fm(W)38 b Fn(is)25 b(unitary)f(and)1386 1230 y Fm(W)14 b(LW)1664 1189 y Fg(\003)1731 1230 y Fl(=)28 b Fm(L)22 b Fl(+)g Fm(E)2093 1245 y Fc(1)2155 1230 y Fi(\000)h Fm(E)2327 1245 y Fc(2)2366 1230 y Fm(:)0 1399 y Fn(It)i(follo)n(ws)e(that)h Fm(E)653 1414 y Fc(2)715 1399 y Fi(\000)f Fm(E)887 1414 y Fc(1)951 1399 y Fn(is)i(an)g(eigen)l (v)n(alue)f(of)h Fm(L)g Fn(and)g Fm(\033)2035 1414 y Fc(p)2078 1399 y Fl(\()p Fm(L)p Fl(\))h Fn(is)e(a)h(subgroup)f(of)h Fj(R)5 b Fn(.)36 b Fa(2)0 1704 y Fo(Pr)n(oof)29 b(of)f(Theor)n(em)i (1.2.)41 b Fn(Theorem)28 b(1.1)g(and)h(the)f(third)g(relation)g(in)g (\(2.1\))g(yield)g(that)g Fm(\033)3181 1719 y Fc(p)3224 1704 y Fl(\()p Fm(L)p Fl(\))35 b(=)f Fm(\033)3566 1719 y Fc(p)3610 1704 y Fl(\()p Fi(L)p Fl(\))p Fn(,)0 1825 y(and)25 b(so)f(it)h(suf)n(\002ces)f(to)h(pro)o(v)o(e)e(that)i Fm(\033)1270 1840 y Fc(p)1314 1825 y Fl(\()p Fi(L)p Fl(\))i(=)g Fi(f)p Fl(0)p Fi(g)p Fn(.)0 1991 y(Let)36 b Fm(E)55 b Fi(2)50 b Fm(\033)467 2006 y Fc(p)510 1991 y Fl(\()p Fi(L)p Fl(\))p Fn(,)39 b Fm(U)795 1955 y Fg(0)819 1991 y Fn(,)g Fm(U)47 b Fn(be)37 b(as)f(in)g(\(2.3\),)j(\(2.4\),)g(and)d Fm(W)63 b Fl(=)49 b Fm(U)10 b(U)2482 1955 y Fg(0)2506 1991 y Fn(.)66 b Fm(W)50 b Fn(is)35 b(unitary)-6 b(,)38 b Fm(W)14 b Fl(\012)49 b Fi(2)h(P)45 b Fn(and)0 2112 y Fm(W)14 b Fi(L)35 b Fl(=)g Fi(L)p Fm(W)14 b Fn(.)42 b(Since)29 b(zero)h(is)e(a)h(simple)f(eigen)l(v)n(alue)g(of)h Fi(L)g Fn(we)g(must)e(ha)n(v)o(e)i Fm(W)14 b Fl(\012)35 b(=)g(e)3087 2076 y Fc(i)p Fe(\022)3146 2112 y Fl(\012)30 b Fn(for)f(some)f(real)0 2232 y(phase)d Fm(\022)s Fn(.)31 b(Since)25 b Fi(P)33 b Fn(is)25 b(a)g(self-dual)f(cone,)h Fm(\022)31 b Fl(=)d(0)c Fn(and)1600 2440 y Fm(U)10 b(J)f(U)h(J)f Fl(\012)30 b(=)e(\012)p Fm(:)0 2648 y Fn(Using)c(that)g Fm(J)9 b Fl(\012)29 b(=)e(\012)e Fn(and)g Fm(U)39 b Fi(2)28 b Fk(M)c Fn(we)h(deri)n(v)o(e)1358 2856 y Fm(J)9 b(U)h Fl(\012)29 b(=)f Fm(U)1776 2815 y Fg(\003)1816 2856 y Fl(\012)g(=)f Fm(J)9 b Fl(\001)2161 2815 y Fc(1)p Fe(=)p Fc(2)2272 2856 y Fm(U)h Fl(\012)1914 3087 y(=)27 b Fm(J)9 b Fl(e)2123 3046 y Fg(L)p