Renormalization and forcing of horseshoe orbits

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Renormalization

and

forcing

of

horseshoe

orbits

Valentín Mendoza

DepartamentodeMatemática,UniversidadeFederaldeViçosa,MG,Brazil

a r t i c l e i n f o a b s t r a c t

Article history:

Received15February2014 Accepted3June2014 Availableonline12June2014 MSC: 37B10 37E30 Keywords: Horseshoeorbits Boylandorder Renormalization

InthispaperwedealwiththeBoylandforcingofhorseshoeorbits.Weprovethat thereexistsasetR ofrenormalizable horseshoeorbitscontainingonly quasi-one-dimensionalorbits, thatis, forthese orbitstheBoylandorder coincideswiththe unimodal order.

1. Introduction

In[2],BoylandintroducedtheforcingrelationbetweenperiodicorbitsofthediskD2.Giventwoperiodic orbitsP andR,wesaythatP forces R,denotedbyP 2R,ifeveryhomeomorphismofD2 containingthe

braidtypeofP mustcontainthebraidtypeofR.Theset ofperiodicorbitsforcedbyP isdenotedbyΣP.

In this paper we are concerned with theforcing of Smale horseshoe periodic orbits. A horseshoe orbitP

is called quasi-one-dimensional ifP forces allorbitR suchthatP 1R,where 1 is theunimodalorder.

In[4],Hallgaveasetofquasi-one-dimensionalhorseshoeorbits,called NBTorbits(Non-Bogus Transition orbits) which are inbijectionwith Q∩ (0,1

2) and havetheproperty thattheirthickintervalmap induced

hasminimalperiodicorbitstructure,thatis,ifP isan NBTorbittheneverybraidtypeofaperiodicorbit of itsthickintervalmap θP isforcedbythebraidtypeof P .

In this paperwe obtainatype oforbits whichare quasi-one-dimensionaltoo althoughtheir associated thickintervalmapsarereducibleinthesenseofThurston[6],thatis,theyareisotopictoreducible homeo-morphismswhichhaveaninvariantsetofnon-homotopicallytrivialdisjointcurves{C1,· · · ,Cn}.Restricted

to thecomponents of D2_{\ {C}

1,· · · ,Cn}, these reducible maps or one of itspower haveminimal periodic

orbitstructure.

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Fig. 1. Dynamics of F .

Theorem1.There existsaset R⊃ NBT ofquasi-one-dimensional horseshoe orbits, thatis, if P ∈ R then ΣP ={R : P 1R}.

Theseorbitsaredeﬁnedusingtherenormalizationoperatorwhichwasintroducedin[3]asthe∗-product. 2. Preliminaries

2.1. Boyland partialorder

LetDn bethepunctureddisk.LetMCG(Dn) bethegroupofisotopyclassesofhomeomorphismsof Dn,

which is called the mapping class group of Dn. Given a homeomorphism f : D2 → D2 of the disk D2

withaperiodicorbitP ,thebraid type ofP ,denotedbybt(P,f ) is deﬁnedasfollows: Takean orientation preservinghomeomorphismh: D2\ P → Dn thenbt(P,f ) istheconjugacyclass[h◦ f ◦ h−1]∈ MCG(Dn)

ofh◦ f ◦ h−1: Dn→ Dn.

LetBT be theunionofalltheperiodicbraidtypesandletbt(f ) bethesetformedbythebraidtypesof theperiodicorbitsoff . Wewillsaythatf : D2_{→ D}2 _{exhibits a}_braid_type_{β if}_there _exists_{an n-periodic}

orbitP forf withβ = bt(P,f ).Nowwecandeﬁnetherelation2onBT.Wesaythatβ1forces β2,denoted

byβ12β2,ifeveryhomeomorphism exhibitingβ1,exhibitsβ2 too.Thenwewillsaythataperiodicorbit P forces anotherperiodicorbitR,denotedbyP 2R,ifbt(P )2bt(R).

In[2],P.Boylandprovedthefollowing theorem.

Theorem2.([2, Theorem9.1]) The relation2 isapartial order. 2.2. Smale horseshoe

TheSmalehorseshoeisamapF : D2_{→ D}2_of_the_disk_which_acts_as_in_{Fig. 1}_._The_set_{Ω =}

j∈ZFj(V0∪ V1) is F -invariant and F|Ω is conjugated to the shift σ on the sequence space of two symbols 0 and 1, Σ2={0,1}Z,where

σ(si)i∈Z= (si+1)i∈Z. (1)

Theconjugacyh: Ω→ Σ2 isdeﬁnedby

h(x) i= 0 if Fi(x)∈ V0 1 if Fi(x)∈ V1 (2)

Tocomparehorseshoeorbitsitisnecessarytodeﬁnetheunimodalorder.ItisatotalorderinΣ+₌_{0,₁_}N

given bythe following rule:Let s= s0s1... and t= t0t1... be sequences inΣ+ such thatsi = ti fori≤ k

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P 1 R if σ (R) 1 cP, ∀n ≥ 1. For every orbit P , there exists a homeomorphism θP thatrealizes the

combinatorics of P . This is obtained fatting the line diagram of P and it is called the tick map induced by P . See[4].

