This is a multi-part message in MIME format. ---------------0204040432580 Content-Type: text/plain; name="02-168.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-168.comments" Published in J. Phys. A: Math. Gen. 30 (1997) 3029-3056; placed here due to renewed interest, in particular in comparison with later work on the topological structure of substitution tilings (e.g., see Anderson and Putnam, Erg. Theory and Dyn. Systems 18 (1998) 509-537). ---------------0204040432580 Content-Type: text/plain; name="02-168.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-168.keywords" quasicrystals, model sets, Fibonacci chain, Penrose tilings, torus parametrization, icosahedral symmetry ---------------0204040432580 Content-Type: application/postscript; name="topar8.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="topar8.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: topar8.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips topar8 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.04.03:1231 %%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: special.pro %! 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g(from)f(the)h(strip)118 700 y(construction)43 b(that)f(are)g(iden)m (ti\014ed)g(in)g(this)f(w)m(a)m(y)-8 b(,)46 b(exactly)d(t)m(w)m(o)g (singular)e(c)m(hains)h(ha)m(v)m(e)i(the)118 833 y(same)35 b(parameter)g(and)g(they)h(di\013er)f(from)f(eac)m(h)i(other)f(in)g(at) g(most)f(t)m(w)m(o)i(p)s(oin)m(ts.)51 b(Also)35 b(these)118 965 y(singular)26 b(c)m(hains)h(form)f(a)g(lo)m(w)m(er-dimensional)e (subset)29 b(\(in)d(a)h(sense)h(to)f(b)s(e)g(made)g(precise)g(later\)) 118 1098 y(of)32 b(the)h(LI-class.)44 b(All)30 b(other)j(c)m(hains)g (are)g(called)e Fv(r)-5 b(e)g(gular)p FC(.)243 1230 y(More)35 b(generally)-8 b(,)35 b(singular)f(patterns)i(deriv)m(ed)g(from)e(the)i (cut-and-pro)5 b(ject)35 b(construction)118 1362 y(with)d(p)s(olytopic) f(windo)m(ws)i(di\013er)f(from)f(eac)m(h)j(other)e(at)g(most)g(on)h(a)f (\014nite)g(n)m(um)m(b)s(er)h(of)f(lo)m(w)m(er-)118 1495 y(dimensional)e(subspaces)35 b(and)e(form)e(\014nite)i(classes.)44 b(\(So)32 b(the)h(am)m(biguities)e(ha)m(v)m(e)i(the)g(\015a)m(v)m(our) 118 1627 y(of)d(a)g(surface)h(e\013ect.\))44 b(W)-8 b(e)30 b(do)g(not)h(completely)e(kno)m(w)i(ho)m(w)g(this)f(is)g(to)g(b)s(e)g (mo)s(di\014ed)f(for)h(more)118 1760 y(general)35 b(shap)s(es)i(of)e (windo)m(ws)h(\(e.g.,)h(spherical)e(or)g(fractal\).)51 b(Ho)m(w)m(ev)m(er,)39 b(if)34 b(the)i(b)s(oundary)g(of)118 1892 y(the)g(windo)m(w)g(has)h(Leb)s(esgue)g(measure)f(0)f(\(and)h(w)m (e)h(will)c(not)j(consider)g(an)m(y)g(situation)e(other)118 2024 y(than)e(that\),)f(the)h(iden)m(ti\014cation)e(of)i(singular)e (patterns)i(is)f(meaningful)f(\(in)g(view)i(of)f(ph)m(ysical)118 2157 y(indistinguishabilit)m(y\))e(and)k(still)d(almost)h(all)g (patterns)i(of)f(the)h(LI-class)g(are)g(regular.)43 b(In)33 b([21)o(])118 2289 y(and)28 b([17])g(the)h(con)m(v)m(en)m(tion)g(is)e (adopted)i(of)e(iden)m(tifying)g(singular)g(patterns)h(in)g(the)g(same) g(class,)118 2422 y(enabling)42 b(the)i(corresp)s(ondence)h(b)s(et)m(w) m(een)h(patterns)e(and)f(parameters)h(to)f(b)s(e)g(regarded)h(as)118 2554 y(one-to-one.)118 2826 y Fo(T)-9 b(orus)37 b(parametrization)118 3023 y FC(W)-8 b(e)38 b(no)m(w)h(c)m(hose)g(a)e(reference)j(p)s(oin)m (t)d Fr(r)j FC(on)e Fq(E)44 b FC(and)38 b(asso)s(ciate)f(with)g(eac)m (h)i(p)s(oin)m(t)e Fr(t)g Fy(2)g Fx(R)3443 2987 y FD(2)3526 3023 y FC(the)118 3155 y(c)m(hain)f(\(or)g(pair)f(of)g(singular)g(c)m (hains\))h(in)g Fq(E)42 b FC(giv)m(en)36 b(b)m(y)h(translating)d Fr(r)39 b FC(to)d Fr(t)p FC(.)54 b(Since)36 b(adding)g(a)118 3288 y(lattice)27 b(v)m(ector)i(in)f Fx(Z)879 3251 y FD(2)944 3288 y FC(to)g Fr(t)g FC(do)s(es)h(not)f(c)m(hange)h(the)g(c)m (hain)f(in)g Fq(E)6 b FC(,)29 b(b)m(y)g(the)g(preceding)f(subsection) 118 3420 y(this)43 b(puts)g(the)h(p)s(oin)m(ts)e(of)h(the)g(torus)g Fx(T)1651 3384 y FD(2)1739 3420 y FC(=)i Fx(R)1926 3384 y FD(2)1972 3420 y Fq(=)p Fx(Z)2090 3384 y FD(2)2169 3420 y FC(in)d(one-to-one)h(corresp)s(ondence)i(with)118 3552 y(the)35 b(regular)f(c)m(hains)h(and)g(singular)f(c)m(hain)h (pairs)f(in)g(the)h(LI-class)g(of)f Fy(F)2815 3567 y Fw(aa)2894 3552 y FC(.)50 b(Adding)35 b(a)f(v)m(ector)118 3685 y Fr(a)e Fy(2)g Fq(E)41 b FC(to)34 b Fr(t)i FC(clearly)e(has)h (the)g(e\013ect)h(of)e(translating)g(the)h(c)m(hain)g(b)m(y)h Fy(\000)p Fr(a)p FC(,)g(but)f(v)m(ectors)h Fr(t)3479 3700 y FD(1)3518 3685 y FC(,)g Fr(t)3622 3700 y FD(2)118 3817 y FC(whose)27 b(di\013erence)f(is)f(not)h(in)f Fq(E)14 b FC(+)8 b Fx(Z)1432 3781 y FD(2)1494 3817 y FC(corresp)s(ond)27 b(to)e(c)m(hains)h(in)f(di\013eren)m(t)h(translation)d(classes.)118 3950 y(Since)37 b Fq(\034)49 b FC(is)36 b(irrational,)f(eac)m(h)i (translation)e(class)j(wraps)f(around)g Fx(T)2670 3914 y FD(2)2750 3950 y FC(in\014nitely)e(man)m(y)i(times)118 4082 y(in)29 b(b)s(oth)h(directions)g(without)f(self-in)m(tersection)g (\(equiv)-5 b(alen)m(t)30 b(to)g(sa)m(ying)g(that)g(the)g(c)m(hains)g (are)118 4215 y(not)j(p)s(erio)s(dic\).)243 4347 y(W)-8 b(e)35 b(call)e(the)i(one-to-one)f(corresp)s(ondence)j(with)d Fx(T)2213 4311 y FD(2)2291 4347 y FC(the)h Fv(torus)i(p)-5 b(ar)g(ametrization)34 b FC(of)g(the)118 4479 y(LI-class.)243 4612 y(Note)25 b(that)g(the)h(parametrization)d(dep)s(ends)k(on)e(the)h (p)s(osition)d(of)i Fr(r)j FC(on)e Fq(E)31 b FC(and)25 b(the)h(p)s(osition)118 4744 y(of)37 b(the)g(windo)m(w)h Fq(W)50 b FC(in)37 b Fq(E)1099 4759 y FD(in)n(t)1187 4744 y FC(:)53 b(if)36 b Fr(r)k FC(is)c(mo)m(v)m(ed)i(to)f Fr(r)28 b FC(+)d Fr(a)37 b FC(on)g Fq(E)43 b FC(the)37 b(parameters)g(are)g(shifted)118 4877 y(b)m(y)f Fy(\000)p Fr(a)g FC(on)f Fx(T)633 4840 y FD(2)675 4877 y FC(,)h(if)e Fq(W)49 b FC(is)34 b(translated)h(b)m(y)h Fr(b)f FC(on)g Fq(E)1970 4892 y FD(in)n(t)2093 4877 y FC(the)h(parameters)f(are)g (shifted)g(b)m(y)h Fy(\000)p Fr(b)f FC(on)118 5009 y Fx(T)181 4973 y FD(2)224 5009 y FC(.)58 b(This)37 b(is)g(analogous)f (to)h(parametrizing)e(a)i(Euclidean)g(line)f(b)m(y)j(the)e(real)g(n)m (um)m(b)s(ers:)54 b(the)118 5141 y(LI-class)27 b(of)f Fy(F)647 5156 y Fw(aa)726 5141 y FC(,)i(lik)m(e)e(the)h(Euclidean)g (line,)f(is)h(a)f(uniform)f(structure)k(with)d(no)h(natural)e(origin.) 