Fe(=)p Fc(2)2247 3087 y Fm(U)h Fl(\012)p Fm(:)0 3293 y Fn(It)25 b(follo)n(ws)e(that)h Fi(L)p Fm(U)10 b Fl(\012)29 b(=)e(0)p Fn(,)e(and)g(since)f Fi(L)p Fm(U)10 b Fl(\012)29 b(=)e Fi(\000)p Fm(E)6 b(U)k Fl(\012)p Fn(,)27 b Fm(E)34 b Fl(=)27 b(0)p Fn(.)k(Hence)25 b Fm(\033)2773 3308 y Fc(p)2817 3293 y Fl(\()p Fi(L)p Fl(\))i(=)h Fi(f)p Fl(0)p Fi(g)p Fn(.)i Fa(2)0 3723 y Fh(Refer)m(ences)0 3987 y Fn([BR])94 b(Brattelli)24 b(O.,)g(Robinson)f (D.)h(W)-9 b(.:)30 b Fb(Oper)o(ator)23 b(Alg)o(ebr)o(as)f(and)i (Quantum)f(Statistical)f(Mec)o(hanics)h(1)p Fn(,)292 4107 y(Springer)n(-V)-11 b(erlag,)25 b(Berlin,)g(second)g(edition)e (\(1987\).)0 4307 y([DJP])60 b(Derezi)8 b(\264)-41 b(nski)25 b(J.,)f(Jak)-5 b Fl(\024)-44 b(s)p Fn(i)6 b(\264)-39 b(c)25 b(V)-13 b(.,)25 b(Pillet)f(C.-A.:)31 b(In)25 b(preparation.)0 4507 y([HN])82 b(Halmos,)21 b(P)-11 b(.R.,)22 b(v)n(on)f(Neumann,)h (J.:)28 b(Operator)21 b(methods)f(in)h(classical)g(mechanics)g(II.)h (Ann.)e(Math.)292 4627 y Fo(43)p Fn(,)k(332)h(\(1940\).)0 4827 y([J])187 b(Jadczyk)22 b(A.)g(Z.:)28 b(On)22 b(some)f(groups)g(of) h(automorphisms)e(of)i(v)n(on)f(Neumann)g(algebras)h(with)f(c)o(yclic) 292 4948 y(and)k(separating)f(v)o(ector)-5 b(.)24 b(Commun.)g(Math.)g (Phys.)g Fo(13)p Fn(,)g(142)h(\(1969\).)0 5147 y([JP])132 b(Jak)-5 b Fl(\024)-44 b(s)p Fn(i)6 b(\264)-39 b(c,)32 b(V)-13 b(.,)31 b(Pillet)f(C.-A.:)41 b(On)30 b(a)h(model)e(for)h (quantum)f(friction)h(III.)g(Er)n(godic)g(properties)g(of)g(the)292 5268 y(spin-boson)23 b(system.)h(Commun.)g(Math.)g(Phys.)g Fo(178)p Fn(,)g(627)g(\(1996\).)p eop %%Page: 6 6 6 5 bop 3730 100 a Fn(6)0 407 y([RR])94 b(Robinson)28 b(D.W)-9 b(.,)30 b(Ruelle)f(D.:)38 b(Extremal)28 b(in)l(v)n(ariant)g (states.)g(Ann.)h(Inst.)f(Henri)h(Poincar)6 b(\264)-39 b(e)29 b Fo(6)p Fn(,)h(299)292 527 y(\(1967\).)0 731 y([W])132 b(W)-8 b(alters,)25 b(P)-11 b(.:)31 b Fb(An)24 b(Intr)l(oduction)f(to)h(Er)l(godic)g(Theory)-5 b(.)25 b Fn(Springer)n(-V)-11 b(erlag,)25 b(Ne)n(w-Y)-11 b(ork)24 b(\(1982\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107090858597--