%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1609.09452v1 [math.DS] 29 Sep 2016 %%CreationDate: Mon Mar 2 01:58:14 2020 %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z Freeorbits.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2020.03.02:0158 %%BeginProcSet: texc.pro 0 0 %! /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 /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}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 0 0 %! 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b(suc)m(h)g(that)g Fk(\036)1407 2313 y Fl(3)1447 2299 y Fp(\()p Fk(a)p Fp(\)\()p Fk(p)p Fp(\))c Fk(>)f Fp(4)21 b(+)f Fk(\013)31 b Fp(and)g Fk(\036)2283 2313 y Fl(3)2322 2299 y Fp(\()p Fk(b)p Fp(\)\()p Fk(p)p Fp(\))d Fk(>)d Fp(4)c(+)f Fk(\014)5 b Fp(.)43 b(De\014ne)31 b Fk(f)k Fj(2)386 2407 y Fp(Homeo)661 2421 y Fl(+)720 2407 y Fp(\()p Fm(R)s Fp(\))d(satisfying)25 b Fk(f)10 b Fp(\()p Fk(x)p Fp(\))25 b(=)g Fk(\036)1635 2421 y Fl(1)1675 2407 y Fp(\()p Fk(a)p Fp(\)\()p Fk(x)p Fp(\))i(if)e Fk(x)g(<)g Fp(0,)j Fk(f)10 b Fp(\()p Fk(x)p Fp(\))25 b(=)g Fk(\036)2644 2421 y Fl(2)2684 2407 y Fp(\()p Fk(a)p Fp(\)\()p Fk(x)p Fp(\))i(if)e Fk(x)g Fj(2)g Fp([1)p Fk(;)15 b Fp(4])386 2515 y(and)37 b Fk(f)10 b Fp(\()p Fk(x)p Fp(\))37 b(=)g Fk(\036)946 2529 y Fl(3)986 2515 y Fp(\()p Fk(a)p Fp(\)\()p Fk(x)p Fp(\))h(if)f Fk(x)g(>)g(p)p Fp(.)62 b(No)m(w)38 b(w)m(e)g(de\014ne)f Fk(g)k Fj(2)c Fp(Homeo)2767 2529 y Fl(+)2826 2515 y Fp(\()p Fm(R)s Fp(\))44 b(satisfying)386 2623 y Fk(g)s Fp(\()p Fk(x)p Fp(\))26 b(=)f Fk(\036)730 2637 y Fl(2)770 2623 y Fp(\()p Fk(a)p Fp(\)\()p Fk(x)p Fp(\))31 b(if)d Fk(x)e(<)f Fp(0,)30 b Fk(g)s Fp(\()p Fk(x)p Fp(\))c(=)f Fk(\036)1741 2637 y Fl(2)1781 2623 y Fp(\()p Fk(b)p Fp(\)\()p Fk(x)p Fp(\))31 b(if)d Fk(x)e Fj(2)e Fp([1)p Fk(;)15 b Fp(4])32 b(and)d Fk(g)s Fp(\()p Fk(x)p Fp(\))d(=)f Fk(\036)3020 2637 y Fl(3)3060 2623 y Fp(\()p Fk(b)p Fp(\)\()p Fk(x)p Fp(\))31 b(if)386 2730 y Fk(x)25 b(>)g(p)p Fp(.)40 b(Finally)25 b(de\014ne)i Fk(f)37 b Fp(and)27 b Fk(g)k Fp(o)m(v)m(er)e([0)p Fk(;)15 b Fp(1])g Fj([)g Fp([4)p Fk(;)g(p)p Fp(])29 b(so)f(that)g Fk(F)13 b(ix)p Fp(\()p Fk(f)d Fp(\))15 b Fj(\\)g Fk(F)e(ix)p Fp(\()p Fk(g)s Fp(\))26 b(=)f Fj(;)p Fp(.)486 2838 y(Consider)31 b Fk( )13 b Fp(:)32 b Fm(F)1047 2852 y Fl(2)1122 2838 y Fj(!)e Fp(Homeo)1519 2852 y Fl(+)1578 2838 y Fp(\()p Fm(R)s Fp(\))39 b(de\014nes)33 b(as)g Fk( )s Fp(\()p Fk(a)p Fp(\))f(=)e Fk(f)42 b Fp(and)33 b Fk( )s Fp(\()p Fk(b)p Fp(\))f(=)d Fk(g)s Fp(.)51 b(Since)386 2946 y Fk(\036)440 2960 y Fl(3)514 2946 y