118 5274 y Fv(T)-7 b(o)36 b(standar)-5 b(dize)37 b(the)g(p)-5 b(ar)g(ametrization)36 b(of)h(the)g(Fib)-5 b(onac)g(ci)36 b(chains)g(we)h(shal)5 b(l)36 b(cho)-5 b(ose)36 b Fr(r)k Fv(to)e(b)-5 b(e)118 5406 y(the)35 b(origin)f(of)h Fq(E)41 b Fv(and)34 b(the)h(interval)f Fq(W)49 b Fv(to)35 b(b)-5 b(e)34 b(symmetric)h(ab)-5 b(out)35 b(the)g(origin)f(of)h Fq(E)3246 5421 y FD(in)n(t)3334 5406 y Fv(.)118 5539 y FC(With)d(this)g(standardization)g Fy(F)1327 5554 y Fw(aa)1438 5539 y FC(itself)f(has)i(parameter)f Fr(t)c FC(=)g(\(0)p Fq(;)17 b FC(0\).)1865 5989 y(4)p eop %%Page: 5 5 5 4 bop 118 171 a Fo(Symmetry)118 367 y FC(The)23 b(only)e(kind)h(of)g (p)s(oin)m(t)f(symmetry)h(p)s(ossible)f(for)h(1D)f(c)m(hains)h(is)g (mirror)e(symmetry)-8 b(,)24 b(whic)m(h)e(w)m(e)118 500 y(shall)33 b(usually)g(refer)h(to)f(as)h Fv(inversion)h(symmetry)f FC(in)f(order)h(to)g(ha)m(v)m(e)h(the)f(same)g(terminology)118 632 y(for)40 b(all)f(dimensions)h(\(\\in)m(v)m(ersion")g(meaning)g(the) h(isometry)f Fr(x)h Fy(7!)g(\000)p Fr(x)q FC(\).)68 b Fy(F)3059 647 y Fw(aa)3178 632 y FC(is)40 b(in)m(v)m(ersion)118 765 y(symmetric.)h(Are)28 b(there)h(other)f(in)m(v)m(ersion)g (symmetric)f(Fib)s(onacci)f(c)m(hains?)42 b(W)-8 b(ell,)28 b(clearly)f(the)118 897 y(Fib)s(onacci)32 b(c)m(hains)i(corresp)s (onding)f(to)h(p)s(oin)m(ts)f Fr(t)h FC(and)g Fy(\000)p Fr(t)g FC(on)f(the)i(torus)e(are)h(mirror)e(images,)118 1029 y(and)23 b(since)h(the)g(torus)f(parametrization)e(is)i(a)g (one-to-one)f(corresp)s(ondence)j(these)g(are)e(the)h(only)118 1162 y(mirror)40 b(image)f(pairs)i(of)g(Fib)s(onacci)f(c)m(hains.)70 b(\(Th)m(us,)46 b(for)41 b(our)g(standard)h(parametrization)118 1294 y(whic)m(h)c(has)g(the)g(reference)h(p)s(oin)m(t)e(in)m(v)m (ersion)h(symmetric)e(in)h Fq(E)44 b FC(and)37 b(the)h(windo)m(w)g(in)m (v)m(ersion)118 1427 y(symmetric)31 b(in)g Fq(E)777 1442 y FD(in)n(t)866 1427 y FC(,)h(in)m(v)m(ersion)g(of)g(Fib)s(onacci)e(c)m (hains)i(in)f Fq(E)38 b FC(deriv)m(es)33 b(from)e(in)m(v)m(ersion)h(in) f Fx(R)3551 1391 y FD(2)3596 1427 y FC(.\))118 1559 y(So)37 b(in)m(v)m(ersion)h(symmetric)e(Fib)s(onacci)f(c)m(hains)j(corresp)s (ond)g(to)f(p)s(oin)m(ts)g Fr(t)g FC(on)h(the)f(torus)h(with)118 1692 y Fr(t)28 b FC(=)g Fy(\000)p Fr(t)33 b FC(i.e.,)f(2)p Fr(t)c FC(=)g Fo(0)p FC(.)43 b(There)34 b(are)f(4)f(suc)m(h)i(p)s(oin)m (ts)1288 1924 y(\(0)p Fq(;)17 b FC(0\))p Fq(;)g FC(\()1598 1884 y FD(1)p 1598 1901 36 4 v 1598 1958 a(2)1642 1924 y Fq(;)g FC(0\))p Fq(;)g FC(\(0)p Fq(;)1956 1884 y FD(1)p 1956 1901 V 1956 1958 a(2)2002 1924 y FC(\))32 b(and)h(\()2310 1884 y FD(1)p 2310 1901 V 2310 1958 a(2)2355 1924 y Fq(;)2408 1884 y FD(1)p 2408 1901 V 2408 1958 a(2)2454 1924 y FC(\))1045 b(\(1\))118 2156 y(that)29 b(form)g(the)h(discrete)g(subgroup)g(of)f (\\2-division)e(p)s(oin)m(ts")i(of)g Fx(T)2600 2119 y FD(2)2643 2156 y FC(,)h(isomorphic)d(to)j Fq(C)3377 2171 y FD(2)3432 2156 y Fy(\002)16 b Fq(C)3595 2171 y FD(2)3634 2156 y FC(.)118 2288 y(The)34 b(4)e(corresp)s(onding)g(symmetric)g(c)m (hains,)h(in)f(the)h(order)g(of)f(the)h(parameters)f(\(1\),)h(are)767 2514 y Fy(F)839 2529 y Fw(aa)1001 2514 y FC(=)113 b Fq(:)17 b(:)g(:)f(baababaabaababaa)p 1975 2530 103 4 v 5 w(babaabaababaab)h(:)g (:)g(:)117 b(;)786 2646 y Fy(F)858 2661 y Fw(a)1001 2646 y FC(=)88 b Fq(:)17 b(:)g(:)f(ababaababaabaaba)p 2001 2662 52 4 v 5 w(baabaababaababa)h(:)g(:)g(:)91 b(;)789 2778 y Fy(F)861 2793 y Fw(b)1001 2778 y FC(=)83 b Fq(:)17 b(:)g(:)f(aabaababaababaab)p 2006 2794 42 4 v 5 w(aababaababaabaa)h(:)g (:)g(:)86 b(;)771 2911 y Fy(F)843 2926 y Fw(ab)1001 2911 y FC(=)112 b Fq(:)17 b(:)g(:)f(baabaababaaba)p Fy(f)1997 2875 y Fw(ab)p 1997 2890 68 4 v 1997 2942 a(ba)2075 2911 y Fy(g)p Fq(abaababaabaab)h(:)g(:)g(:)115 b(:)3537 2711 y FC(\(2\))118 3128 y(Here)33 b Fy(F)420 3143 y Fw(ab)524 3128 y FC(is)f(a)h(pair)e(of)h(singular)f(c)m(hains)i(but)g(the)g (other)g(3)f(c)m(hains)h(are)g(regular.)118 3400 y Fo(In\015ation)118 3596 y FC(In)25 b(addition)d(to)i(in)m(v)m(ersion)g(symmetry)-8 b(,)26 b(the)f(Fib)s(onacci)d(c)m(hains)i(sho)m(w)i(another)e(feature)g (t)m(ypical)118 3729 y(of)37 b(quasip)s(erio)s(dic)f(patterns,)k (namely)d(an)h(in\015ation/de\015ation)d(symmetry)-8 b(.)59 b(Let)38 b(us)g(explain)118 3861 y(this)26 b(in)f(the)h(pro)5 b(jection)26 b(picture.)41 b(In)27 b(Fig.)d(1)i(the)g(original)d(strip) i Fq(S)2564 3876 y FD(1)2630 3861 y FC(represen)m(ting)i(a)e(Fib)s (onacci)118 3994 y(c)m(hain)35 b Fy(F)44 b FC(\()p Fy(F)604 4009 y Fw(aa)718 3994 y FC(in)34 b(fact\))h(is)f(enclosed)i(in)e(a)h (strip)f Fq(S)2045 4009 y Fw(\034)2088 3994 y FC(,)i Fq(\034)46 b FC(times)35 b(as)g(wide.)50 b(The)36 b(lattice)e(p)s(oin)m (ts)118 4126 y(in)h Fq(S)295 4141 y Fw(\034)375 4126 y FC(giv)m(e)h(a)f(re\014nemen)m(t)i(of)f(the)g(c)m(hain)g(giv)m(en)g (b)m(y)h Fq(S)2150 4141 y FD(1)2190 4126 y FC(.)54 b(This)36 b(re\014nemen)m(t)h(is)f(itself)e(a)i(scaled)118 4258 y(v)m(ersion)d(of)f(a)h(Fib)s(onacci)d(c)m(hain.)44 