Volume 2010, Article ID 323069,31pages doi:10.1155/2010/323069
Research Article
A Poincar ´e Formula for the Fixed Point Indices of the Iterates of Arbitrary Planar Homeomorphisms
Francisco R. Ruiz del Portal
1and Jos ´e M. Salazar
21Departamento de Geometr´ıa y Topolog´ıa, Facultad de CC.Matem´aticas, Universidad Complutense de Madrid, Madrid 28040, Spain
2Departamento de Matem´aticas, Universidad de Alcal´a, Alcal´a de Henares, Madrid 28871, Spain
Correspondence should be addressed to Francisco R. Ruiz del Portal,r portal@mat.ucm.es Received 11 November 2009; Accepted 1 March 2010
Academic Editor: Marl`ene Frigon
Copyrightq2010 F. R. Ruiz del Portal and J. M. Salazar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetU ⊂ R2 be an open subset andf : U → R2 be an arbitrary local homeomorphism with Fixf {p}. We compute the fixed point indices of the iterates offatp, iR2fk, p, and we identify these indices in dynamical terms. Therefore, we obtain a sort of Poincar´e index formula without differentiability assumptions. Our techniques apply equally to both orientation preserving and orientation reversing homeomorphisms. We present some new results, especially in the orientation reversing case.
1. Introduction
There is abundant literature about the fixed point index of a homeomorphism f, in a neighborhood of an isolated fixed point and the local dynamical behavior of f. There are results in both directions, that is, boundsor explicit computationfor the fixed point index from dynamical properties off and conversely how the knowledge of the fixed point index is used to describe the dynamics locally.
One can notice that due to the systematic use of Brouwer’s translation arcs theorem see 1 or 2, most of the known results are limited to orientation preserving homeomorphisms.
It is well known that the classical Poincar´e index formula relates the index of a planar vector field with the elliptic and hyperbolic regions in a neighborhood of a critical point.
Such a formula, for the iterates of an arbitrary homeomorphism, will give a geometric interpretation of the fixed point indices of the iterates, it could help to attack some open problems and it will provide simple proofs of many of the strongest theorems in the subject.
This is the main goal of this article.
The Ulam’s problem about the existence of minimal homeomorphisms in the multipunctured plane was solved completely in the negative by Le Calvez and Yoccoz in 3. The main technique in the proof of their theorem is the computation of the fixed point index of all iterates of an orientation preserving homeomorphism in a neighborhood of a fixed pointpwhich is an isolated invariant set, neither an attractor nor a repeller. Given an orientation preserving local homeomorphismf : U ⊂ R2 → R2, they carry out a detailed local study, near the fixed pointp. Then they prove the existence of integersr, q≥1 such that
iR2 fk, p
⎧⎨
⎩
1−rq if k∈rN,
1 if k /∈rN. 1.1
The authors, in4, using Conley index ideas, gave, in a quite simple way, a general theorem extending the above result to arbitrary local homeomorphisms. In particular, iff reverses the orientation, there are integersδ∈ {0,1,2}andqsuch that
iR2 fk, p
⎧⎨
⎩
1−δ ifk odd,
1−δ−2q ifk even. 1.2
Later, Le Calvez extended his theorem with Yoccoz to arbitrary isolated fixed points of orientation preserving planar homeomorphisms. Again the fixed point indices of the iterations of the homeomorphism have periodical behavior. Le Calvez, in5, uses in a very clever way the nice Carath´eodory’s prime ends theorysee6,7. The idea of applying the compactification of Carath´eodory to study planar dynamical problems is not new. It was introduced by P´erez-Marco in8and it was used more recently by the first author, in9, to prove that the index of arbitrary stable planar fixed points is equal to 1.
On the other hand, Baldwin and Slaminka, in10, dealt with the problem of relating the fixed point index of an orientation and area preserving homeomorphism around an isolated fixed pointpand the number of branches in which the stable/unstable “manifold”
ofpdecomposes. The results of Baldwin and Slaminka were improved by Le Roux, in11, where the fixed point index is used not only to detect stable/unstable branches but also Leau- Fatou petals aroundp. The authors, in12, gave a stable/unstable “manifold” theorem for arbitrary planar homeomorphisms near a fixed point admitting nice filtration pairs.
There are some papers dedicated to the study of the analogous problem in dimension 3. See13–16and its references.
The computation of the fixed point index of any iteration of any planar homeomor- phism at an isolated fixed point laying in an isolated invariant compactum was done by the authors in4,12. As we said above, whenpdoes not belong to any isolated invariant compactum and the homeomorphism is orientation preserving, Le Calvez improved a result of Brown, see17, showing that the sequence of indices is periodic. We will find with our methods the same formula for orientation preserving homeomorphisms and we shall solve the problem also for orientation reversing homeomorphisms. The main fact to obtain our results is the existence of special classes of filtration pairs in the Carath´eodory’s prime ends compactification that will allow us to by-pass the technical problem that occurs if the fixed point does not lay in an isolated invariant compactum.