Fp(has)35 b(a)g(free)g(orbit)f(and)g Fk(\036)1402 2960 y Fl(3)1476 2946 y Fp(has)h(no)f(global)g(\014xed)g (p)s(oin)m(t,)i(for)e(an)m(y)h Fk(g)h Fj(2)d Fm(F)3079 2960 y Fl(2)3148 2946 y Fj(\000)22 b(f)p Fk(e)p Fj(g)386 3054 y Fp(there)41 b(exists)g Fk(x)j Fj(2)f Fm(R)50 b Fp(greater)42 b(than)f Fk(p)g Fp(suc)m(h)g(that)h Fk(\036)2305 3068 y Fl(3)2345 3054 y Fp(\()p Fk(g)s Fp(\)\()p Fk(x)p Fp(\))j Fj(6)p Fp(=)e Fk(x)f Fp(and)e(therefore)386 3162 y Fk( )s Fp(\()p Fk(g)s Fp(\)\()p Fk(x)p Fp(\))28 b Fj(6)p Fp(=)d Fk(x)p Fp(.)40 b(This)29 b(implies)e(that)k Fk( )j Fp(is)c(a)g(faithful)e(action.)486 3270 y(No)m(w,)k(the)f(fact)h(that)f Fk( )k Fp(has)c(no)f(global)h(\014xed)f(p)s(oin)m(ts)g(implies)e(that)k (giv)m(en)f Fk(x)26 b Fj(2)g Fm(R)386 3378 y Fp(there)41 b(exists)g Fk(g)46 b Fj(2)d Fm(F)1133 3392 y Fl(2)1220 3378 y Fp(so)e(that)h Fk( )s Fp(\()p Fk(g)s Fp(\)\()p Fk(x)p Fp(\))j Fk(<)e Fp(0)e(and)f(therefore)i Fk( )s Fp(\()p Fk(g)2818 3345 y Fg(\000)p Fl(1)2913 3378 y Fk(ag)s Fp(\)\()p Fk(x)p Fp(\))j(=)e Fk(x)386 3486 y Fp(whic)m(h)29 b(pro)m(v)m(es)i(that)g Fk( )j Fp(has)c(no)g(free)h(orbit.)486 3594 y(It)25 b(remains)g(to)h(c)m(hec)m(k)i(the)e(minimalit)m(y)d(of)i Fk( )s Fp(.)40 b(Observ)m(e)26 b(that)g(giv)m(en)g(an)m(y)g Fk(x)f Fj(2)g Fp([1)p Fk(;)15 b Fp(2])386 3702 y(it)33 b(is)f(clear)h(that)h(the)g Fk( )i Fp(orbit)d(of)g Fk(x)g Fp(is)g(dense)f(on)i([1)p Fk(;)15 b Fp(2].)51 b(No)m(w,)35 b(since)d Fk( )s Fp(\()p Fk(a)p Fp(\)\([1)p Fk(;)15 b Fp(2]\))26 b Fj(\\)386 3810 y Fp([1)p Fk(;)15 b Fp(2])27 b Fj(6)p Fp(=)e Fj(;)30 b Fp(and)f Fk( )s Fp(\()p Fk(b)p Fp(\)\([1)p Fk(;)15 b Fp(2]\))23 b Fj(\\)18 b Fp([1)p Fk(;)d Fp(2])27 b Fj(6)p Fp(=)e Fj(;)31 b Fp(w)m(e)f(can)g(deduce)f (that)h Fm(F)2689 3824 y Fl(2)2734 3810 y Fk(:)p Fp([1)p Fk(;)15 b Fp(2],)33 b(the)c(union)386 3918 y(of)35 b(the)g Fk( )k Fp(orbits)33 b(of)j(p)s(oin)m(ts)d(in)h([1)p Fk(;)15 b Fp(2],)38 b(is)c(a)h(connected)h(set.)55 b(Also,)36 b(since)f Fk( )j Fp(has)d(no)386 4026 y(global)i(\014xed)f(p)s(oin)m (ts)h Fm(F)1217 4040 y Fl(2)1263 4026 y Fk(:)p Fp([1)p Fk(;)15 b Fp(2])39 b(is)e(un)m(b)s(ounded)d(in)j(b)s(oth)f(directions)g (and)h(therefore)386 4134 y Fm(F)435 4148 y Fl(2)481 4134 y Fk(:)p Fp([1)p Fk(;)15 b Fp(2])27 b(=)e Fm(R)s Fp(.)46 b(Finally)-8 b(,)29 b(an)m(y)i(orbit)f(accum)m(ulates)h(on)f ([1)p Fk(;)15 b Fp(2])32 b(and)e(therefore)g(on)h Fm(R)38 b Fp(as)386 4241 y(claimed.)386 4428 y SDict begin H.S end 386 4428 a 386 4428 a SDict begin 13 H.A end 386 4428 a 386 4428 a SDict begin [/View [/XYZ H.V]/Dest (remark.3) cvn /DEST pdfmark end 386 4428 a Black Fu(Remark)24 b(3.)p Black 34 w Fn(Sinc)-5 b(e)25 b(any)g(action)g(on)g Fm(R)32 b Fn(c)-5 b(an)25 b(b)-5 b(e)25 b(se)-5 b(en)24 b(as)h(an)g(action)g(on)g Fk(S)2902 4395 y Fl(1)2966 4428 y Fp(=)g Fm(R)5 b Fj([)r(f1g)386 4536 y Fn(with)34 b(a)g(glob)-5 b(al)35 b(\014xe)-5 b(d)35 b(p)-5 b(oint,)35 b(this)f(is)g(also)h(an)f(example)h(of)f(how)g(The)-5 b(or)g(em)p 1 0 0 TeXcolorrgb 3001 4536 a SDict begin H.S end 3001 4536 a 1 0 0 TeXcolorrgb Fn(1)p 1 0 0 TeXcolorrgb 3047 4477 a SDict begin H.R end 3047 4477 a 3047 4536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.1) cvn H.B /ANN pdfmark end 3047 4536 a Black 34 w Fn(c)g(an)34 b(fail)386 4644 y(when)f(the)g(action)h(is)e(not)i(minimal.)386 4843 y SDict begin H.S end 386 4843 a 386 4843 a SDict begin 13 H.A end 386 4843 a 386 4843 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 386 4843 a 431 4950 a Fp(4.)47 b Fo(Fur)-6 b(ther)32 b(pr)n(oper)-6 b(ties)32 b(of)i(minimal)g(a)n(ctions)f(without)h(free)f(orbits)386 5112 y SDict begin H.S end 386 5112 a 386 5112 a SDict begin 13 H.A end 386 5112 a 386 5112 a SDict begin [/View [/XYZ H.V]/Dest (remark.4) cvn /DEST pdfmark end 386 5112 a Black Fu(Remark)e(4.)p Black 39 w Fn(If)e Fp(\000)h Fn(is)f(a)h(non-cyclic)g(gr)-5 b(oup)31 b(acting)e(minimal)5 b(ly)31 b(and)g(faithful)5 b(ly)30 b(with-)386 5220 y(out)j(fr)-5 b(e)g(e)33 b(orbits)g(c)-5 b(on)34 b(the)f(cir)-5 b(cle,)32 b(then)h(it)g(is)g(non-ab)-5 b(elian.)486 5407 y Fp(T)d(o)24 b(see)h(this,)g(consider)e(an)i(elemen)m(t)f Fk(f)35 b Fj(2)25 b Fp(\000)p Fj(nf)p Fk(e)p Fj(g)h Fp(suc)m(h)e(that)h(Fix)o (\()p Fk(f)10 b Fp(\))24 b(is)g(non-empt)m(y)-8 b(.)386 5515 y(If)29 b(\000)f(w)m(ere)i(ab)s(elian,)e(the)h(set)h(Fix\()p Fk(f)10 b Fp(\))29 b(w)m(ould)e(b)s(e)i(in)m(v)-5 b(arian)m(t)28 b(b)m(y)h(all)f(elemen)m(ts)i(of)f(\000,)g(so)386 5623 y(the)h(action)h(w)m(ould)e(not)i(b)s(e)f(minimal.)p Black Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 378 185 a SDict begin H.S end 378 185 a 378 185 a SDict begin H.R end 378 185 a 378 185 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 378 185 a Black Black 386 259 a Fl(6)743 b(J.)24 b(BR)n(UM,)e(M.)h(MAR)-6 b(T)1822 243 y(\023)1827 259 y(INEZ,)23 b(AND)g(R.)g(POTRIE)p Black 386 459 a SDict begin H.S end 386 459 a 386 459 a SDict begin 13 H.A end 386 459 a 386 459 a SDict begin [/View [/XYZ H.V]/Dest (prop.1) cvn /DEST pdfmark end 386 459 a Black Fu(Prop)s(osition)43 b(1.)p Black 45 w Fn(If)38 b Fp(\000)g Fn(is)h(a)f(c)-5 