b(T)-8 b(o)32 b(see)i(this,)e(consider)h(the)g(action)f(of)g(the)h(matrix)1535 4530 y Fq(M)66 b FC(=)1826 4390 y Fn(\022)1916 4464 y FC(1)97 b(1)1916 4596 y(1)g(0)2128 4390 y Fn(\023)2218 4530 y Fq(:)1292 b FC(\(3\))118 4812 y(It)36 b(has)f(an)h(eigen)m(v)m (ector)1015 4776 y FD(2)1091 4812 y Fr(e)d FC(=)f(\()p Fq(\034)6 b(;)17 b FC(1\))35 b(with)g(eigen)m(v)-5 b(alue)35 b Fq(\034)47 b FC(in)34 b(the)i(direction)f(of)g Fq(E)6 b FC(,)36 b(an)f(eigen-)118 4944 y(v)m(ector)43 b Fr(e)475 4959 y FD(in)n(t)607 4944 y FC(=)g(\()p Fy(\000)p FC(1)p Fq(=\034)6 b(;)17 b FC(1\))41 b(with)g(eigen)m(v)-5 b(alue)41 b Fy(\000)p FC(1)p Fq(=\034)53 b FC(in)41 b(the)h(direction)f(of)g Fq(E)3043 4959 y FD(in)n(t)3132 4944 y FC(,)j(and,)g(b)s(eing)118 5077 y(in)m(tegral)34 b(and)h(unimo)s(dular,)f(acts)i(as)g(an)f (automorphism)e(on)j(the)f(lattice)f Fx(Z)3014 5040 y FD(2)3051 5077 y FC(.)52 b(So)35 b Fq(M)10 b(S)3432 5092 y Fw(\034)3512 5077 y FC(is)35 b(a)118 5209 y(strip)40 b(of)f(the)h(same)g(width)g(and)g(direction)e(as)j Fq(S)1981 5224 y FD(1)2060 5209 y FC(and)f(represen)m(ts)i(a)e(Fib)s(onacci)e(c)m (hain)i Fy(F)3612 5173 y Fu(0)3634 5209 y FC(.)118 5341 y(Since)f Fq(M)49 b FC(maps)39 b(the)g(lattice)e(p)s(oin)m(ts)h(in)g Fq(S)1741 5356 y Fw(\034)1823 5341 y FC(one-to-one)g(on)m(to)h(the)g (lattice)e(p)s(oin)m(ts)h(of)g Fq(M)10 b(S)3590 5356 y Fw(\034)3634 5341 y FC(,)118 5474 y(preserv)m(es)43 b(the)e(pro)5 b(jection)40 b(direction)f(and)h(stretc)m(hes)i (distances)f(in)f(the)g(direction)f Fq(E)47 b FC(b)m(y)41 b(a)p 118 5561 1418 4 v 230 5622 a Fs(2)268 5653 y FB(F)-7 b(or)27 b(later)f(con)n(v)n(enience,)h(w)n(e)g(do)g(not)h(use)f(unit)i (v)n(ectors)d(here.)1865 5989 y FC(5)p eop %%Page: 6 6 6 5 bop 118 171 a FC(factor)41 b Fq(\034)11 b FC(,)45 b(the)d(c)m(hain)g(giv)m(en)g(b)m(y)g(the)h(lattice)d(p)s(oin)m(ts)h (in)g Fq(S)2353 186 y Fw(\034)2438 171 y FC(is)g(the)i(c)m(hain)e Fy(F)3071 134 y Fu(0)3136 171 y FC(shrunk)i(b)m(y)g(a)118 303 y(factor)32 b Fq(\034)11 b FC(.)243 435 y(The)32 b(t)m(w)m(o)h(features)f(of)g(b)s(eing)f(lo)s(cally)e(isomorphic)h(to)h Fy(F)42 b FC(and)32 b(scaling)e(to)h(a)h(re\014nemen)m(t)h(of)118 568 y(it)f(\(ev)m(en)m(tually)h(up)g(to)f(a)g(shift\))g(mak)m(e)h Fy(F)1656 532 y Fu(0)1711 568 y FC(an)g Fv(in\015ation)f FC(of)g Fy(F)10 b FC(.)243 700 y(Ho)m(w)31 b(is)f(the)h(torus)f (parameter)g(of)g Fy(F)1627 664 y Fu(0)1681 700 y FC(related)g(to)g (that)g(of)g Fy(F)10 b FC(?)42 b(W)-8 b(ell,)30 b(if)f Fy(F)40 b FC(has)31 b(parameter)118 833 y Fr(t)40 b FC(then)g(the)h(p)s (oin)m(t)e(on)g(the)h(cen)m(treline)g(of)f Fq(S)1810 848 y Fw(\034)1893 833 y FC(that)h(pro)5 b(jects)41 b(on)m(to)e(the)h (origin)e(of)h Fq(E)46 b FC(is)39 b Fr(t)28 b FC(+)118 965 y(\()p Fq(\034)11 b(=)p FC(2)307 883 y Fy(p)p 390 883 49 4 v 82 x FC(5\))p Fr(e)532 980 y FD(in)n(t)620 965 y FC(,)33 b(so)g Fy(F)882 929 y Fu(0)937 965 y FC(has)g(parameter) 1233 1204 y Fq(M)10 b FC(\()p Fr(t)23 b FC(+)1611 1137 y Fq(\034)p 1547 1181 181 4 v 1547 1283 a FC(2)1596 1201 y Fy(p)p 1679 1201 49 4 v 82 x FC(5)1738 1204 y Fr(e)1793 1219 y FD(in)n(t)1882 1204 y FC(\))k(=)h Fq(M)10 b Fr(t)23 b Fy(\000)2347 1137 y Fr(e)2402 1152 y FD(in)n(t)p 2329 1181 181 4 v 2329 1283 a FC(2)2378 1201 y Fy(p)p 2460 1201 49 4 v 2460 1283 a FC(5)2519 1204 y Fq(:)991 b FC(\(4\))118 1468 y(As)39 b Fy(F)49 b FC(v)-5 b(aries,)40 b(the)f(in\015ation)e(giv) m(en)i(b)m(y)g(enclosing)f(the)i(strip)e(for)g Fy(F)48 b FC(in)38 b(a)h(strip)f Fq(\034)50 b FC(times)38 b(as)118 1601 y(wide)h(with)f(the)h(lo)m(w)m(er)f(edges)i(coinciding)d(is)h (represen)m(ted)j(on)d Fx(T)2580 1564 y FD(2)2661 1601 y FC(b)m(y)i Fr(t)e Fy(7!)g Fq(M)10 b Fr(t)27 b FC(+)f Fr(c)p FC(,)40 b(where)118 1733 y Fr(c)28 b FC(=)f Fy(\000)p Fr(e)433 1748 y FD(in)n(t)522 1733 y Fq(=)p FC(2)620 1651 y Fy(p)p 702 1651 V 702 1733 a FC(5)32 b(is)h(indep)s(enden)m(t)g (of)f Fr(t)q FC(.)243 1865 y(Cho)s(osing)i(a)h(di\013eren)m(t)g (\014xed)h(o\013set)f(for)f(the)i(wide)f(strip)f(relativ)m(e)g(to)h(to) g(the)g(narro)m(w)g(one)118 1998 y(leads)42 b(to)g(a)g(di\013eren)m(t)h (in\015ation)d(on)i(the)h(LI-class)f(whic)m(h)h(is)f(again)e(represen)m (ted)45 b(on)d Fx(T)3473 1962 y FD(2)3558 1998 y FC(b)m(y)118 2130 y Fr(t)j Fy(7!)f Fq(M)10 b Fr(t)29 b FC(+)g Fr(c)p FC(,)45 b(but)d(with)g(a)g(di\013eren)m(t)h(v)-5 b(alue)42 b(of)f Fr(c)q FC(.)72 b(The)43 b(ab)s(o)m(v)m(e)g(re\014nemen)m(t)h (prop)s(ert)m(y)f(of)118 2263 y(the)36 b(in\015ation)f(corresp)s(onds)i (to)f(the)g(demand)g(that,)g(in)g(our)f(picture,)i(the)g(wider)f(strip) f(m)m(ust)118 2395 y(enclose)e(the)g(narro)m(w)m(er)h(one.)243 2528 y(In)i(terms)g(of)g(Fib)s(onacci)e(c)m(hains,)j(the)g(in\015ation) d(illustrated,)h(giv)m(en)i(b)m(y)g Fr(c)c FC(=)h Fy(\000)p Fr(e)3317 2543 y FD(in)n(t)3405 2528 y Fq(=)p FC(2)3503 2445 y Fy(p)p 3586 2445 V 83 x FC(5)o(,)118 2660 y(corresp)s(onds)g(to) e(the)h(substitution)f(rule)1532 2886 y Fq(b)c Fy(7!)g Fq(a;)114 b(a)28 b Fy(7!)f Fq(ab:)1291 b FC(\(5\))118 3113 y(The)34 b(in\015ation)c(giv)m(en)j(b)m(y)h Fr(c)27 b FC(=)h Fr(e)1331 3128 y FD(in)n(t)1420 3113 y Fq(=)p FC(2)1518 3031 y Fy(p)p 1600 3031 V 1600 3113 a FC(5,)33 b(corresp)s(onds)g(to)g(the)g(rule)1532 3339 y Fq(b)28 b Fy(7!)g Fq(a;)114 b(a)28 b Fy(7!)f Fq(ba:)1291 b FC(\(6\))118 3566 y(More)33 b(generally)-8 b(,)32 b Fr(c)27 b FC(=)h Fq(\014)6 b Fr(e)1102 3581 y FD(in)n(t)1190 3566 y Fq(=)p FC(2)1288 3483 y Fy(p)p 1371 3483 V 83 x FC(5)32 b(with)g Fy(j)p Fq(\014)6 b Fy(j)27 b(\024)h FC(1)k(has)h(the)g(e\013ect)1410 3792 y Fq(b)28 b Fy(7!)g Fq(a;)114 b(a)28 b Fy(7!)f Fq(ab)33 b FC(or)g Fq(ba;)1168 