Roughly speaking, if a fixed pointpdoes not lay in arbitrary small isolated compacta, we can consider any discJcontainingpin its interior and takeKp, the component containing
pof the maximal invariant set contained inJ. By using the Carath´eodory’s compactification ofS2\Kp, we work in a disc and we can compute the index atpfrom the local indicesin semidiscs of the fixed prime ends that now will admit isolating blocks. The existence of such isolating blocks around the fixed prime ends not only provides a simple technique to compute the index of the iterations of arbitrary homeomorphisms but also allows to identify such indices in a geometrical way. Given a discJthe existence of isolating blocks, around the fixed points that appear in the compactification, allows to find dynamical objectsgeneralized stable/unstable branches and generalized attracting/repelling petals whose definitions we will precise laterwhich are the keys for the computations of the indices.
Essentially, the index of the homeomorphism atponly provides “optimal” dynamical information if p admits isolating blocks. Otherwise, the set of indices of the induced homeomorphism in the Carath´eodory’s compactification ofS2\Kpat the new fixed points provides much more information than the index atp.
The main goals of this paper are the following:
aThe first goal is to provide a general geometrical method to compute the fixed point index of the iterations of an arbitrary local homeomorphism at an isolated fixed point;
bGiven any Jordan domainJ, InvclJ, f∩∂J/∅and an isolating block,N, is a neighborhood that isolates the fixedor periodicalprime ends of the component of InvclJ, fcontainingp, to prove thatJandNdetermine canonically a number of generalized unstablestablebranches and generalized repellingattractingpetals around the fixed pointseeDefinition 2.6. Their number depends onJ andNbut their difference depends just on the germ off;
cThe third goal is to provide some dynamical consequences. We shall give new and short proofs of some known results and new theorems in the orientation reversing framework.
The paper is organized as follows: in Section 2 we start with some preliminary definitions. We will dedicate subsections to recall the results we will need in the special case where the fixed point is an isolated invariant set and to give a brief presentation of the Carath´eodory’s prime ends theory. At the end of the section, we give the statement of the main results. Section 3is devoted to the computation of the fixed point indices of the iterations of arbitrary planar homeomorphisms at an isolated fixed point. InSection 4, we will give the proof of the main theorems and the dynamical meaning of the indices. First we shall study the case where the homeomorphism has a finite number of periodic prime ends.
The general case follows easily from this previous simpler case seeRemark 2.12. Finally Section 5contains the proofs of a number of corollaries of our techniques.
2. Preliminary Definitions and Results. The Main Construction and the Statement of the Principal Results
2.1. Preliminary Definitions
GivenA ⊂B⊂ N, clA, clBA, intA, intBA,∂Aand∂BAwill denote the closure of A, the closure ofAinB, the interior ofA, the interior ofAinB, the boundary ofA, and the boundary ofAinB, respectively.
Let U ⊂ X be an open set. By a (local) semidynamical system, we mean a local homeomorphism f : U → X. The invariant part ofN, InvN, f, is defined as the set of allx∈Nsuch that there is a full orbitγwithx∈γ⊂N.
InvN, f resp., Inv−N, fwill denote the set of allx∈Nsuch thatfjx∈Nfor everyj∈Nresp.,f−jxis well defined and belongs toNfor everyj ∈N.
A compact setS ⊂ X is invariant if fS S. A compact invariant setS is isolated with respect tof if there exists a compact neighborhoodNofSsuch that InvN, f S. The neighborhoodNis called an isolating neighborhood ofS.
An isolating blockNis a compactum such that clintN Nandf−1N∩N∩fN⊂ intN. Isolating blocks are a special class of isolating neighborhoods.
We consider the exit set of Nto be defined as
N−
x∈N:fx/∈intN
. 2.1
If X is a locally compact ANR absolute neighborhood retract for metric spaces, iXf, S will denote the fixed point index off in a small enough neighborhood of S. The reader is referred to the text of18–22for information about the fixed point index theory.
An isolated fixed pointpis said to be indifferent if for every small enough discDsuch thatp∈intD, InvD, f∩∂D/∅.
An isolated fixed pointpis accumulated ifp∈clPerf|V\{p}for every neighborhood V ofp.
2.2. Strong Filtration Pairs
The next definition is based on the notion of filtration introduced by Franks and Richeson, in 23. It is the key for the direct computation of the fixed point index of any iteration of any homeomorphism of the plane.
Definition 2.1. Letf : U ⊂ R2 → R2 be a local homeomorphism. Suppose thatL ⊂ N is a compact pair contained in the interior ofU. The pairN, Lis said to be a strong filtration pair forfprovidedNandLare each the closure of their interiors and
1Nand∂N\Lare homeomorphic to a disc andS1, respectively.
2clN\Lis an isolating neighborhood.
3fclN\L⊂intN i.e.,Lis a neighborhood ofN−inN.
4For any componentLi ofL,∂NLiis an arc and there exists a topological discBi such that∂NLi⊂Bi⊂Li,Bi∩N−/∅, andfBi∩clN\L ∅.
Theorem 2.2see4,12. Letf:U⊂R2 → fU⊂R2be a homeomorphism. Suppose that there exists a strong filtration pair,N, L, forfand letKInvclN\L, f. Then, there are an absolute retract for metric spaces,D0, containing a neighborhoodV ⊂R2ofK, a finite subset{q1, . . . , qm} ⊂ D0, and a mapf:D0 → D0such thatf|V f|V and for everyk∈N, Fixfk⊂K∪{q1, . . . , qm}.