2.3. Renormalizedhorseshoe orbits

Let P and Q be two horseshoe periodicorbits with codes cP = Aan−1, where A = a0a1· · · an−2, and cQ= b0b1· · · bm−2bm−1 and periodsn andm, respectively.

Deﬁnition 3(Renormalization operator).Wewill writeP∗ Q forthenm-periodicorbitwithcode

c_P∗Q=

Ab1Ab2· · · Abm−2Abm−1 if (A) is even Ab1Ab2· · · Abm−2Abm−1 if (A) is odd

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where (A)=n−2_i=0 ai andbi= 1− bi.

The orbit P ∗ Q is called the renormalization of P and Q. If an orbit S satisﬁes S = P ∗ Q for some

P,Q∈ Σ2,itissaidthatS is renormalizable.Alsowewill denote P1∗ P2∗ · · · ∗ Pk =

· · ·(P1∗ P2)∗ P3

· · ·.

Example 4.IfP andQ have codescP = 101 andcQ= 1001 thenP∗ Q has codecP∗Q= 100101101100. 2.4. NBT orbits

There are atype of horseshoe orbitsfor which theBoyland partial order is well-understood.They are constructed inthefollowing way.Given arationalnumberq = m_n ∈ Q := Q∩ (0,1₂),letLq bethestraight

line segmentjoining(0,0) and (n,m) in R2_. _Then_construct_a_ﬁnite _word_c

q= s0s1· · · sn asfollows:

si=

1 if Lq intersects some line y = k, k∈ Z, for x ∈ (i − 1, i + 1)

0 otherwise (4)

It followsthatcq ispalindromicandhastheform:

cq= 10μ1120μ212· · · 120μm−1120μm1. (5)

We willdenote Pq to theperiodicorbitsofperiodn+ 2 whichhavethecodescq10, whenthedistinction is

notimportantandletNBT ={Pq: q∈ Q}. In[4],Hall provedthefollowing result.

Theorem 5.Letq,q∈ Q. Then

(i) Pq isquasi-one-dimensional, thatis, Pq1R =⇒ Pq2R. (ii) q q⇐⇒ (cq01)∞1(cq01)∞⇐⇒ (cq01)∞2(cq01)∞

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Fig. 2. The image θP∗Q(Ci) when P = 1001.

So theorem above says that the Boyland order restricted to the NBT orbits is equal to the unimodal order.

3. Forcingofrenormalizableorbits

ForprovingTheorem 1weshallneedthefollowingresult: Theorem6.LetP = A0

1 andQ be periodicorbits. Then

ΣP∗Q={R : P 2R} ∪ {P ∗ R : Q 2R}. (6)

Toprovetheresultaboveitwillbeneededtwo lemmaswhoseproofsareleft tothereader.

Lemma7.Leti,j∈ {1,· · · ,n− 1} be positiveintegerswithi= j andTM = P∗ Q andTm= σn(TM). Then (a) If(A) is eventhen A0∞1Tm1TM 1A1∞,

(b) If(A) is oddthen A1∞1Tm1TM 1A0∞,

(c) σi(P )1σj(P )⇐⇒ [σi(TM)1σj(TM) and σi(Tm)1σj(Tm)].

Lemma8.LetP = A0

1 andQ be twoperiodicorbits. If i,j ∈ {0,· · · ,m− 1},withi= j,then

σi(Q) >1σj(Q)⇐⇒ σin(P∗ Q) >1σjn(P∗ Q).

Proofof Theorem 6. LetθP∗Q be thethickmap induced byP ∗ Q.First wesee thatthe onlyiterates of P∗ Q satisfyingTm1σi(P ∗ Q)1TM aretheiteratesσin(P ∗ Q),with 0≤ i≤ m− 1;sothere existsa

curveCn−1 containingthese orbitsdisjoint from theothersand bounding aregion Dn−1. ByLemma 7(c)

and noting that σi_(T

m) and σi(TM) have the same initial symbol for i ∈ {1,· · · ,n− 1}, it follows that {θi