b FC(\(7\))118 4019 y(the)38 b(prop)s(ortion)e (of)h Fq(a)p FC('s)h(that)f(c)m(hange)i(to)e Fq(ba)h FC(b)s(eing)f(\()p Fq(\014)31 b FC(+)25 b(1\))p Fq(=)p FC(2.)58 b(Only)37 b(coun)m(tably)g(man)m(y)h(of)118 4151 y(these)i(in\015ations)d(are)h(lo)s(cal,)g(in)g(the)h(sense)h (that)e(whether)i(a)e(giv)m(en)h Fq(a)g FC(c)m(hanges)g(to)g Fq(ab)g FC(or)f Fq(ba)118 4284 y FC(can)30 b(b)s(e)g(determined)g(b)m (y)h(lo)s(oking)d(at)h(the)i(part)f(of)f Fy(F)39 b FC(within)29 b(a)h(b)s(ounded)h(distance)f(of)f(that)h Fq(a)p FC(.)243 4416 y(In)i(particular,)f(the)h(symmetric)f(in\015ation)g(giv)m(en)h(b) m(y)h Fr(c)28 b FC(=)f Fo(0)p FC(,)32 b(discussed)i(in)e([17)o(],)h(in) e(whic)m(h)118 4548 y(half)39 b(the)h Fq(a)p FC('s)g(c)m(hange)h(to)e Fq(ab)i FC(and)e(half)g(to)g Fq(ba)p FC(,)j(is)e(non-lo)s(cal.)62 b(It)40 b(can)g(b)s(e)g(describ)s(ed)g(b)m(y)h(the)118 4681 y(rule)34 b Fq(a)e Fy(7!)f Fq(x)584 4696 y Fu(\000)p Fw(j)676 4681 y Fq(x)731 4696 y Fw(j)803 4681 y FC(in)j(\(7\),)h(where) h Fq(x)1445 4696 y Fu(\000)p Fw(j)1537 4681 y Fq(x)1592 4696 y Fu(\000)p Fw(j)t FD(+1)1791 4681 y Fq(:)17 b(:)g(:)f(x)1977 4696 y Fu(\000)p FD(1)2072 4681 y Fq(ax)2178 4696 y FD(1)2234 4681 y Fq(:)h(:)g(:)f(x)2420 4696 y Fw(j)t Fu(\000)p FD(1)2547 4681 y Fq(x)2602 4696 y Fw(j)2674 4681 y FC(is)35 b(the)g(shortest)h(segmen)m(t)118 4813 y(of)k Fy(F)50 b FC(cen)m(tred)42 b(on)f(the)g Fq(a)f FC(to)g(b)s(e)h(in\015ated)f (for)g(whic)m(h)h Fq(x)2249 4828 y Fu(\000)p Fw(j)2382 4813 y Fy(6)p FC(=)g Fq(x)2554 4828 y Fw(j)2591 4813 y FC(.)67 b(It)41 b(is)f(non-lo)s(cal)e(b)s(ecause)118 4946 y(a)43 b(Fib)s(onacci)d(c)m(hain)j(has)g(arbitrarily)d(long)i (symmetric)f(neigh)m(b)s(ourho)s(o)s(ds)i(cen)m(tred)h(on)f Fq(a)p FC('s.)118 5078 y(In)g(particular,)h(for)e(the)h(cen)m(tral)g Fq(a)g FC(of)f Fy(F)1697 5093 y Fw(a)1781 5078 y FC(the)h(symmetry)g (nev)m(er)i(breaks)e(and)g(the)h(rule)e(is)118 5211 y(undecidable,)33 b(resulting)e(in)h(in\015ation)f(to)h(a)g(singular)f(c)m(hain)i(with)f (t)m(w)m(o)h(p)s(ossibilities.)243 5343 y(In\015ations)46 b(deriv)m(ed)i(from)e(other)h(v)-5 b(alues)47 b(of)f Fr(c)p FC(,)51 b(with)46 b(a)h(non-zero)g(part)f(parallel)f(to)h Fr(e)p FC(,)118 5475 y(consist)35 b(of)e(one)i(of)e(these)j(follo)m(w)m (ed)d(b)m(y)i(a)f(translation.)46 b(Note)34 b(that)g(the)h(decomp)s (osition)d(in)m(to)118 5608 y(substitution)d(and)g(translation)f(is)h (not)g(unique:)42 b(for)29 b(example,)h(the)g(in\015ation)d(\(6\))i(is) g(iden)m(tical)118 5740 y(to)f(\(5\))f(translated)h(b)m(y)h(the)f (length)g(of)f Fq(a)p FC(.)43 b(This)28 b(non-uniqueness)h(re\015ects)h (the)e(fact)g(that)g Fr(c)g FC(can)1865 5989 y(6)p eop %%Page: 7 7 7 6 bop 118 171 a FC(b)s(e)34 b(written)g(in)f(man)m(y)h(w)m(a)m(ys)i (as)e Fq(\013)q Fr(e)23 b FC(+)g Fq(\014)6 b Fr(e)1691 186 y FD(in)n(t)1780 171 y Fq(=)p FC(2)1878 88 y Fy(p)p 1960 88 49 4 v 1960 171 a FC(5)34 b(\(mo)s(d)f Fx(Z)2371 134 y FD(2)2408 171 y FC(\),)h(where)h Fy(j)p Fq(\014)6 b Fy(j)29 b(\024)i FC(1)i(to)h(guaran)m(tee)118 303 y(the)f (re\014nemen)m(t)h(prop)s(ert)m(y)-8 b(,)33 b(but)g Fq(\013)28 b Fy(2)g Fx(R)43 b FC(arbitrary)-8 b(.)43 b(Ho)m(w)m(ev)m(er,)35 b(distinct)c(p)s(oin)m(ts)i Fr(c)f FC(of)g Fx(T)3418 267 y FD(2)3493 303 y FC(giv)m(e)118 435 y(distinct)g(in\015ations.)243 568 y(The)h(ab)s(o)m(v)m(e)h(discussion)e(mak)m(es)i(the)f(follo)m (wing)c(de\014nition)j(natural.)118 725 y Fo(De\014nition)96 b FC(An)29 b Fv(in\015ation)e FC(on)h(the)g(LI-class)g(of)f(Fib)s (onacci)f(c)m(hains)i(is)g(a)f(mapping)g Fx(T)3358 689 y FD(2)3428 725 y Fy(!)h Fx(T)3619 689 y FD(2)118 858 y FC(of)k(the)h(form)1635 990 y Fr(t)28 b Fy(7!)f Fq(A)p Fr(t)c FC(+)f Fr(c)p Fq(;)1393 b FC(\(8\))118 1176 y(where)35 b Fr(c)29 b Fy(2)g Fx(T)639 1140 y FD(2)715 1176 y FC(and)34 b Fq(A)f FC(is)g(a)g(non-singular)e(2)23 b Fy(\002)g FC(2)33 b(in)m(teger)g(matrix)f(with)h(eigen)m(v)m(ectors)i Fr(e)e FC(and)118 1309 y Fr(e)173 1324 y FD(in)n(t)262 1309 y FC(.)55 b(A)36 b Fv(family)i(of)g(in\015ations)e FC(is)g(a)g(set)h(of)f(in\015ations)f(that)h(share)h(the)g(same)f (matrix)f Fq(A)h FC(but)118 1441 y(ha)m(v)m(e)e Fr(c)f FC(running)f(through)g(all)f(p)s(oin)m(ts)h(of)g Fx(T)1761 1405 y FD(2)1804 1441 y FC(.)243 1599 y(This)g(de\014nition)g(mak)m(es) h(plain)e(that)h(eac)m(h)i(family)c(of)i(in\015ations,)f(lik)m(e)h(the) h(LI-class)g(itself,)118 1731 y(is)27 b(parametrized)g(b)m(y)h Fx(T)990 1695 y FD(2)1033 1731 y FC(.)42 b(As)28 b(men)m(tioned)f(ab)s (o)m(v)m(e,)i(the)f(in\015ation)e(\(giv)m(en)h(b)m(y)i(a)e(certain)g Fr(c)p FC(\))h(ma)m(y)118 1863 y(b)s(e)j(lo)s(ok)m(ed)f(at)h(in)e (\(coun)m(tably)i(man)m(y\))g(di\013eren)m(t)g(w)m(a)m(ys)h(in)e(the)h (space)h Fq(E)k FC(as)31 b(a)g(transformation)118 1996 y(on)i(the)g(c)m(hains.)243 2128 y(In)45 b(our)g(presen)m(t)i(example,) g(the)f(situation)d(is)i(as)g(follo)m(ws.)80 b(Giv)m(en)45 b Fr(c)k Fy(2)g Fx(T)3151 2092 y FD(2)3194 2128 y FC(,)g(there)c(are) 118 2261 y(coun)m(tably)37 b(man)m(y)f(v)-5 b(alues)36 b Fr(c)1186 2225 y Fu(0)1244 2261 y Fy(2)e Fx(R)1410 2225 y FD(2)1492 2261 y FC(with)i Fr(c)1769 2225 y Fu(0)1827 2261 y FC(=)e Fq(\013)q Fr(e)24 b FC(+)h Fq(\014)6 b Fr(e)2296 2276 y Fw(int)2392 2261 y Fq(=)p FC(2)2490 2178 y Fy(p)p 2573 2178 V 83 x FC(5)33 b Fy(\021)i Fr(c)i FC(\(mo)s(d)e Fx(Z)3185 2225 y FD(2)3222 2261 y FC(\),)i Fy(j)p Fq(\014)6 b Fy(j)33 b(\024)i FC(1.)118 2393 y(Ev)m(ery)43 b(c)m(hoice)e(of)g Fr(c)876 2357 y Fu(0)940 2393 y FC(represen)m(ts)j (a)d(di\013eren)m(t)g(view)g(of)g(the)g(same)g(in\015ation,)h(where)g Fq(\014)k FC(tells)118 2526 y(us)38 b(the)f(prop)s(ortion)f(of)g Fq(a)p FC('s)i(that)f(c)m(hange)h(to)f Fq(ab)p FC(,)h(and)g Fy(\000)p Fq(\013)q(\034)11 b(=)2440 2447 y Fy(p)p 2523 2447 223 4 v 79 x Fq(\034)34 b FC(+)22 b(2)37 b(giv)m(es)g(the)g (distance)h(of)118 2658 y(the)c(in\015ation)e(\014xed)j(p)s(oin)m(t)e (to)g(the)h(origin.)44 b(\(Though)34 b(there)h(is)e(no)g(natural)g(c)m (hoice,)h(w)m(e)h(shall)118 2790 y(usually)d(tak)m(e)h(the)g(represen)m (tativ)m(e)i(with)d(minimal)c Fy(j)p Fq(\013)q Fy(j)p FC(.