Moreover,
aiffpreserves the orientation, then
iR2 fk, K
⎧⎨
⎩
1−rq ifk∈rN,
1 ifk /∈rN, 2.2
wherek∈N,qis the number of periodic orbits offin{q1, . . . , qm}, andris their period;
biffreverses the orientation, then
iR2 fk, K
⎧⎨
⎩
1−δ ifk odd,
1−δ−2q ifk even, 2.3
where δ ∈ {0,1,2} and qare the number of fixed points and period two orbits off in {q1, . . . , qm}, respectively.
Definition 2.3. Under the setting of the above theorem, the integerrr 2 iffis orientation reversingis called the period of the strong filtration pairN, L.
We conclude this subsection with the next theorem that resumes the main results of 12. We will construct a family of branches of the stable and unstable “manifolds” associated to a fixed pointpwhich admits a strong filtration pairN, L. The minimum number of elements of these families depends on the fixed point indexiR2fr, pwith r being the period of the strong filtration pairN, L. In order to make the paper as self-contained as possible, we will sketch the proof which contains some ingredients we will need in the future.
Theorem 2.4. Letf : U ⊂ R2 → fU ⊂ R2 be a homeomorphism with p being an isolated fixed point off, and let us assume that there is a strong filtration pair of periodr,N, L, such that p∈intN\L,L /∅,fjclN\L⊂Uforj ∈ {1, . . . , r}and Fixfr∩clN\L {p}. Let us suppose that the connected component ofKInvclN\L, fwhich containspisKp{p}. Then there exist trivial shape continuaS1, . . . , Ss, U1, . . . , Usin clN\L, withs 1−iR2fr, p, such that:
1 si1Si ⊂ Kp and si1Ui ⊂ K−p, withKp and Kp− being the connected components of KInvclN\L, fandK−Inv−clN\L, fwhich containp;
2Si∩SjUi∩Uj{p}for alli /jandSi∩Uj{p}for alli, j∈ {1, . . . , s};
3frSi ⊂ Si,f−rUi ⊂ Ui, and
n∈Nf−nrUi
n∈NfnrSi {p} for everyi ∈ {1, . . . , s};
4the setsSi∩∂clN\LandUi∩∂clN\Lalternate in∂clN\L.
Proof. IfL L1∪ · · · ∪Lm, let us consider the quotient spaceNLobtained from clN\Lby identifying each∂NLito a pointqifori1, . . . , m. Take the projection mapπ : clN\L → NLand the retractionr:N → clN\L. The map
f π◦r◦f◦π−1:NL\
q1, . . . , qm
−→NL 2.4
induces in a natural way a continuous map f : NL → NL. It is easy to see that f{q1, . . . , qm} ⊂ {q1, . . . , qm}. Let θ {p1, . . . , ps} be the biggest subset of{q1, . . . , qm} on whichf acts as a permutation. It is clear thatθ is an attractor forf is locally constant for everypi∈θ. LetAbe the region of attraction ofθ,
A
x∈NL: there isn0 such that fn0
x∈θ
2.5 and letApjbe the component ofAcontainingpj ∈ θ. Let us observe thatK−p andKpare trivial shape continua such that limk→ ∞f−kx pfor everyx ∈Kp−and limk→ ∞fkx p for everyx∈Kpsee the Main Theorem in12for a proof. Then it is not difficult to see that p∈clApjfor allj1, . . . , s.
LetKi
n∈NfnrclApifori ∈ {1, . . . , s}. SincefrclApi ⊂ clApi, it is clear thatKi is a continuum with{p, pi} ⊂ Ki frKi ⊂ clApi. Then we have that
i∈{1,...,s}Ki\ {pi}⊂K−p, then∂NLi∩K−p/∅for alli1, . . . , s.
Let us define the continuumUiπ−1Ki∩Kp−. We have thatUiis negatively invariant forfrand containsp.
On the other hand,Ui∩K{p}. In fact, since
n∈Nf−nrUiis an invariant continuum forfrwhich containsp, then
n∈Nf−nrUi⊂Kp{p}. Ifx∈Ui∩K, thenx∈
n∈Nf−nrUi⊂ Kp{p}.
Let us see thatUi has a trivial shape. In fact, ifUihas a holeV, then there area∈V andn0 ∈Nsuch thatfrn0a∈intLiandfrna∈clN\Lfor alln∈Z,n < n0. Then it is immediate thata∈Uiwhich is a contradiction.
Let us prove that Ui ⊂ π−1Api∪ {p}. If x ∈ Ui \ {p}, then there existsn0 ∈ N such thatfnrx ∈ clN\Lfor all integern < n0 andfn0rx ∈ intLi if this is not true, x∈Kand we havexp. Then it follows thatx∈π−1Api. As a corollary, we obtain that Ui π−1Api∪ {p}∩K−p.
It is obvious thatUi∩∂clN\L⊂∂NLi.
If we repeat this construction fori∈ {1, . . . , s}, we obtainU1, . . . , UswithUi∩Uj{p}
for everyi /j.
Let us construct the setsS1, . . . , Ss. Let us consider the setθ {p1, . . . , ps} withpi−1 adjacent topithere is an arcγ ⊂π∂N\Ljoiningpi−1withpisuch thatγ∩θ{pi−1, pi}.