b(ountable)39 b(gr)-5 b(oup)40 b(acting)e(minimal)5 b(ly)40 b(and)f(faithful)5 b(ly)386 567 y(without)34 b(fr)-5 b(e)g(e)33 b(orbits)h(on)f(the)h(cir) -5 b(cle,)33 b(then)g(it)g(c)-5 b(ontains)34 b(a)f(fr)-5 b(e)g(e)34 b(gr)-5 b(oup)34 b(in)e(two)i(gener-)386 675 y(ators.)p Black 386 839 a(Pr)-5 b(o)g(of.)p Black 43 w Fp(A)31 b(result)e(conjectured)i(b)m(y)g(Gh)m(ys)g(and)f(later)h(pro) m(v)m(ed)g(b)m(y)g(Margulis)e(\(see)j([)p 0 1 0 TeXcolorrgb 3266 839 a SDict begin H.S end 3266 839 a 0 1 0 TeXcolorrgb Fp(M)p 0 1 0 TeXcolorrgb 3349 777 a SDict begin H.R end 3349 777 a 3349 839 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Margulis) cvn H.B /ANN pdfmark end 3349 839 a Black Fp(])386 947 y(or)h([)p 0 1 0 TeXcolorrgb 525 947 a SDict begin H.S end 525 947 a 0 1 0 TeXcolorrgb Fp(N)p 0 1 0 TeXcolorrgb 593 884 a SDict begin H.R end 593 884 a 593 947 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Navas) cvn H.B /ANN pdfmark end 593 947 a Black Fp(]\),)i(states)f(that)g (an)m(y)f(group)f(of)h(circle)g(homeomorphisms)e(either)h(preserv)m(es) h(a)386 1055 y(probabilit)m(y)d(measure)j(on)f Fk(S)1399 1022 y Fl(1)1472 1055 y Fp(or)g(con)m(tains)h(a)g(free)g(group)f(in)g (t)m(w)m(o)i(generators.)49 b(If)32 b(\000)386 1162 y(acts)f(without)f (free)g(orbits,)g(it)g(m)m(ust)g(b)s(e)g(non-ab)s(elian.)486 1270 y(Supp)s(ose)35 b(there)j(is)f(a)h(\000-in)m(v)-5 b(arian)m(t)37 b(probabilit)m(y)f(measure)h Fk(\026)p Fp(.)63 b(Since)36 b(the)i(action)386 1378 y(is)i(minimal,)i(it)f(m)m (ust)g(ha)m(v)m(e)i(full)c(supp)s(ort)g(and)i(no)g(atoms.)75 b(There)41 b(is)f(an)h(home-)386 1486 y(omorphism)g(sending)g Fk(\026)i Fp(to)g(the)g(Leb)s(esgue)g(measure.)78 b(This)41 b(means)i(\000)g(m)m(ust)f(b)s(e)386 1594 y(conjugated)h(to)h(a)f (group)f(of)g(rotations,)47 b(and)42 b(therefore)h(ab)s(elian,)h(whic)m (h)d(giv)m(es)i(a)386 1702 y(con)m(tradiction.)2370 b Ff(\003)386 1866 y SDict begin H.S end 386 1866 a 386 1866 a SDict begin 13 H.A end 386 1866 a 386 1866 a SDict begin [/View [/XYZ H.V]/Dest (prop.2) cvn /DEST pdfmark end 386 1866 a Black Fu(Prop)s(osition)43 b(2.)p Black 45 w Fn(If)38 b Fp(\000)g Fn(is)h(a)f(c)-5 b(ountable)39 b(gr)-5 b(oup)40 b(acting)e(minimal)5 b(ly)40 b(and)f(faithful)5 b(ly)386 1974 y(without)32 b(fr)-5 b(e)g(e)31 b(orbits)g(on)g(the)g (cir)-5 b(cle,)31 b(then)g(it)g(c)-5 b(ontains)32 b(fr)-5 b(e)g(e)31 b(ab)-5 b(elian)31 b(gr)-5 b(oups)32 b(of)f(arbi-)386 2082 y(tr)-5 b(arily)34 b(lar)-5 b(ge)34 b(r)-5 b(ank.)p Black 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