\))243 2923 y(Changing)33 b Fr(c)h FC(b)m(y)h(a)e(scalar)h(m)m (ultiple)d(of)i Fr(e)h FC(merely)g(c)m(hanges)h(the)f(translation)e (comp)s(onen)m(t)118 3055 y(of)41 b(the)h(in\015ation,)g(but)f(other)h (c)m(hanges)g(to)f Fr(c)h FC(c)m(hange)g(the)g(in\015ation)d(more)i (fundamen)m(tally)-8 b(.)118 3188 y(Due)33 b(to)f(the)h(conditions)f (men)m(tioned)g(ab)s(o)m(v)m(e,)h Fq(A)g FC(in)f(fact)g(m)m(ust)h(tak)m (e)g(the)g(form)1255 3420 y Fq(A)56 b FC(=)f Fq(r)25 b Fy(\001)d FC(1)-62 b Fz(1)22 b FC(+)g Fq(s)g Fy(\001)g Fq(M)108 b(;)72 b(r)m(;)17 b(s)28 b Fy(2)g Fx(Z)1010 b FC(\(9\))118 3652 y(where)50 b Fq(M)59 b FC(is)48 b(the)h(matrix)e (of)h Fq(\034)11 b FC(-in\015ation)47 b(as)h(in)g(Eq.)h(\(3\).)91 b Fq(A)49 b FC(represen)m(ts)i(an)d(in)m(v)m(ertible)118 3784 y(in\015ation/de\015ation)39 b(symmetry)i(if)f(and)h(only)g(if)f Fq(A)i FC(=)g Fy(\006)p Fq(M)2424 3748 y Fw(m)2533 3784 y FC(with)f Fq(m)h Fy(2)h Fx(Z)p FC(.)66 b(W)-8 b(e)42 b(call)d(the)118 3917 y(eigen)m(v)-5 b(alue)32 b(of)g Fq(A)h FC(in)f Fq(E)38 b FC(the)33 b(in\015ation)e Fv(multiplier)p FC(.)243 4049 y(The)43 b(in\015ation,)h(giv)m(en)f(b)m(y)g(its)f (represen)m(tation)i(on)f Fx(T)2291 4013 y FD(2)2333 4049 y FC(,)j(generates)e(an)e(in\014nite)g(Ab)s(elian)118 4181 y(group)33 b(isomorphic)e(with)h Fq(C)1181 4196 y Fu(1)1288 4181 y FC(whic)m(h)i(acts)f(on)g(the)g(LI-class.)44 b(This)33 b(giv)m(es)g(rise)f(to)h(some)f(kind)118 4314 y(of)45 b(generalized)g(symmetry)g([1],)j(but)e(not)f(to)g(a)g(p)s(oin) m(t)f(symmetry)i(b)s(ecause)g(it)f(cannot)g(b)s(e)118 4446 y(represen)m(ted)i(as)d(an)g(isometry)-8 b(.)78 b(In)44 b(the)h(sequel,)j(w)m(e)d(shall)e(call)f(c)m(hains)j(\(or)e (patterns\))i Fq(I)3582 4410 y Fw(n)3629 4446 y FC(-)118 4579 y(symmetric)32 b(if)f(they)j(are)f(in)m(v)-5 b(arian)m(t)31 b(under)i Fq(n)p FC(-fold)f(in\015ation.)41 b(Sim)m(ultaneously)-8 b(,)31 b(w)m(e)j(shall)d(use)118 4711 y(the)c(same)f(sym)m(b)s(ol)g(to) g(denote)h(the)g(group)g(generated)g(b)m(y)g Fq(I)2318 4675 y Fw(n)2365 4711 y FC(,)h(where)g Fq(I)34 b FC(stands)27 b(for)f(the)h(sp)s(ecial)118 4844 y(in\015ation)36 b(under)j (consideration)e(\(i.e.)g Fq(M)49 b FC(in)37 b(this)g(case\).)60 b(Note)38 b(that,)h(as)f(groups,)i(they)e(are)118 4976 y(all)31 b(isomorphic)f(with)j(the)g(in\014nite)e(cyclic)h(group)h Fq(C)2083 4991 y Fu(1)2157 4976 y FC(.)118 5248 y Fo(Applications)i(of) j(the)f(torus)h(parametrization)118 5445 y FC(The)25 b(usefulness)h(of)e(the)h(torus)f(parametrization)e(stems)j(from)e(the) h(simple)f(b)s(eha)m(viour)h(of)g(what)118 5577 y(migh)m(t)31 b(b)s(e)i(called)e(the)i(\\a\016ne)g(torus)g(op)s(erators")f(\(8\).) 1865 5989 y(7)p eop %%Page: 8 8 8 7 bop 118 171 a Fo(F)-9 b(undamen)m(tal)38 b(F)-9 b(act)86 b Fv(A)n(n)46 b(op)-5 b(er)g(ator)46 b(of)h(the)f(form)g(\(8\),)j(wher) -5 b(e)46 b Fq(A)h Fv(is)f(any)h(non-singular)118 303 y FC(2)23 b Fy(\002)g FC(2)36 b Fv(inte)-5 b(ger)36 b(matrix)g(with)g Fy(j)17 b FC(det)f Fq(A)p Fy(j)30 b FC(=)f Fq(d)p Fv(,)36 b(is)g(a)g Fq(d)p Fv(-fold)f(c)-5 b(over)36 b(of)f Fx(T)2678 267 y FD(2)2757 303 y Fv(by)h(itself,)g(in)g(the)g(sense)118 435 y(that,)f(for)g(e)-5 b(ach)34 b Fr(w)c Fy(2)e Fx(T)992 399 y FD(2)1035 435 y Fv(,)35 b(ther)-5 b(e)35 b(ar)-5 b(e)35 b(exactly)f Fq(d)h Fv(distinct)g(solutions)f Fr(t)28 b Fy(2)g Fx(T)2901 399 y FD(2)2979 435 y Fv(of)1625 641 y Fq(A)p Fr(t)23 b FC(+)f Fr(c)27 b FC(=)h Fr(w)s Fq(:)1333 b FC(\(10\))118 846 y(W)-8 b(e)40 b(giv)m(e)g(a)g(pro)s(of)f(of)g(this) g(in)h(App)s(endix)g(A.)g(Note)g(that)f(the)i(n)m(um)m(b)s(er)f(of)f (solutions)g(is)h(not)118 979 y(only)30 b(indep)s(enden)m(t)h(of)e Fr(w)k FC(but)d(also)f(indep)s(enden)m(t)j(of)d(the)h(constan)m(t)h (term)f Fr(c)g FC(of)f(the)i(op)s(erator.)243 1111 y(Here)j(are)g(some) g(immediate)d(applications)g(of)j(the)g(F)-8 b(undamen)m(tal)32 b(F)-8 b(act.)47 b(In)34 b(in)m(terpreting)118 1244 y(them)d(for)f(Fib) s(onacci)e(c)m(hains)k(w)m(e)f(need)h(to)e(b)s(ear)h(in)f(mind)f(that)i (the)g(t)m(w)m(o)g(singular)e(c)m(hains)i(are)118 1376 y(attac)m(hed)41 b(to)e(one)h(parameter)f(and)h(m)m(ust)g(b)s(e)g (treated)g(as)g(a)f(single)g(en)m(tit)m(y)-8 b(.)65 b(As)41 b(ab)s(o)m(v)m(e,)h(the)118 1509 y(follo)m(wing)30 b(equations)j(are)f (to)h(b)s(e)f(understo)s(o)s(d)h(mo)s(d)f Fx(T)2174 1472 y FD(2)2217 1509 y FC(.)118 1662 y Fv(How)44 b(many)f(Fib)-5 b(onac)g(ci)43 b(chains)f(ar)-5 b(e)44 b(inversion)f(symmetric?)169 b FC(As)43 b(w)m(e)g(ha)m(v)m(e)h(seen,)i(these)118 1794 y(corresp)s(ond)36 b(to)f(p)s(oin)m(ts)g Fr(t)e Fy(2)f Fx(T)1270 1758 y FD(2)1348 1794 y FC(with)j(2)p Fr(t)d FC(=)g Fo(0)q FC(.)51 b(T)-8 b(aking)35 b Fq(A)d FC(=)g(2)24 b Fy(\001)g FC(1)-62 b Fz(1)35 b FC(and)g Fr(c)d FC(=)g Fo(0)k FC(tells)e(us)i(that)118 1927 y(there)d(are)g Fy(j)17 b FC(det\(2)22 b Fy(\001)f FC(1)-62 b Fz(1)p FC(\))p Fy(j)27 b FC(=)h(4)k(of)g(these.)118 2080 y Fv(Is)44 b(in\015ation)g(of)f(Fib)-5 b(onac)g(ci)43 b(chains)g(invertible?)170 b FC(Since)43 b Fy(j)17 b FC(det)g Fq(M)10 b Fy(j)45 b FC(=)g(1)d(there)i(is)e(exactly)118 2213 y(one)35 b(solution)e(of)h Fq(M)10 b Fr(t)25 b FC(+)e Fr(c)31 b FC(=)g Fr(w)37 b FC(for)d(eac)m(h)i Fr(w)r