Denominator bounds in Thompson-like groups and flows

0.12 MB·English

by  Danny Calegari

DENOMINATOR BOUNDS IN THOMPSON-LIKE GROUPS AND FLOWS
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0 DANNYCALEGARI
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ABSTRACT. LetT denoteThompson’sgroupofpiecewise2-adiclinearhomeo-
n
morphismsofthecircle. GhysandSergiescushowedthattherotationnumberof
a everyelementofTisrational,buttheirproofisveryindirect.Wegivehereashort,
J directproofusingtraintracks,whichgeneralizestoelementsofPL+(S1)withra-
5 tionalbreakpointsandderivativeswhicharepowersofsomefixedinteger,and
2 alsotocertainflowsonsurfaceswhichwecallThompson-like. Wealsoobtainan
explicitupperboundonthesmallestperiodofafixedpointintermsofdatawhich
] canbereadofffromthecombinatoricsofthehomeomorphism.
S
D
.
h
t 1. INTRODUCTION
a
m
In [3], Ghys and Sergiescustudied Thompson’s group T of homeomorphisms
[ of the circle froma number of points of view. This group wasintroduced in un-
published notes by Thompson, and is defined as the subgroup of Homeo+(S1)
3
v consistingofhomeomorphismstakingdyadicrationalstodyadicrationalswhich
3 arepiecewiselinear,wherethebreakpointsaredyadicrationals(i.e. numbersof
7 theformp2qforp,q∈Z),andwherethederivativesarealloftheform2qforq ∈Z.
5
One of the main theorems in [3] is that the rotation number of every element
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of T is rational. RecallPoincare´’s definition [8] of rotationnumber for anelement
0
6 g ∈Homeo+(S1). Letg˜beanyliftofgtoHomeo+(R),anddefine
0 g˜n(0)
/ rot(g˜)= lim
h n→∞ n
at Thenrot(g)=rot(g˜) (mod Z);i.e. rot(g)∈S1. Differentliftsg˜1,g˜2ofgsatisfy
m
rot(g˜1)−rot(g˜2)∈Z
:
v sothisiswell-defined.
i
X In[2],Ghys says“theproof(ofrationality)isveryindirectandthereisaneed
for a better proof”. One such argument was given by I. Liousse [6],[7]. We also
r
a recentlylearnedthatV.Kleptsynhasanapproachtounderstandingrationalityin
Thompson’s group using automata, which is distinct from, but not unrelated to,
theapproachinthispaper;see[5].
The argumentof Ghys–Sergiescuis a proof by contradiction: they show there
isamorphismρ : T →Diffeo∞(S1)whichissemi-conjugatetothenatural(topo-
logical) action, and which has an exceptional minimal set. Rotation number is
invariantunder semi-conjugacy. Thereforethe existence of anelementof T with
irrationalrotationnumberwouldcontradictDenjoy’stheorem(thateveryC2 dif-
feomorphismofS1withanirrationalrotationnumberhasdenseorbits). Liousse’s
proof is more straightforward, but is still nonconstructive, and is still a proof by
contradiction(toDenjoy’sinequalities).
Date:1/25/2007,Version0.08.
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Itiswell-knownandeasytoshow(seee.g.[8])thatanelementg ∈Homeo+(S1)
hasaperiodicpointof(least)periodq ifandonlyifitsrotationnumberisp/qfor
somecoprimepairofintegersp,q. Inthisnote,wesupplyadirectproofofratio-
nality of rotation number for certain PL homeomorphisms by directly finding a
periodicpoint. Infact,ourargumentappliesmoregenerallythantheargumentof
[3],thoughnotmoregenerallythantheargumentof[7]. Ontheotherhand,since
it is constructive, we obtain explicit bounds on the denominator of the rotation
numberintermsofthecombinatoricsoftheoriginalmap,whicharenotobtained
ineither[3]or[7].
Themainrationalitytheorem,provedin§2,isasfollows:
TheoremA. LettbeanelementofPL+(S1)mappingrationalstorationals,withrational
break points, and with derivatives of the form nq for q ∈ Z and some fixed n. Then the
rotationnumberoftisrational.