Ifpi−1pi is the arc inπ∂clN\Lwhich makes adjacentpi−1andpi, we have that there is a componentKpi ⊂Kpof∂Apiseparatingpi fromθ\pisee the Main Theorem in12 withKpi∩pi−1pi/∅.
LetBibe the connected component of clN\L\Ui−1∪Uiwhich containsπ−1Kpi∩ pi−1pi. Then we defineSi Bi∪ {p}∩Kp. Following the steps given with the family{Ui}, it is easy to prove the analogous properties for the family{Si}.
2.3. Carath ´eodory’s Prime Ends
LetB ⊂ Cbe the unit open disc and letf : B → G ⊂ C∪ {∞}be an onto and conformal mapping. The problem whether f admits an extension to clB B ∪S1, by defining fz limx→zfxforz∈ S1, has a topological answer. Indeed,f admits that an extension if and only if∂Gis locally connected. The problem whetherf has an injective extension has also an answer of topological nature:f has an injective extension if and only if∂Gis a Jordan curveCarath´eodory’s Theorem, see 24. If ∂G is locally connected but not a
Jordan curve, there are points of ∂Gthat have several preimages. The situation becomes much more complicated if∂Gis not locally connected. Carath´eodory introduced the notion of prime end to describe this setting. The pointsz ∈S1 correspond one-to-one to the prime ends ofGand the limitfzexists if and only if the prime end has only one pointPrime End Theorem, see24.
LetD ⊂ R2be a simply connected open domain containing the point at infinity such that∂Dcontains more than one point. Then∂Dis bounded. A cross-cut is a simple arc, C, lying inD, except in the end points, with different extremities. IfCis a cross-cut of D then D \C has exactly two components A1 and A2 such that D ∩∂A1 D ∩∂A2
C\ {end points}.
A sequence{Cn}of mutually disjoint cross-cuts and such that eachCnseparatesCn−1 andCn1is called a chain. A chain of cross-cuts induces a nested chain of domainsbounded by eachCn· · ·Dn1 ⊂ Dn· · ·. Each chain of cross-cuts defines an end. Two chains of cross- cuts,{Cn}and {Cn}, are equivalent if for anyn ∈ Nthere is mnsuch thatDm ⊂ Dn and Dm ⊂ Dn for everym > mn. Equivalent chains of cross-cuts are said to induce the same end. IfPandQare ends represented by chains of cross-cuts{CPn}and{CQn}such that for everyn,DPm ⊂DQnform > mn, we say thatP dividesQ. A prime endP is an end which cannot be divided by any other.
LetPbe a prime end. The set of points ofPis the intersectionE
n∈NclDPnwhere {DPn}is the sequence of domains bounded by any sequence of cross-cuts representingP.
A principal point ofPis a limit point of a chain of cross-cuts representingPtending to a point.
The setHP ⊂Eof principal points of a prime endPis a continuumcompact connected set see6or7for details.
Each chain of cross-cuts inducing a prime endPdetermines a basis of neighborhoods ofP. We obtain in this way a topology in the set of prime ends ofD. More precisely, ifPis the set of prime ends ofDandD∗is the disjoint union ofDandP, we can introduce a topology in D∗in such a way that it becomes homeomorphic to the closed disk and the boundary being composed by the prime ends. It is enough to define a basis of neighborhoods of a prime end P ∈P. Given the sequence of domains{DPn}, we produce a basis of neighborhoods{Un} ofP inD∗. EachUnis composed by the points inDPn and by the prime endsQsuch that DQm⊂DPnformlarge enough.
IfS2is the 2-sphereR2∪ {∞}and∞ ∈D⊂S2is a simply connected open domain, the natural compactification, due to Carath´eodory, see6, ofDobtained by attaching toDa set homeomorphic to the one-dimensional sphereS1is called the prime ends compactification ofD.
We identifyR2Cand we consider a conformal homeomorphismg :D → S2\BwhereB is the discB{z∈C:|z| ≤1}. Now a one-dimensional sphereS1is attached toDusingg.
Each point ofS1corresponds to a prime end ofD.
2.4. The Main Construction
Letf : U → W be a local homeomorphism withU, W ⊂ R2 open subsets and letp be a nonaccumulated and indifferent fixed point in a small enough Jordan domain J with {p}
being the unique periodic orbit contained in clJand such thatKp∩∂J/∅forKpbeing the connected component ofKInvclJ, fwhich containsp. We will suppose thatp∈∂Kp e.g., ifpis not stable andJis small enough, thenp∈∂Kp.
Remark 2.5. Let us observe that, givenpbeing a non-accumulated and indifferent fixed point, ifiR2fk, p/1 for somek∈N, then we can select a Jordan domainJ, as above, withp∈∂Kp.
In fact, ifp ∈ intKpfor every small enough Jordan domainJ, thenpis stable forfkand iR2fk, p 1see9,25.
It is easy to see that the setKp ⊂clJhas a trivial shape, that is,KpandR2\Kpare connected.
We follow with some of the most important notions of the paper: the generalized stable/unstable branches and generalized attracting/repelling petals. The first ones are essentially branches, in a classical sense, for the map that our homeomorphism f induces in the compactification ofR2\Kpat a fixed prime end.