FC(,)f(so)g(ev)m(ery)i(in\015ation)32 b(in)i(the)h(family)d(\(for)i(an)m(y)h Fr(c)118 2345 y FC(whatev)m(er\))30 b(is)e(a)f(one-to-one)h(corresp)s(ondence)i(from) d Fx(T)2177 2309 y FD(2)2248 2345 y FC(to)h(itself)f(and)h(eac)m(h)h (Fib)s(onacci)e(c)m(hain)118 2478 y(comes)33 b(from)e(a)h(unique)i (precursor)f(under)h(the)f(in\015ation.)118 2631 y Fv(How)k(many)f(Fib) -5 b(onac)g(ci)35 b(chains)h(ar)-5 b(e)36 b(in\015ation-invariant?)146 b FC(W)-8 b(e)34 b(need)i(to)e(coun)m(t)h(the)g(solu-)118 2764 y(tions)21 b(of)h Fq(M)10 b Fr(t)r FC(+)p Fr(c)28 b FC(=)f Fr(t)q FC(,)d(whic)m(h)e(can)g(b)s(e)h(rewritten)f(as)g(\()p Fq(M)11 b Fy(\000)p FC(1)-62 b Fz(1)q FC(\))p Fr(t)p FC(+)p Fr(c)29 b FC(=)e Fo(0)p FC(.)40 b(Since)22 b Fy(j)17 b FC(det\()p Fq(M)11 b Fy(\000)p FC(1)-62 b Fz(1)q FC(\))p Fy(j)27 b FC(=)h(1)118 2896 y(there)36 b(is)g(precisely)g(1)f (in\015ation-in)m(v)-5 b(arian)m(t)32 b(c)m(hain.)52 b(F)-8 b(or)35 b(the)h(non-lo)s(cal)d(symmetric)i(in\015ation)118 3028 y Fr(c)28 b FC(=)f Fo(0)j FC(this)e(is)h Fy(F)739 3043 y Fw(aa)818 3028 y FC(;)h(for)f(the)g(lo)s(cal)e(in\015ation)h Fr(c)f FC(=)h(\()2024 2989 y FD(1)p 2024 3006 36 4 v 2024 3063 a(2)2069 3028 y Fq(;)17 b FC(0\))27 b(=)h(\()p Fr(e)15 b Fy(\000)g Fr(e)2587 3043 y FD(in)n(t)2675 3028 y FC(\))p Fq(=)p FC(2)2811 2946 y Fy(p)p 2894 2946 49 4 v 82 x FC(5)29 b(it)f(is)g(the)i(singular)118 3161 y(c)m(hain)j Fy(F)448 3176 y Fw(ab)519 3161 y FC(.)243 3293 y(As)38 b(w)m(ell)g(as)g(eac)m(h)i(in\015ation)c Fr(c)i FC(ha)m(ving)h(a)f(unique)g(\014xed)i(p)s(oin)m(t,)f(namely)e Fr(a)g FC(=)h Fy(\000)p Fq(M)10 b Fr(c)q FC(,)40 b(for)118 3426 y(eac)m(h)d(p)s(oin)m(t)e Fr(a)h FC(of)f Fx(T)875 3390 y FD(2)954 3426 y FC(there)i(is)e(a)h(unique)g(in\015ation)e(that) i(lea)m(v)m(es)h(it)e(in)m(v)-5 b(arian)m(t,)36 b(namely)f(that)118 3558 y(giv)m(en)e(b)m(y)g Fr(c)28 b FC(=)g Fr(a)22 b Fy(\000)g Fq(M)10 b Fr(a)q FC(.)118 3712 y Fv(How)32 b(many)g(Fib)-5 b(onac)g(ci)30 b(chains)i(have)f(in\015ation)h(orbits)g (of)f(length)h(2?)140 b FC(Or,)30 b(in)e(other)i(w)m(ords,)118 3844 y(ho)m(w)f(man)m(y)g(Fib)s(onacci)d(c)m(hains)j(are)g Fq(I)1522 3808 y FD(2)1561 3844 y FC(-symmetric?)41 b(The)29 b(relev)-5 b(an)m(t)29 b(equation)f(is)g(\()p Fq(M)3315 3808 y FD(2)3369 3844 y Fy(\000)14 b FC(1)-62 b Fz(1)p FC(\))p Fr(t)15 b FC(+)118 3977 y Fr(c)28 b FC(=)f Fo(0)q FC(.)42 b(Since)30 b Fy(j)17 b FC(det)o(\()p Fq(M)999 3940 y FD(2)1055 3977 y Fy(\000)f FC(1)-62 b Fz(1)q FC(\))p Fy(j)27 b FC(=)g(1)j(there)g(is)f(only)g(1)g(solution.)41 b(But)30 b(this)f(m)m(ust)g(represen)m(t)j(the)118 4109 y(in\015ation-in)m(v)-5 b(arian)m(t)29 b(c)m(hain,)k(so)f(there)i(is)e (no)g(c)m(hain)h(with)f(an)g(in\015ation)f(orbit)h(of)g(length)g(2.)118 4263 y Fv(How)g(many)g(Fib)-5 b(onac)g(ci)31 b(chains)g(ar)-5 b(e)32 b Fq(I)1551 4226 y FD(3)1590 4263 y Fv(-symmetric?)139 b FC(Since)30 b Fy(j)17 b FC(det\()p Fq(M)2828 4226 y FD(3)2884 4263 y Fy(\000)f FC(1)-62 b Fz(1)q FC(\))p Fy(j)27 b FC(=)h(4)h(there)h(are)118 4395 y(4)36 b(c)m(hains)g(left)f (in)m(v)-5 b(arian)m(t)35 b(b)m(y)h(triple)f(in\015ation.)52 b(There)37 b(is)e(only)h(1)f(in\015ation-in)m(v)-5 b(arian)m(t)32 b(c)m(hain,)118 4527 y(so)h(the)g(remaining)d(3)i(form)g(a)g(single)g (orbit)f(of)h(length)g(3.)243 4681 y(W)-8 b(e)39 b(can)g(sa)m(y)h(more) f(ab)s(out)f(what)i(these)g(last)e(4)h(c)m(hains)g(are.)63 b(The)40 b(symmetric)e(in\015ation)118 4813 y(with)h Fr(c)g FC(=)g Fo(0)g FC(acting)f(on)i(a)e(symmetric)h(c)m(hain)g(giv)m (es)h(another)f(symmetric)f(c)m(hain.)64 b(\(This)39 b(is)118 4946 y(clear)44 b(either)g(directly)g(from)f(the)i(strip)f (construction)h(or)f(from)f(the)i(observ)-5 b(ation)43 b(that)i Fq(M)118 5078 y FC(comm)m(utes)d(with)g Fy(\000)p FC(1)-62 b Fz(1)p FC(,)45 b(the)d(in)m(v)m(ersion.\))73 b(So)42 b(the)g(set)h(of)f(4)g(in)m(v)m(ersion)g(symmetric)f(c)m(hains) i(is)118 5211 y(stabilized)29 b(b)m(y)i(this)f(in\015ation.)41 b(In)30 b(view)h(of)e(the)i(foregoing,)e(one)h(m)m(ust)h(b)s(e)f (in\015ation-in)m(v)-5 b(arian)m(t)118 5343 y(and)37 b(the)h(others)f(a)g(3-cycle)g(under)h(in\015ation.)55 b(In)37 b(fact)g Fy(F)2307 5358 y Fw(aa)2422 5343 y FC(is)g(in)m(v)-5 b(arian)m(t)35 b(under)j(symmetric)118 5475 y(in\015ation)31 b(and)i(\()p Fy(F)803 5490 y Fw(b)836 5475 y Fq(;)17 b Fy(F)952 5490 y Fw(a)993 5475 y Fq(;)g Fy(F)1109 5490 y Fw(ab)1181 5475 y FC(\))32 b(is)g(a)g(3-cycle.)243 5608 y(Are)g(there)h(other)g(in\015ations)e(in)h(the)g(family)e(that)i (comm)m(ute)g(with)g(in)m(v)m(ersion?)44 b(If)32 b(so)h(then)118 5740 y Fq(M)10 b FC(\()p Fy(\000)p Fr(t)r FC(\))30 b(+)g Fr(c)48 b FC(=)g Fy(\000)p FC(\()p Fq(M)10 b Fr(t)32 b FC(+)e Fr(c)p FC(\))44 b(for)g(all)f Fr(t)p FC(,)48 b(giving)43 b Fr(c)48 b FC(=)g Fy(\000)p Fr(c)p FC(,)g(so)c Fr(c)h FC(m)m(ust)g(b)s(e)f(one)h(of)f(the)h(2-)1865 5989 y(8)p eop %%Page: 9 9 9 8 bop 118 171 a FC(division)26 b(p)s(oin)m(ts)i(of)f Fx(T)932 134 y FD(2)975 171 y FC(.)42 b(These)30 b(come)d(from)g(the)h (p)s(oin)m(ts)g(of)f Fx(R)2416 134 y FD(2)2489 171 y FC(that)h(are)g(cen)m(tres)i(of)d(in)m(v)m(ersion)118 303 y(symmetry)45 b(of)f(the)g(lattice)f Fx(Z)1267 267 y FD(2)1304 303 y FC(.)79 b(W)-8 b(e)45 b(ha)m(v)m(e)g(already)f(seen)i (that)e Fr(c)k FC(=)g(\()2900 264 y FD(1)p 2900 280 36 4 v 2900 337 a(2)2945 303 y Fq(;)17 b FC(0\))44 b(giv)m(es)h(a)f(lo)s (cal)118 435 y(in\015ation)39 b(that)i(\014xes)h Fy(F)1031 450 y Fw(ab)1103 435 y FC(.)68 b(The)42 b(statistics)f(for)f(all)f (four)i(of)f(these)i(in\015ations)e(are)h(giv)m(en)g(in)118 568 y(T)-8 b(able)46 b(1.)86 b(There,)52 b(w)m(e)47 b(also)f(list)f (the)i(\014xed)h(p)s(oin)m(ts)e