This theorem is also proved in [7]. We remark that some hypothesis on t is
necessary, since thereareexamplesof elements of PL+(S1) mapping rationalsto
rationals,withrationalbreakpointsandrationalderivatives,whoserotationnum-
bers are irrational. However, it is possible that the conditions on t could still be
relaxedfurther(c.f.Question4.6). See[7]andalso[4]or[1].
Givent ∈ PL+(S1)asinthestatementofthetheorem,wedefinetheheightoft
asfollows. LetmbetheleastintegersuchthatasubdivisionofS1intomintervals
of the form p,p+1 is Markov for t; i.e. t either takes a strip of nk consecutive
m m
intervalsline(cid:2)arlytoa(cid:3) singleintervalbycontraction,ortakesasingleintervaland
stretchesitlinearlyovernkconsecutiveintervalsbyexpansion,forvariousk.Then
defineheight(t)=m.
Remark 1.1. Note that m as we have defined it is bounded by the least common
multipleofthedenominatorsofthebreakpointsoftandtheirimages(assuming
thereisatleastonebreakpoint). Thisremarkjustifiestheexistenceofmandgives
adirectestimateforitssize.
In § 3 we obtain a straightforward bound on denominator of rotationnumber
asfollows:
TheoremA’. LettbeanelementofPL+(S1)mappingrationalstorationals,withratio-
nalbreakpoints,andwithderivativesoftheformnq forq ∈Zandsomefixedn. Suppose
wehave
height(t)=m
Thenthasaperiodicpointofperiodatmostnm·m.
Ofcourse,TheoremA’impliesTheoremA.
Finally,in§4weconstructsomeexampleswithperiodicpointswithlongperi-
ods,complementingtheestimateinTheoremA’.
1.1. Acknowledgements. While writingthispaper,Iwaspartiallysupportedby
aSloanResearchFellowship, andNSFgrantDMS-0405491. I’mgratefulforcom-
ments fromE´tienne Ghys and Collin Bleak, and for some substantial corrections
bytheanonymousreferee.IamalsogratefultoIsabelleLiousseforhercomments
DENOMINATORBOUNDSINTHOMPSON-LIKEGROUPSANDFLOWS 3
on an earlyversion of this paper, and to her and Victor Kleptsyn for forwarding
metheirrelevantpreprints.
2. TRAIN TRACKS
ToproveTheoremA,itsufficestoshowthattasinthestatementofthetheorem
hasaperiodicorbit.
Throughout this section, for concreteness and easeof exposition, we will con-
centrate on the case n = 2. This includes (but is more general than) the case of
the classical Thompson group, although our argument goes through essentially
verbatimforgeneraln.
We fix t as in the statement of the theorem with rational break points and all
derivativespowersof2. Hereisasummaryoftheproof. Weassociatetotatrain
trackwhichcarriesitsdynamics;byanalyzingthecombinatoricsofthetraintrack
weshowthatwecaneithersplitoffacircleinwhichcasesomepoweroftisequal
to the identity on some segment, or else we can find an attracting cycle in which
case some (possibly negative) power of t has a periodic orbit which is attracting
onatleastoneside. Thiswillcompletetheproof.
Train tracks are introduced in [9] as a combinatorial tool for studying one di-
mensionaldynamicsonsurfaces(similarobjectswereintroducedearlierbyDehn
andNielsen). Atraintrackτ isagraphwithaC1 combingateveryvertexwhich
comes with an embedding in a surface. The vertices of τ are called the switches.
Wenowshowhowtoassociateatraintrackτ inatorustoourelementt.
By the defining properties of t, there is a least integer m such that there is a
Markovpartitionfortofthesimpleform
S1 =I1∪I2∪···∪Im
whereeachI haslength 1/m. The elementt actsintwo ways: by takinga strip
i
of 2k consecutive intervals and mapping them linearly to a single interval (con-
traction), orbytakingasingle intervalandstretchingitoutlinearlyover2k con-
secutive intervals(expansion), for variouspositive integers k. Recall thatwe are
callingmtheheightoft.