Let p ∈ J be an indifferent and non-accumulated fixed point for f in the above conditions. Given the open domainS2\Kp, for each open arcγ ⊂Jwith end-pointsa, b∈Kp
we do not exclude the possibility a bsuch that γ ∩Kp ∅, we call Dγ the bounded connected component ofR2\γ∪Kp. The setDγis an open ball contained inJ.
Definition 2.6. A continuumUp⊂clJis a generalized unstable branch forfatpif:
iUp∩Kpis an invariant continuum contained in∂Kpsuch thatp ∈Up∩Kpand Up\Kp⊂Jis nonempty and has trivial shape components;
iif−1Up⊂Upand
n∈Nf−nUp Up∩Kp;
iiithere exists an open ballDγassociated to an open arcγ, as above, withUp⊂clDγ, Up∩γa compact set, and such that:
athe setUpis locally maximal, that is, ifUp⊂clDγsatisfies conditionsiand ii, thenUp⊂Up;
bfor every open neighborhood V of Up, there exists x ∈ Dγ ∩ V with f−nxx/∈clDγfor somenx∈N.
In an analogous way, we define generalized stable branchesSpforf atp. We only have to replacefbyf−1iniiandiii.
A setRpis a generalized repelling petal forfatpif:
iRpclDγ⊂clJwithDγbeing an open ball associated to an open arcγ, as above, such thatclγ γ∪ {q1, q2}withp /∈ {q1, q2};
iif−1Rp ⊂ Rp and
n∈Nf−nRp ⊂ ∂Kp is an invariant continuum forf which containsp.
In an analogous way, we define generalized attracting petals forfatp. We only have to replacefbyf−1inii.
Remark 2.7. The stable and unstable branches in the classical sense associated to f at p and constructed in the proof ofTheorem 2.4, are, of course, particular cases of generalized unstable and stable branches if we consider the mapf fr andKp{p}. It is easy to obtain an adequate arcγj ⊂clN\Lfor each unstablestablebranchUj.
LetU be a Jordan domain such that clJ ⊂ U ⊂ U ⊂ S2 and letf :S2 → S2 be a homeomorphism such thatf|U f. The Carath´eodory’s compactification ofS2\Kpis a disc obtained by gluingS1toS2\Kpwhich we callD. The homeomorphismf|S2\Kp :S2\Kp → S2\Kpcan be extended to a homeomorphismf:D → D. Let us denoteD\S2\Kp ∂D
and let us consider the set of prime ends obtained from the accessible pointsKp∩∂J by arcs onU\clJand which we callPKp∩∂J⊂∂D.
Iffis orientation preserving and there exist periodic orbits forf|∂D, then all of them have the same periodr. Iffis orientation reversing, thenf|∂Dhas exactly two fixed points and period two periodic orbits.
Let us see that the compact sets Perf|∂DandPKp∩∂Jare disjoint. LetP0 be a prime end inPKp∩∂Jassociated with a pointp0 ∈ Kp∩∂J. ThenP0/∈Perf|∂D. In fact, if this is not true,P0is a fixed prime end forfr r 2 iffis orientation reversing and, since p0 is accessible by an arcγp0 ⊂ U\clJsuch that clγp0\γp0 {p0}, then the principal points of the fixed prime endP0 are the continuum, invariant forfr,HP0 {p0} HP0 ⊂clγp0\γp0{p0}. Then,p0must be a fixed point forfr. But this is a contradiction.
Remark 2.8. Note that both fand the set of fixed prime ends of fdepend on the Jordan domainJsuch that InvclJ, f∩∂J/∅. SeeExample 2.9.
Example 2.9. Let us consider the dynamical system ofFigure 1, which gives us a homeomor- phismfofR2withpbeing a non-accumulated and indifferent fixed point.
The Jordan domainsJ1andJ2ofFigure 1are such that InvclJ1, f K1pis a “petal”
which containspand such thatK1p∩∂J1/∅. On the other hand, InvclJ2, f K2pare two “petals” which containpand such thatK2p∩∂J2/∅.
The mapsf:D → Dhave the dynamical behavior inFigure 2.
The mapfforJ1has, in∂D, a fixed prime endp1and the mapfforJ2has, in∂D, two fixed prime ends{p1, p2}.
Following with the main construction, there are two possible situations:
aPerf|∂Dis a finite set ofnpoints;
bPerf|∂Dis an infinite set of points.
Let us suppose that we are in casea.Remark 2.12permit us to reduce casebto case aby identifications to points of adequate intervals in∂D. Iffis an orientation preserving homeomorphism, we have that n qr for certain q, r ∈ Nwith r being the period of the periodic orbits of f|∂D and q the number of periodic orbits. On the other hand, if fis orientation reversing, we obtainqperiodic orbits of period 2 and two fixed points in∂D. It is obvious thatn2q2.
Let us suppose thatD ⊂ S2 and let us denote byfs :S2 → S2the homeomorphism obtained by pasting along∂Da symmetric copy off:D → D.
The next lemma is needed for the computation of the fixed point indexiR2fk, pby using strong filtration pairs.