Fr(a)h FC(of)f(the)h(in\015ations)f (whic)m(h)h(coincide)118 700 y(with)29 b(the)g(torus)g(parameters)g(of) f(the)i(corresp)s(onding)e(in\015ation)f(in)m(v)-5 b(arian)m(t)28 b(Fib)s(onacci)f(c)m(hains.)118 833 y(F)-8 b(urthermore,)29 b(w)m(e)h(giv)m(e)f(the)g(more)g(sp)s(ecial)f(information)d(of)k(whic)m (h)g(fraction)f(of)h Fq(a)p FC('s)g(in\015ate)g(to)118 965 y Fq(ba)k FC(for)e(the)h(represen)m(tativ)m(e)i(sp)s(eci\014ed)e(b) m(y)h(the)f(distance)g(of)g(the)g(in\015ation)e(\014xed)j(p)s(oin)m(t)e (to)g(the)118 1098 y(origin)g(in)g Fq(E)6 b FC(.)p 193 1336 3395 4 v 191 1584 4 249 v 321 1483 a Fm(c)p 495 1584 V 588 1423 a Fz(In\015.)30 b(\014xed)588 1547 y(p)s(oin)m(t)g Fm(a)p 1057 1584 V 1150 1423 a Fz(Prop)s(ortion)f(of)h Fp(a)p Fz('s)1150 1547 y(that)h(in\015ate)f(to)h Fp(ba)p Fz(.)p 1938 1584 V 2037 1423 a(Shift)2037 1547 y(in)e Fp(E)p 2322 1584 V 2428 1423 a Fz(Action)h(on)g(the)2428 1547 y(symmetric)f(c)m(hains.)p 3243 1584 V 3287 1483 a(Lo)s(cal?)p 3585 1584 V 193 1588 3395 4 v 191 1712 4 125 v 244 1675 a(\(0)p Fp(;)15 b Fz(0\))p 495 1712 V 233 w(\(0)p Fp(;)g Fz(0\))p 1057 1712 V 554 w(1)p Fp(=)p Fz(2)p 1938 1712 V 543 w(0)p 2322 1712 V 211 w(\()p Fl(F)2465 1689 y Fw(aa)2545 1675 y Fz(\))106 b(\()p Fl(F)2786 1690 y Fw(b)2821 1675 y Fp(;)15 b Fl(F)2926 1689 y Fw(a)2968 1675 y Fp(;)g Fl(F)3073 1690 y Fw(ab)3146 1675 y Fz(\))p 3243 1712 V 178 w(No)p 3585 1712 V 191 1836 V 239 1799 a(\()284 1763 y FD(1)p 284 1778 36 4 v 284 1830 a(2)329 1799 y Fp(;)g Fz(0\))p 495 1836 4 125 v 218 w(\()712 1763 y FD(1)p 712 1778 36 4 v 712 1830 a(2)758 1799 y Fp(;)808 1763 y FD(1)p 808 1778 V 808 1830 a(2)853 1799 y Fz(\))p 1057 1836 4 125 v 588 w(0)p 1938 1836 V 460 w(\()p Fp(\034)2066 1766 y FD(2)2106 1799 y Fp(=)p Fz(2\))p Fm(e)p 2322 1836 V 88 w Fz(\()p Fl(F)2469 1814 y Fw(ab)2541 1799 y Fz(\))106 b(\()p Fl(F)2782 1813 y Fw(a)2825 1799 y Fp(;)15 b Fl(F)2930 1814 y Fw(b)2965 1799 y Fp(;)g Fl(F)3070 1813 y Fw(aa)3150 1799 y Fz(\))p 3243 1836 V 163 w(Y)-8 b(es)p 3585 1836 V 191 1960 V 239 1923 a(\(0)p Fp(;)370 1887 y FD(1)p 370 1902 36 4 v 370 1954 a(2)415 1923 y Fz(\))p 495 1960 4 125 v 222 w(\()717 1887 y FD(1)p 717 1902 36 4 v 717 1954 a(2)762 1923 y Fp(;)15 b Fz(0\))p 1057 1960 4 125 v 524 w(1)p Fp(=)p Fz(2)p Fp(\034)p 1938 1960 V 400 w Fz(\()p Fp(\034)2066 1890 y FD(4)2106 1923 y Fp(=)p Fz(2\))p Fm(e)p 2322 1960 V 103 w Fz(\()p Fl(F)2484 1937 y Fw(a)2526 1923 y Fz(\))106 b(\()p Fl(F)2767 1938 y Fw(b)2802 1923 y Fp(;)15 b Fl(F)2907 1938 y Fw(ab)2980 1923 y Fp(;)g Fl(F)3085 1937 y Fw(aa)3165 1923 y Fz(\))p 3243 1960 V 159 w(No)p 3585 1960 V 191 2101 4 141 v 234 2047 a(\()279 2012 y FD(1)p 279 2027 36 4 v 279 2079 a(2)325 2047 y Fp(;)375 2012 y FD(1)p 375 2027 V 375 2079 a(2)420 2047 y Fz(\))p 495 2101 4 141 v 217 w(\(0)p Fp(;)803 2012 y FD(1)p 803 2027 36 4 v 803 2079 a(2)848 2047 y Fz(\))p 1057 2101 4 141 v 545 w Fp(\034)10 b(=)p Fz(2)p 1938 2101 V 413 w(\()p Fp(\034)2066 2014 y FD(3)2106 2047 y Fp(=)p Fz(2\))p Fm(e)p 2322 2101 V 107 w Fz(\()p Fl(F)2488 2062 y Fw(b)2523 2047 y Fz(\))105 b(\()p Fl(F)2763 2061 y Fw(a)2806 2047 y Fp(;)15 b Fl(F)2911 2061 y Fw(aa)2991 2047 y Fp(;)g Fl(F)3096 2062 y Fw(ab)3168 2047 y Fz(\))p 3243 2101 V 156 w(No)p 3585 2101 V 193 2105 3395 4 v 500 2420 a FC(T)-8 b(able)32 b(1:)43 b(In)m(v)m(ersion)34 b(symmetric)e(in\015ations)f(of)h(the)h(Fib)s(onacci)e(LI-class.)243 2668 y(These)44 b(are)e(the)g(only)g(in\015ations)f(in)g(the)h(family)e (that)i(stabilize)e(the)j(set)g(of)e(symmetric)118 2801 y(c)m(hains.)53 b(Only)35 b(the)h(second)h(of)f(them,)g Fr(c)d FC(=)g(\()1830 2761 y FD(1)p 1830 2778 36 4 v 1830 2835 a(2)1875 2801 y Fq(;)17 b FC(0\),)36 b(is)f(lo)s(cal.)50 b(It)36 b(is)f(in)m(teresting)g(to)h(note)g(that)118 2933 y(the)43 b(cub)s(es)g(of)f(these)h(in\015ations)e(are)h(all)e(the) i(same)g(and)h(all)d(equal)i(to)f(the)i(translationless)118 3066 y(in\015ation)30 b(giv)m(en)i(b)m(y)h(the)g(symmetric)e(lo)s(cal)e (rule)j Fq(b)c Fy(7!)f Fq(aba)p FC(,)33 b Fq(a)28 b Fy(7!)f Fq(ababa)p FC(,)34 b(whic)m(h)e(\014xes)i(all)c(four)118 3198 y(symmetric)i(c)m(hains.)243 3330 y(W)-8 b(e)33 b(see)i(here)f(a)f(connection)g(b)s(et)m(w)m(een)i(symmetric)e (patterns)h(and)f(short)h(in\015ation)d(orbits)118 3463 y(that)41 b(will)d(feature)j(in)f(higher-dimensional)d(patterns)k(to)s (o.)67 b(What)41 b(has)g(happ)s(ened)g(is)g(this.)118 3595 y(There)32 b(is)d(a)h(small)e(n)m(um)m(b)s(er)j(of)f(symmetric)f (c)m(hains.)43 b(This)31 b(set)g(of)f(c)m(hains)g(is)g(stabilized)f(b)m (y)i(the)118 3728 y(symmetric)j(in\015ation)g(\(since)h(it)f(comm)m (utes)h(with)g(in)m(v)m(ersion\))g(and)h(hence)g(decomp)s(oses)g(in)m (to)118 3860 y(short)28 b(in\015ation)d(orbits.)41 b(The)28 b(set)g(of)e(symmetric)h(c)m(hains)g(is)g(a)g(group)g(on)g(the)g(torus) h(and)f(hence)118 3992 y(stabilized)33 b(b)m(y)i(the)g(corresp)s (onding)f(translations.)47 b(Consequen)m(tly)36 b(it)d(is)h(stabilized) f(not)h(only)118 4125 y(b)m(y)i(the)g(symmetric)f(in\015ation)e(but)j (also)e(b)m(y)i(the)g(in\015ation)e(got)g(b)m(y)j(com)m(bining)c(it)h (with)h(eac)m(h)118 4257 y(of)d(these)i(translations.)42 b(F)-8 b(ortuitously)g(,)31 b(one)i(of)f(these)i(in\015ations)d(is)h (lo)s(cal.)118 4529 y Fo(Coun)m(ting)37 b(in\015ation)f(orbits)118 4726 y FC(The)g(answ)m(ers)g(to)e(the)h(last)f(three)h(questions)h(ab)s (o)m(v)m(e)f(immediately)c(generalize:)47 b(the)35 b(n)m(um)m(b)s(er) 118 4858 y(of)42 b(Fib)s(onacci)f(c)m(hains)i(with)f Fq(I)1284 4822 y Fw(n)1331 4858 y FC(-symmetry)g(is)h Fy(j)17 b FC(det)o(\()p Fq(M)2259 4822 y Fw(n)2336 4858 y Fy(\000)29 b FC(1)-62 b Fz(1)p