Themappingtorusoftisliterallya(twodimensional) toruswhichwedenote
by F. In F we construct an oriented train track τ by gluing intervals. We take
one oriented interval e for each I , and we glue the intervals together at their
i i
endpoints in a pattern determined by the dynamics of t. As a convention, we
thinkofthecircleS1 asbeingembedded“horizontally”inF,andtheedgese as
i
beingembedded“vertically”.Ift(I )∩I hasnonemptyinterior,thenweidentify
i j
thepositive endpointof e tothe negativeendpointofe . The orientationonthe
i j
e determines the combing at the vertices. In this way we obtain an orientable
i
traintrackτ, which comeswith anaturalembeddinginF. We emphasizethata
singleedgeofτ mightconsistofmanyintervals. Wedonotneedtokeeptrackof
intervalsinthissection,buttheywillbeimportantin§3whenwetrytoestimate
periodsofperiodicpoints.
Ateachswitchofτ weeitherhave2kincomingedgesandoneoutgoingedgefor
eachcontraction,oroneincomingedgeand2koutgoingedgesforeachexpansion.
4 DANNYCALEGARI
Notice that k may vary from switch to switch. We remedy this in the following
way:aswitchwith2k+1incidentedgescanbesplitopenlocallyto2k−1switches,
eachwith3incidentedges.Wesaythatweareresolvingthe(highvalence)switches
bythisprocess;seeFig.1foranexample.
FIGURE 1. Splitopena5-valentswitchtothree3-valentswitches
Werefertotheresolvedtraintrackasτ0.Notethateveryswitchofτ0is3-valent.
Weremarkthatτ0couldwellbedisconnected(forthatmatter,τ itselfcouldbedis-
connected). Notice that τ0 is contained in F in such a way that the edges are all
transversetoafoliationofF bymeridians,sothatanorientededgeofτ0pointsin
awell-defineddirectionaroundF. Noticetoothatthe(unparameterized)dynam-
ics of t can be recovered completely from the combinatorics of τ0: we associate
to each edge a Euclidean rectangle of width 1 foliated by vertical lines. At each
switchweattachthe(foliated)mappingcylinderofthelinearmap[0,1]→[0,2]to
thehorizontalboundariesoftheedges. Thisgivesafoliatedsurfacewithbound-
arywhichcomeswithanembeddinginF,andwhichcanalsobearrangedsothat
leavesaretransversetothefoliationofF bymeridians;bycollapsingcomplemen-
taryregionswegetpreciselythe(foliated)mappingtorusofthehomeomorphism
t;callthisoperationtherealizationofthecombinatorialtraintrackτ0.
Now, ateachswitchv thereisawelldefinedcontractingdirectionwhichpoints
alongthe edge which isisolated onits side of v. Lete(v)be this (directed)edge,
andleta(v)betheotherendpointofe(v). Soe(v)pointsfromv toa(v). Suppose
thereissomevsuchthate(a(v))isthesameunderlyingedgease(v),butwiththe
opposite orientation. In this case, if e denotes the underlying (undirected)edge,
we calle a sink. The key point is thatsinks canbe split open togive a new train
trackwhoserealizationstillrecoversthedynamicsoft. Thereisonlyone(obvious)
waytodothis;seeFig.2foranillustration.
FIGURE 2. Sinkscanbesplitopentogiveanewtraintrackwith
thesamerealization
Letτibetheresultofsplittingτi−1openalongasink,ifoneexists.Wesplitopen
allsinksuntiltherearenoneleft. Eachsplittingreducesthenumberofverticesby
two,sothisprocessmusteventuallyterminate. Wedenotethesinklesstraintrack
′ ′
weultimatelyobtainbyτ .Notethattherealizationofeveryτ ,andthereforeofτ ,
i
DENOMINATORBOUNDSINTHOMPSON-LIKEGROUPSANDFLOWS 5
stillrecoversthedynamicsoft. Afterthesplitting,somecomponentofτ′mightbe
acirclewithnoswitches;insuchacasewesaywehavesplitoffacircle. Evidently
therealizationofacircleisfoliatedbyperiodicorbits. Thishappensexactlywhen
somepoweroftfixesanonemptyintervalinS1.
′
Otherwise,thereisstillsomeswitchv. Sinceτ hasonlyfinitelymanyswitches,
the sequence v,a(v),a2(v)... is eventually periodic. Since τ′ is sinkless, the ori-
ented edges e(ai(v)) all point in the same direction around F. It follows that if
v,a(v),...,an(v)=visaperiodicsequence,theunion
γ(v):=e(v)∪e(a(v))∪···∪e(an−1(v))
isanembeddedcircle in τ′, oriented coherentlyby the orientation on eachedge,
andalwayspointinginthesamedirectionaroundF.