P∗Q(Cn−1)}n−1i=1 hasthe samecombinatoricsas P . For i= 0,· · · ,n− 2,letCi = θi+1_P∗Q(Cn−1) beacurve

which bounds a domain Di. It is possible to deﬁne θP∗Q such that θn_P∗Q(Cn−1) = Cn−1. Then the line

diagram of {D0,· · · ,Dn−1} is as the line diagram of P and then θP∗Q has the same behaviour than θP

intheexteriorof∪Di.Since θP∗Q canbereduced byafamilyofcurves, wewill needstudy theThurston

representativeofθP∗QrestrictedtoD2\ ∪Ci.AsθP andθP∗Qhavethesamecombinatoricsintheexterior

of∪Di,theyhavethesameThurstonrepresentativeintheexteriorof∪Di.SoP andP∗ Q forcethesame

periodicorbitsintheexteriorof∪Di.Then{R : P 2R}⊂ ΣP∗Q.SeeFig. 2.

It isclearthatto ﬁndwhatorbitsareforcedbyP∗ Q in∪Di,itis enoughtostudy θPn∗Q restrictedto D_n−1.ByLemma 8, theline diagramofθn

P∗QinsideDn−1 isthesameasthelinediagram ofQ when(A)

iseven,and itisﬂippedwhen(A) is odd.SeeFig. 3.Soθn

P∗Q hasthesamecombinatoricsthanθQ.

As inLemma 8,we canprovethatQ2 R⇐⇒ P ∗ Q2 P∗ R.SoΣP∗Q ={R : P 2 R}∪ {P ∗ R : Q2R}. 2

Remark9.Theorem 6saysusthattolookfortheorbitsthatareforcedbyP∗ Q itisenoughtolookforthe orbitsthatare forcedbyP and theorbitsthatare forcedbyQ. Sowe canstudy thethickmaps induced

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Fig. 3. The image θn

P∗Q(Cn−1).

byP andQ separately.Every ofthesethickmapscanbe reducedusingmethodsto determineitsminimal representative,e.g.[1,5].

Corollary 10.LetP1,P2,· · · ,Pk be NBTorbits. Then (a) ΣP1∗···∗Pk=

k

j=1{P1∗ · · · ∗ Pj−1∗ R : Pj1R},and (b) ΣP1∗···∗Pk={R : P1∗ · · · ∗ Pk1R}.

Proof. Item (a) follows directly from Theorem 6. Foritem (b) it is enoughto provethatif P andQ are

quasi-one-dimensionalhorseshoeorbitsthenP∗ Q isaquasi-one-dimensionalorbittoo. Supposethat(A)

is even.FromTheorem 6,

Σ_P∗Q={R : P 1R} ∪ {P ∗ R : Q 1R}. (7)

HenceitfollowsthatΣ_P∗Q⊂ {S : P ∗ Q1S}.Wehavetoprovetheinclusion{S : P ∗ Q1S}⊂ ΣP∗Q.

If P 1 S then S ∈ ΣP∗Q. Let S withcS = s0s1· · · sk−1 be aperiodicorbit with P 1 S 1 P∗ Q. By

Lemma 7(a),(A0)∞1c∞S 1(A1)∞.ThisimpliesthatcS = Asn−1sn· · · and

(0A)∞1σn−1

c∞_S = sn−1sn· · · 1(1A)∞,

and σn_(c

S∞) 1 (A0)∞. In the other hand σn(cS∞) 1 cP∗Q∞. Then σn(cS∞) = As2n−1· · · and then cS = AsnAs2n−1· · ·. Continuing this process, it follows that S = P ∗ R where cR = sn−1s2n−1· · ·. So P ∗ R 1P∗ Q whichimpliesthatR1Q.SoS∈ {P ∗ R : Q1R} andtheproofisﬁnished. 2

Now weproceedto proveTheorem 1.

Proof of Theorem 1. Let {Pj}j∈N be the set of NBT orbits and consider the space NN of sequences of

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RJ = ∞

k=1

{Pj1∗ · · · ∗ Pjk} (8)

andR=_{J ∈N}NRJ.ByCorollary 10(b),everyorbitofR isquasi-one-dimensional. 2

Acknowledgements

I would like to thanks to Toby Hall and Dylene de Barros for their comments which contributed to improvethispaper.

References

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[3]P.Collet,J.-P.Eckmann,IteratedMapsontheIntervalasDynamicalSystems,Birkhauser,Boston,1980. [4]T.Hall,Thecreationofhorseshoes,Nonlinearity7(1994)861–924.

[5]H.Solari,M.Natiello,Minimalperiodicorbitsstructureof2-dimensionalhomeomorphisms,J.NonlinearSci.15 (3)(2005) 183–222.

[6]W.Thurston,Onthegeometry anddynamicsofdiﬀeomorphisms ofsurface,Bull., NewSer.,Am.Math.Soc.19(1988) 417–431.