FC(\))p Fy(j)p FC(.)74 b(F)-8 b(urthermore,)44 b(there)f(is)g(a)118 4991 y(uniform)k(w)m(a)m (y)i(of)f(computing)f(these)j(determinan)m(ts.)90 b(Since)49 b(the)f(eigen)m(v)-5 b(alues)49 b(of)e Fq(M)59 b FC(are)118 5123 y Fq(\034)54 b FC(and)42 b(its)g(algebraic)e(conjugate)j Fq(\034)1485 5087 y Fu(0)1552 5123 y FC(=)h Fy(\000)p FC(1)p Fq(=\034)54 b FC(and)42 b(the)h(determinan)m(t)f(of)g(a)g (matrix)e(is)i(the)118 5256 y(pro)s(duct)33 b(of)f(its)g(eigen)m(v)-5 b(alues,)33 b(w)m(e)g(ha)m(v)m(e)1114 5488 y(det)q(\()p Fq(M)1392 5446 y Fw(n)1461 5488 y Fy(\000)23 b FC(1)-62 b Fz(1)p FC(\))83 b(=)g(\()p Fq(\034)1964 5446 y Fw(n)2033 5488 y Fy(\000)23 b FC(1\)\(\()p Fq(\034)2349 5446 y Fu(0)2372 5488 y FC(\))2410 5446 y Fw(n)2479 5488 y Fy(\000)g FC(1\))822 b(\(11\))1714 5645 y(=)83 b(N)1946 5660 y FD(2)1985 5645 y FC([)p Fq(\034)2065 5604 y Fw(n)2135 5645 y Fy(\000)22 b FC(1])p Fq(;)1151 b FC(\(12\))1865 5989 y(9)p eop %%Page: 10 10 10 9 bop 118 171 a FC(where,)34 b(for)e Fq(\013)c FC(=)g Fq(r)d FC(+)d Fq(s\034)39 b Fy(2)28 b Fx(Q)11 b FC(\()p Fq(\034)h FC(\),)38 b(N)1498 186 y FD(2)1537 171 y FC([)p Fq(\013)q FC(])32 b(is)g(the)h Fv(norm)f FC(of)h Fq(\013)g FC(de\014ned)h(as)792 370 y(N)865 385 y FD(2)905 370 y FC([)p Fq(\013)q FC(])55 b(=)g Fq(\013)q(\013)1334 329 y Fu(0)1412 370 y FC(=)g(\()p Fq(r)25 b FC(+)d Fq(s\034)11 b FC(\)\()p Fq(r)25 b FC(+)d Fq(s\034)2189 329 y Fu(0)2213 370 y FC(\))55 b(=)h Fq(r)2485 329 y FD(2)2546 370 y FC(+)22 b Fq(r)s(s)g Fy(\000)g Fq(s)2904 329 y FD(2)2960 370 y Fq(:)501 b FC(\(13\))118 570 y(Here,)31 b Fx(Q)12 b FC(\()p Fq(\034)f FC(\))34 b(=)27 b Fy(f)p Fq(a)16 b FC(+)g Fq(b\034)41 b Fy(j)27 b Fq(a;)17 b(b)28 b Fy(2)g Fx(Q)12 b Fy(g)33 b FC(=)28 b Fx(Q)11 b FC(\()1725 487 y Fy(p)p 1814 487 49 4 v 1814 570 a FC(5\))29 b(is)h(the)g(quadratic)f (\014eld)h(of)f(degree)i(2)e(o)m(v)m(er)i(the)118 702 y(rationals)36 b(generated)i(b)m(y)g Fq(\034)11 b FC(.)59 b(Note)37 b(that)g(N)1782 717 y FD(2)1822 702 y FC([)p Fq(\013)q FC(])g(is)g(an)g(in)m(teger)h(if)e Fq(r)m(;)17 b(s)37 b FC(in)g(\(13\))f(are)i(in)m(tegers.)118 834 y(F)-8 b(or)31 b(details)f(on)h(this)g(and)h(ho)m(w)g(to)f(form)m (ulate)f(the)h(cut-and-pro)5 b(ject)32 b(sc)m(heme)h(systematically)118 967 y(in)f(n)m(um)m(b)s(er)h(theoretic)f(terms)1232 931 y FD(3)1304 967 y FC(w)m(e)i(refer)f(to)f([22].)243 1099 y(The)h(n)m(um)m(b)s(er)g Fq(a)849 1114 y Fw(n)929 1099 y FC(of)f(Fib)s(onacci)f(c)m(hains)i(with)f Fq(I)2055 1063 y Fw(n)2102 1099 y FC(-symmetry)g(is)g(th)m(us)i(giv)m(en)e(b)m(y) 645 1299 y Fq(a)696 1314 y Fw(n)771 1299 y FC(=)c Fy(j)p FC(N)975 1314 y FD(2)1015 1299 y FC([)p Fq(\034)1095 1258 y Fw(n)1165 1299 y Fy(\000)22 b FC(1])p Fy(j)27 b FC(=)h Fy(\000)p FC(N)1650 1314 y FD(2)1689 1299 y FC([)p Fq(\034)1769 1258 y Fw(n)1839 1299 y Fy(\000)22 b FC(1])28 b(=)f Fq(\034)2198 1258 y Fw(n)2268 1299 y FC(+)22 b(\()p Fq(\034)2457 1258 y Fu(0)2481 1299 y FC(\))2519 1258 y Fw(n)2588 1299 y Fy(\000)g FC(1)g Fy(\000)h FC(\()p Fy(\000)p FC(1\))3060 1258 y Fw(n)3107 1299 y Fq(;)354 b FC(\(14\))118 1498 y(whic)m(h)30 b(di\013ers)f(from)f Fq(\034)961 1462 y Fw(n)1038 1498 y FC(b)m(y)i(less)g(than)g(2.)42 b(Being)28 b(the)i(ro)s(ots)f(of)g(the)h(equation)f Fq(x)3093 1462 y FD(2)3148 1498 y Fy(\000)16 b Fq(x)g Fy(\000)g FC(1)28 b(=)g(0,)118 1631 y Fq(\034)44 b FC(and)33 b Fq(\034)447 1595 y Fu(0)503 1631 y FC(b)s(oth)f(satisfy)h Fq(x)1093 1595 y Fw(n)1168 1631 y FC(=)28 b Fq(x)1327 1595 y Fw(n)p Fu(\000)p FD(1)1486 1631 y FC(+)22 b Fq(x)1639 1595 y Fw(n)p Fu(\000)p FD(2)1777 1631 y FC(,)33 b(and)f(hence)i Fq(a)2348 1646 y Fw(n)2428 1631 y FC(satis\014es)f(the)g(recurrence) 1243 1830 y Fq(a)1294 1855 y Fw(n)1369 1830 y FC(=)28 b Fq(a)1524 1855 y Fw(n)p Fu(\000)p FD(1)1683 1830 y FC(+)22 b Fq(a)1832 1855 y Fw(n)p Fu(\000)p FD(2)1992 1830 y FC(+)g(1)g Fy(\000)g FC(\()p Fy(\000)p FC(1\))2462 1789 y Fw(n)2509 1830 y Fq(:)952 b FC(\(15\))118 2030 y(This)27 b(allo)m(ws)e(us)i(to)f(compute)g(it)f(v)m(ery)j(quic)m(kly)f (starting)e(from)g(the)i(v)-5 b(alues)26 b Fq(a)2947 2045 y FD(1)3014 2030 y FC(=)i Fq(a)3169 2045 y FD(2)3236 2030 y FC(=)g(1)e(whic)m(h)118 2162 y(w)m(e)39 b(already)f(ha)m(v)m(e.) 61 b(The)38 b(v)-5 b(alues)38 b(up)h(to)e Fq(n)g FC(=)g(15)g(are)h(giv) m(en)g(in)f(T)-8 b(able)38 b(2.)59 b(\(In)39 b(terms)e(of)h(the)118 2295 y(Fib)s(onacci)e(n)m(um)m(b)s(ers)j Fq(f)1016 2310 y Fw(n)1101 2295 y FC(w)m(e)g(ha)m(v)m(e)h Fq(a)1532 2310 y Fw(n)1616 2295 y FC(=)d Fq(f)1777 2310 y Fw(n)p FD(+1)1940 2295 y FC(+)26 b Fq(f)2090 2310 y Fw(n)p Fu(\000)p FD(1)2253 2295 y Fy(\000)g FC(1)g Fy(\000)g FC(\()p Fy(\000)p FC(1\))2736 2258 y Fw(n)2783 2295 y FC(,)39 b(since)g(this)f(sequence) 118 2427 y(satis\014es)33 b(the)g(same)g(recurrence)h(and)f(has)g(the)g (same)f(starting)g(v)-5 b(alues.\))243 2559 y(As)27 b(already)f (glimpsed)f(in)h(the)h(cases)h Fq(n)g FC(=)f(2)f(and)h Fq(n)h FC(=)g(3,)f(the)g(c)m(hains)g(with)f(in\015ation)f(orbits)118 2692 y(of)31 b(length)f Fq(n)i FC(are)f(those)h(in)m(v)-5 b(arian)m(t)29 b(under)j Fq(n)p FC(-fold)e(in\015ation)f(but)j(not)e (under)i Fq(m)p FC(-fold)e(in\015ation)118 2824 y(for)g(an)m(y)h(prop)s (er)g(factor)f Fq(m)g FC(of)g Fq(n)p FC(.)43 b(Let)31 b Fq(b)1600 2839 y Fw(n)1678 2824 y FC(b)s(e)f(the)h(n)m(um)m(b)s(er)g (of)f(c)m(hains)h(in)m(v)-5 b(arian)m(t)29 b(under)i Fq(n)p FC(-fold)118 2957 y(in\015ation)g(but)i(not)f(under)i Fq(m)p FC(-fold)d(in\015ation)g(for)h(an)m(y)h Fq(m)28 b(<)f(n)p FC(.)44 b(Then)1478 3156 y Fq(b)1519 3171 y Fw(n)1622 3156 y FC(=)55 b Fq(a)1804 3171 y Fw(n)1873 3156 y Fy(\000)1981 3061 y Fn(X)1983 3257 y Fk(m