Wecallγ(v)anattractingcycle. Noticethatallthewayaroundtherealizationof
anattractingcycle,thelinearmapattheswitchesiscontracting,andthereforethe
realizationofanattractingcyclecontainsaperiodicorbitwhichiscontractingonat
leastoneside.Dependingontheorientationofγ(v),thiscorrespondstoaperiodic
orbitwhichisattractingonatleastonesideeitherfortorfort−1. Itfollowsthatt
hasaperiodicorbit,andthereforetherotationnumberrot(t)isrational.
If we replace 2 by n above, then switches in τ are nk + 1 valent, and each
such switch can be resolved to a union of nk−1 switches, each of valence n+1,
n−1
toproduceτ0. Therealizationofτ0 isobtainedbygluingfoliatedEuclideanrect-
angles associated to each edge by attaching the mapping cylinder of the linear
map[0,1]→[0,n]ateachswitch. Theonlypointthatneedsstressingisthatsinks
of τi−1 can still be split open to produce τi, since the two attaching maps in the
realizationassociatedtotheendpointsofasinkarestillinversetoeachother. This
provesTheoremAingeneral.
Remark 2.1. If a general element t ∈ PL+(S1) has derivatives and break points
which are rational, we can still associate a train track to t where at every switch
therearepincomingand1outgoingedge,andtherealizationisobtainedbygluing
the mapping cylinder of the linear map [0,1] → [0,p], for varying p. Note that
multiplicationbyp/qcanberealizedasthecompositionofmultiplicationbyp,and
division byq. The problemis thatone might havea sink where the valencesare
different at the two endpoints. There is no obvious combinatorical simplification
whichcanbedoneinsuchacase,evenwhenthederivativesarealloftheformrk
forsomefixedr,wherer =p/qisrationalbutnotintegral(seeQuestion4.6).
Remark2.2. LetSbeaclosedorientablesurface,andletτ ⊂Sbeanorientabletrain
trackwitheveryswitchofvalence3.Therealizationofτ givesapossiblysingular
foliationonSwithorientableleaves,whichdefinesan(unparameterized)flowon
SthatwesayisThompson-like.Moregenerally,wecallsuchaflowThompson-like
ifeveryswitchhasvalencen+1withnedgesononeside,forsomefixedn. The
argumentinthissectionshowsthateveryorbitinaThompson-like flowiseither
periodic,oraccumulatesonaperiodicorbit.
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3. BOUNDING DENOMINATORS
To prove the stronger Theorem A’ we must analyze the complexity of a split
off circle or an attracting cycle. To do this, we must relate the complexity of the
sinklesstraintrackobtainedbytheargumentof§2totheoriginaltraintrack.
Let τ denote the original train track. We distinguish between edges of τ and
intervals which correspond to the original e , and which each wrap exactly once
i
aroundthemappingtorus. Bydefinition, anedgeofthetraintrackhasbothver-
ticesatswitches;asingleedgemaybecomposedofmanyintervals. Theperiodof
theperiodiccyclewefinallyidentifywillbeequaltothenumberofintervalsthat
itcontains.
Thetraintrackτ hasexactlymintervals,andhasswitchesofvalencenk+1for
variousk.
Weresolveaswitchofvalencenk+1to nk−1 switchesofvalencen+1,thereby
n−1
creating nk−n newedges. Werefertothesenewedgesasinfinitesimaledges;since
n−1
theresolutionisperformedlocally,weassumetheinfinitesimaledgesareasshort
aswe like, and no consecutive sequence of them islong enough to wraparound
themappingtorus. Sowecanstillmeasuretheperiodofaperiodiccyclebycount-
ingthenumberofintervalsitcontains,andignoringtheinfinitesimaledges. Ob-
servethatafterthisresolutionweobtainatraintrackwhichwecallτ0 withevery
switchofvalencen,inwhichthereareatmost2mswitches, andexactlyminter-
vals.