Lemma 2.10. Given a fixed pointp1offs
k|∂D, (k∈rNiffis orientation preserving), there is a pair N1, L1which is in one of the following two situations.
a N1, L1is a strong filtration pair forfs
k:S2 → S2, in a neighborhood ofp1. The period of the strong filtration pair is 1 iffis orientation preserving or 2 iffreverses the orientation.
bThe pairN1, L1has the properties (1), (2), and (3) of strong filtration pairs withL1being a disc with a hole,∂N1L1S1andN1⊂intfskN1.
J1
J2
p
Figure 1
p1
fforJ1
a
p1 p2
fforJ2
b Figure 2
Proof. Given a fixed pointp1offsk|∂D, let us see that there exists the pairN1, L1forfskin S2withp1∈InvclN1\L1,fs
k.
Take a small enough arc a, b ⊂ ∂D with p1 ∈ a, b and such that Inva, b,fk|∂D p1. The set a, b is an isolating block for fk|∂D. Let us consider a small enough discMinD with M∩∂D a, band Fixfk|M {p1}. Since the space of components of InvM,fkis a zero-dimensional compactum, it is easy to construct a disc M1 ⊂Msuch thata, b ⊂M1and InvM,fk∩∂DM1 ∅. If we choose the discN ⊂S2 obtained by joiningM1 with its reflected disc on∂D,M2, we have thatNis an isolating neighborhood forfsk.
It is not difficult to construct a discN1 ⊂intN,N1symmetric with respect to∂D, and isolating block forfsksee12,26, with∂N1∩InvN,fsk ∅andp1Fixfsk|N1.
If there is not a discB⊂N1such thatp1∈intBandB⊂intfskB, then there exists a strong filtration pairN1, L1forfskwithL1being a finiteperhaps emptyunion of disjoint
discssee4,12. By the symmetry property with respect to∂Doffsk, it is immediate that the period of the generalized filtration pair is 1 iffs
kis orientation preserving and 2 iffs kis orientation reversingsee4. Therefore, we are in the conditions ofa.
On the other hand, it there exists the above discB, we obtain in an easy way the pair N1, L1of the caseb.
Definition 2.11. We are interested, for each fixed pointpioffsk|∂D, in the pairsNi∩D, Li∩ D Ni, Liwhich we call strong filtration pairs adapted toDforpi. Let us observe that the pairNi, Lihas the properties of the strong filtration pairs forfk:D → Dat each fixed point pi ∈∂D. We will suppose without loss of generality that each arcγi ∂DNicorresponds inJto an arc with two end points inKp.
There are three possible cases.
iIfLi∅, thenfkNi⊂intDNiand we say thatNiis an attracting petal associated tofkatpi.
iiIf ∂NiLi S1, then Ni ⊂ intDfkNiand we say that Ni is a repelling petal associated tofkatpi.
iiiIfNi, Liis a strong filtration pair withLi/∅, given the sets of stable and unstable branches {Sj} and {Uj}of Ni, Liassociated to fs
k at pi seeTheorem 2.4, we select the subsets of branches{Sm}and{Um}which are contained inNi\∂D∪ {pi}. We call{Sm}and{Um}stable and unstable branches ofNi, Liassociated tofk atpi.
Remark 2.12. If Perf|∂D is not a finite set of pointswe supposed before, we can select a finite and disjoint union I I1 ∪ · · · ∪In, of closed arcs of ∂D, with fI I, such that Perf|∂D ⊂ I and PKp∩∂J∩I ∅. Let us identify each component ofI to a point. We obtain a new disc which we call D again. If f : D → D is the new induced homeomorphism, we have that Perf|∂Dis a finite set and the construction of the strong filtration pairs adapted toD is also validseeFigure 3. It is obvious that this construction depends on the choice of the setI.
Example 2.13. Let us consider the dynamical system ofFigure 4. We obtain a homeomorphism fofR2withpbeing a non-accumulated and indifferent fixed point and InvclJ, f Kpan infinite family of petals which containpin their boundary.
The dynamic of the mapfin D is given inFigure 5a. We have an infinite family of fixed prime ends fixed points for f in ∂D. If we consider the two invariant arcs for f, I1 and I2, of Figure 5a and make an identification of them to points p1 and p2, we obtain a new homeomorphism which we call in the same way f : D → D. This homeomorphism has only two fixed points in∂Dand we are in casea; seeFigure 5b.
The new mapfhas a repelling point in p2 and an unstable branch in p1. Let us observe that the choice of the invariant intervals which contain the fixed prime ends,I I1∪I2, is not unique. We can selectI with an arbitrary family of intervals of this type which gives us a different dynamic forfand a different set of fixed points in∂Dfor the identification map.
pi
Casei
pi
Caseii
LiD
pi
Caseiii
Li D U1
S1
S2
Figure 3
Kp
p
J
Figure 4
Definition 2.14. Given a Jordan domainJ, a set of strong filtration pairs adapted toDis a finite collection of pairs{Ni∩D, Li∩D}iassociated to the family{pi}iof fixed points offsk|∂D. Let us observe that this set depends on the choice ofJand, if Perf|∂Dis not finite, on the choice of the setIsuch that, after an identification, it transforms Perf|∂Din a finite set .
I1 I2
a
p1 p2
b Figure 5
2.5. The Statement of the Principal Results
To conclude this section, we summarize below the main results of this article.