Splitting open a sink produces a new train track with n−1 new edges and 2
fewerswitches. Eachedgeisaconcatenationofintervalsandinfinitesimaledges,
eachofwhichisreplacedbyn−1parallelcopiesaftersplittingopen. Ifweletτ be
i
obtainedfromτi−1 bysplittingopenasinglesink,theniftherearemi−1 intervals
inτi−1,theedgewhichissplitopencontainsatmostmi−1intervals,andtherewill
beatmostn·mi−1 intervalsinτi. Since m0 = m, andwe split openatotalof at
mostmsinks,itfollowsthatifτ′istheultimatesinklesstraintrack,thenumberof
intervalsinτ′ isatmostnm·m.
Since τ′ is sinkless, it contains an embeddedcircle or attracting cycle, and we
aredone.
4. EXAMPLES
In this section we give some examples. Note thatthe argumentsin § 3 do not
usetheembeddingofτ inatorusF. Someoftheexamplesinthissectioncanbe
realizedinatorus,andsomecannot.
Example4.1. Themaptmightbearotationoforderm. Thetraintrackτ isacircle
madeupofmintervals.
Example 4.2. Fix some k. Let m = 2k + s + 2 and define t to be linear on the
followingintervals:
0 2k 2k 2k+1
t: , → ,
(cid:20)m m(cid:21) (cid:20)m m (cid:21)
2k 2k+s 2k+1 2k+s+1
t: , → ,
(cid:20)m m (cid:21) (cid:20) m m (cid:21)
DENOMINATORBOUNDSINTHOMPSON-LIKEGROUPSANDFLOWS 7
2k+s 2k+s+1 2k+s+1 2k+s+1+2k −1 2k−1
t: , → , = ,
(cid:20) m m (cid:21) (cid:20) m m (cid:21) (cid:20) m m (cid:21)
−1 0 2k−1 2k
t: , → ,
(cid:20) m m(cid:21) (cid:20) m m(cid:21)
Theassociated traintrackhastwo switchesof valence2k +1 boundinga sinkof
lengths+1(exceptwhens = 0,k = 1). Thetwobushysidesoftheswitchesare
gluedtogether with a“twist”. Whenτ iscompletelysplit open, the resultτ′ isa
singlecirclecontaining2k+2k·(s+1)+1intervals,sotheperiodis2k·(s+2)+1.
Ifwechooses = 2k−2,thentisactuallycontainedinThompson’sgroupT,and
theorderoftheperiodicpointisO(m2).
Example4.3. We now describea Thompson-like flow with verylongperiodic or-
bits.
We define a traintrack τ depending on three
s+ parameters.Atypicalexampleisillustratedin
a3 thefigure.Thereisamiddleedgeecontaining
a2 r1 intervals. There are a nested sequence of
a1 r2 switches on either side of e, each 3-valent.
Thea onthetopleftaregluedtothea onthe
i i
bottom right of the figure. Finally, there are
two“bushy”switchess±eachofwhichisr3+
e 1valent.Theextremepointsoftheseswitches
aregluedwithatwist. Notem=r1+3r2+r3.
Each time we split open a sink, the length of
a1 the innermost edge doubles. We do this r2
times,givinganedgeoflengthr1·2r2. When
a2 wesplitopens±,theresultisasinglecircleof
s− a3 lengthr1r3·2r2. Ifwesetr2 = O(m)thenthe
lengthoftheperiodicorbitisatleastexponen-
tialinm.
The train track in Example 4.3 can be embedded in a surface of genus O(m).
ItsexistencemeansthatonecannotimprovetheboundinTheoremA’verymuch
without using more detailedinformation aboutthe embeddingof the traintrack
inatorus.
Thissuggestssomeobviousquestions:
Question4.4. Isthereapolynomialbound(inm)forthedenominatorofrotationnumber
foranelementina(generalized)Thompson’scirclegroup?Whataboutaquadraticbound?
Question4.5. HowdoesthelengthofthesmallestperiodicorbitinaThompson-likeflow
dependongenus?
Finally, the following question seemstobe open, and hardto addressdirectly
withourmethods:
Question 4.6. Let r = p/q be a non-integral rational number. Let t be an element of
PL+(S1)mappingrationalstorationals,withrationalbreakpoints,andwithderivatives
oftheformrs fors∈Z. Istherotationnumberoftrational?
8 DANNYCALEGARI
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