Letf : U → W be a local homeomorphism withU, W ⊂ R2open subsets and letp be a non-accumulated, indifferent fixed point. Ifpis stable, that is, if there exists a basis of neighborhoods{Un}n∈Nofpsuch thatfUn⊂Unfor alln∈N, we obtainiR2fk, p 1 for allk∈Nsee9,25.
We are interested in the relation between the fixed point index of the iterations offat pand the local dynamics atp, withpbeing a nonstable fixed point.
Main Theorem 1 Poincar´e formula: Orientation preserving case. Let f : U → W be an orientation preserving local homeomorphism with p being an unstable, non-accumulated, and indifferent fixed point. Let us select a Jordan domainJsuch thatp∈J⊂clJ⊂UwithKp∩∂J/∅, and let {Ni ∩ D, Li ∩D}i be a set of strong filtration pairs adapted to D, the Carath´eodory’s compactification ofS2\Kp. Then there existr ∈Nandrp, up, sp, ap∈rNsuch that
iR2 fk, p
⎧⎨
⎩
1 ifk /∈rN,
1−uprp1−spap ifk∈rN,
⎧⎪
⎨
⎪⎩
1 ifk /∈rN,
11 2
rpap
−
upsp
ifk∈rN.
2.6
We have the following dynamical interpretation: there areup spgeneralized unstable stable branches and rpap generalized repelling attracting petals for fr at p see Definition 2.6. They are negatively positively invariant for fr and f−1f acts as a
permutation on them. Let us observe that the numbers{up, rp, sp, ap}depend onJ and the set of strong filtration pairs but their differences depend only on the germ off.
Remark 2.15. The last result gives us, as a corollary, a theorem due to Le Calvez see5 which says that ifpis a non-accumulated, indifferent fixed point, there existr≥1 andq∈Z such that
iR2 fk, p
⎧⎨
⎩
1 if k /∈rN,
q if k∈rN. 2.7
On the other hand, iff preserves or contracts expandsareas, thenrp 0 ap 0 and we obtain a corollary which improves a result of Simonsee27about the existence of a bound for the fixed point index of the area and orientation preserving homeomor- phisms at an isolated fixed point. More precisely, if f preserves or contracts areas then iR2f, p≤1.
From the above considerations, given an orientation preserving homeomorphismh: U ⊂ R2 → R2 which preserves a measure supported in the open sets, such that Fixh Perh {0} and iR2hk,0 1 for everyk ∈ Z, it is natural to ask if 0 must be a stable in the past or in the futurefixed point. The famous example of Anosov and Katok,28, is a counterexample to this problem. They produced a diffeomorphism of the disc which preserves natural measures and it is ergodic. This map is constructed inductively as a limit of an appropriate sequence of diffeomorphisms. In the next sectionseeExample 3.3, we will exhibit an explicit, very simple and geometric example of an orientation and area preserving homeomorphismh:R2 → R2 such that Fixh Perh {0}, 0 is stable neither forhnor forh−1, and the fixed point indicesiR2hk,0 1 for everyk ∈Z. Moreover, there will not be h-invariant subsets of positive finite measure.
For the orientation reversing case, we prove the following theorem.
Main Theorem 2 Poincar´e formula: Orientation reversing case. Letf : U → W be an orientation reversing local homeomorphism withp being an unstable, non-accumulated, indifferent fixed point. Let us select a Jordan domainJsuch thatp∈J⊂clJ⊂U, withKp∩∂J/∅, and let {Ni∩D, Li∩D}ibe a set of strong filtration pairs adapted toD, the Carath´eodory’s compactification ofS2\Kp. Then there existup, up, rp, rp∈Nwithup≤up,rp≤rp,uprp≤2 such that
iR2 fk, p
⎧⎨
⎩
1−uprp ifk even,
1−up−rp ifk odd, 2.8
and with the following dynamical meaning: there areupgeneralized unstable branches forf2atpwith up ≤ 2of them negatively invariant forf(f−1sends each of theupgeneralized unstable branches to a subset of itself). In the same way, there arerpgeneralized repelling petals forf2 atpandrp ≤2 of them are negatively invariant forf.
As in the orientation preserving case, we have similar formulas involving generalized stable branches and generalized attracting petals.
Remark 2.16. As a corollary, iR2f, p ∈ {−1,0,1} for f an orientation reversing local homeomorphism andpa non-accumulated fixed point. This is Bonino’s theoremsee29 whenpis non-accumulated.
Remark 2.17. The Main Theorem for orientation reversing homeomorphisms says that iR2f2n, pis constant. Then it solves Problem 7.3.9. of21.
Theorem 2.18. Letf :U → Wbe an arbitrary local homeomorphism with Fixf {p}being an indifferent fixed point, such thatiR2fr, p 1−m <1 for somer∈N(r2 iffreverses orientation).
Then there existmunstable (stable) branches, in the classical sense,{Ui}{Si}forfratpsuch that:
1f−1andfact as permutations in{Ui}and{Si}, respectively;
2limn→ ∞f−nx pfor everyx∈Ui, limn→ ∞fny pfor everyy∈Si;
3there exists a closed discDp⊂J, withp∈intDp, mi1Ui∪Si⊂Dp, in such a way that the intersection of the stable and unstable branches with∂Dpalternates in∂Dp. Each generalized repelling attracting petal contains p in its boundary. As a corollary of the Main Theorems for both orientation preserving and orientation reversing homeomorphisms, we will obtain the following resultsee11for the orientation preserving case.
Theorem 2.19Petal’s theorem. Letf : U → W be an arbitrary local homeomorphism withp being a non-accumulated and isolated fixed point such thatiR2fr, p 1m > 1 for somer ∈N.
Then there existmgeneralized repelling petals{Ri}andmgeneralized attracting petals{Ai}forfr atpsuch that:
1intAi∩intAj intRi∩intRj ∅for alli /j, and intAi∩intRj ∅for alli, j;
2the mapf(f−1) acts as a permutation in{Ai}({Ri});
3limn→ ∞f−nx pfor everyx∈Ri, limn→ ∞fny pfor everyy∈Ai;
4the sequences {f−nrRi}n∈N and {fnrAi}n∈N determine ends containing p and
n∈Nf−nrRiand
n∈NfnrAiarefr-invariant continua containingp;
5there is a Jordan curve γ aroundpsuch thatγ intersects alternatively the sets{Ai}and {Ri}, withγ∩Aiandγ∩Ribeing closed arcs.
Remark 2.20. Using the petal’s theorem, one can prove the following consequences that extend a theorem due to Le Calvezsee5.
Iff:U → Wis a local homeomorphism such that Fixf {p}and 1/iR2fr, p>1−q for somer∈Nr2 iffreverses orientation, take a discJsuch thatp∈intJ⊂clJ⊂U.
We have the following two properties.
aIf there existqgeneralized stable branches forfr atp, then there exists a domain V1 ⊂ Usuch that the domains of the sequence{fnV1}n∈N are well defined and disjoint.
bIf there exist q generalized unstable branches forfr at p, then there exists a domainV2 ⊂ U such that the domains of the sequence{f−nV2}n∈N are well defined and disjoint.
As a corollary, ifiR2fr, p>1 for somer∈N, there exist domainsV1, V2⊂Usuch that the domains of the sequences{fnV1}n∈Nand{f−nV2}n∈Nare well defined and disjoint.
The last remark can be applied to the following interesting situation: let M be an oriented compact 2-dimensional manifold with boundary and letf : U ⊂ M → Mbe an orientation preserving homeomorphism. Letp ∈ ∂M∩Ube an isolated fixed point off.
Denote by DM the double of the manifoldM and Df : DM → DM the natural map induced byf.
Then,
aifpis a saddle point off|∂MandiDMDf, p>0, then there exist domainsV1, V2⊂ Usuch that the domains of the sequences{fnV1}n∈Nand{f−nV2}n∈Nare well defined and disjoint;
bifpis an attractor off|∂MandiDMDf, p>−1, then there exists a domainV1⊂U such that the domains of the sequence{fnV1}n∈Nare well defined and disjoint;
cifpis a repeller off|∂MandiDMDf, p >−1, then there exists a domainV2 ⊂ U such that the domains of the sequence{f−nV2}n∈Nare well defined and disjoint.
Note that in this particular setting, since p is isolated using Brouwer’s lemma on translation arcs, it is not necessary to assume thatiDMDf, p/1.
For orientation and area preserving homeomorphisms in surfaces, we have the following Nielsen type theoremsee30for the particular case whereMis a disc.
Corollary 2.21. LetMbe an oriented compact 2-dimensional manifold with boundary and letf : M → Mbe an area and orientation preserving homeomorphism such thatf|∂M hasnattracting fixed points andnrepelling fixed points. Thenf has, at least,n Λffixed points in intMwhere Λfdenotes the Lefschetz number off. As a consequence, ifMis the 2-dimensional disc, we have thatfhas, at least,n1 fixed points in intM.
Restricting ourselves to orientation reversing homeomorphisms and using the fact that iR2f, p∈ {−1,0,1}, we shall produce a sharp theorem. The proof will be obtained easily by using the previous results.
Theorem 2.22. Let f : U → W be an orientation reversing local homeomorphism withpbeing a non-accumulated, indifferent fixed point, andiR2f2, p/1. Then there areup generalized unstable branches andrpgeneralized repelling petals forf2atpsuch thatiR2f2, p 1−uprpand
athe generalized unstable (stable) branches and the generalized repelling (attracting) petals are negatively (positively) invariant forf2;
b.1ifiR2f2, p 1m > 1, thenrp ≥ mand there are, at least,mgeneralized attracting petals forf2atp(mof the generalized attracting petals alternate withmof the generalized repelling petals aroundp);
b.2ifiR2f2, p 1−m <1, thenup≥mand there are, at least,mgeneralized stable branches forf2atp(mof the generalized stable branches alternate withmof the generalized unstable branches aroundp);
c.1ifiR2f, p 1, then there are neither generalized repelling petals nor generalized unstable branches forf2atp, negatively invariant forf. On the other hand, there are two generalized attracting petals or two generalized stable branches or a generalized stable branch and a generalized attracting petal forf2atp, positively invariant forf.
The numbersupandrpare even. Therefore,iR2f2, pis odd;
