On noncontractible periodic orbits for surface homeomorphisms
Abstract.
In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of noncontractible periodic orbits. We show that, if is such a homeomorphism, and if is its lift to the universal covering of that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) The set of fixed points for projects to a closed subset which contains an essential continuum, (2) has noncontratible periodic points of every sufficiently large period, or (3) there exists an uniform bound such that, if projects to a contractible periodic point, then the orbit of has diameter less or equal to . Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.
Key words and phrases:
Surface homeomorphisms, pseudo–Anosov, non–contractible periodic orbits1. Introduction
Let be a compact manifold and be an isotopy between the identity and a map . A fixed point for is called contractible if the loop is homotopic, in , to a point.
The study of the existence of contractible fixed points and contractible periodic orbits for homeomorphisms of manifolds has a long history in mathematics. In the particular case that is a sympletic manifold, is the time one map of a periodic Hamiltonian flow and has finitely many fixed points, the Conley conjecture on the existence of contractible periodic orbits of arbitrary large period was a great motivation for the fields of topological dynamics on surfaces ([Fr92, FH03, LC05, LC06]) and sympletic dynamics (see [Hi09, Gi10] for some recent results).
Lately, the study of the existence of noncontractible periodic points has been the focus of significant work in sympletic geometry ([GL00, BPS03, We06, Gü12, SW13]) with great developments, usually employing Floer homology techniques. This note is concerned with the existence of such points for general homeomorphisms of surfaces. Our intention is to understand the dynamical restrictions that are implied whenever all –periodic points are contractible. Our main result is:
Theorem 1.
Let be a compact orientable surface, a homeomorphism isotopic to the identity and a lift of to the universal covering that commutes with the deck transformations. Then either:

is essential,

there exists a deck transformation such that, for every with sufficiently small, there exists such that , or

there exists such that, if is periodic, then the orbit of has diameter less then .
and a direct consequence is:
Corollary 2.
Let be a compact orientable surface, a homeomorphism isotopic to the identity with finitely many fixed points, and a lift of to the universal covering that commutes with the deck transformations. Then either:

there exists such that, if is periodic, then the orbit of has diameter less then or,

has non–contractible periodic orbits of arbitrary large period.
The previous theorem has some other interesting consequences. Whenever a homeomorphism of a compact orientable surface leaves invariant a measure of full support (in which case the set of nonwandering points of , is the whole surface), the prevailing picture of the dynamics is that of a phase space partitioned in two different regions: one of points contained in periodic topological disks, often denoted as “elliptic islands”, and another connected compact set where the dynamics is transitive. Recent works ([Ja11, KT13a, AZ13]) have shown that for torus homeomorphisms this description is often valid. A topological description of the “islands” involves the notion of inessential points for the dynamics, those which are either wandering or contained in periodic topological disks. Denote by the set of inessential points, and by the set of nonwandering points. Inessential nonwandering points in many ways behave as periodic points, and the conclusions of Theorem 1 can be adapted to them:
Theorem 3.
Let be a compact orientable surface, a homeomorphism isotopic to the identity and a lift of to the universal covering that commutes with the deck transformations. Then either:

is essential,

has non–contractible periodic orbits,

there exists such that, if and , then the orbit of has diameter less than .
Another interesting consequence is obtained for irrotational homeomorphisms of compact surfaces, i.e., those homeomorphisms homotopic to the identity for which every point have null rotation vector (a precise definition of irrotacional homeomorphism is given in the next section). These homeomorphisms are in some sense the simplest possible for the rotational viewpoint, and for them all periodic orbits must be contractible. Irrotational homeomorphisms that are also area preserving are necessarily Hamiltonian in the sense of [LC06], and the dynamics may be very complicated when the set of fixed points is large (see [KT13b]). Apart from this case, a reasonable question is:
Question 4.
If is a compact surface, is irrotational and preserves a measure of full support, and is contained in a topological disk, is there such that all orbits have diameter less then ?
Theorem 1 shows this to be true if periodic points are dense in . When is the torus, the main result of [KT13c], Theorem 14, indicates this is probably true, and Theorem 3 allows us to improve it:
Theorem 5.
Let be an irrotational homeomorphisms preserving a measure of full support, and its lift to . If is contained in a topological disk, then there exists such that the diameter of every orbit by is uniformly bounded in the direction.
The next section introduces notation and the main background needed for the proof of Theorem 1, NielsenThurston theory and Le Calvez work on equivariant Brouwer lines, and the proofs are done in the last section.
2. Preliminar results
In what follows will be a compact orientable surface, possibly with boundary, its universal covering and the covering projection. We denote the set of deck transformations of by . We endow with a metric and denote the induced metric in by .
Following [KT13a], an open set is called inessential if every closed loop in is homotopically trivial in . A set is called inessential if it has an inessential neighborhood, otherwise is called essential. A connected set is called bounded if whenever is a connected components of , then is bounded in , in which case the diameter of is defined as the diameter of . While open inessential sets need not be bounded, it is immediate that any closed inessential set is contained in a closed bounded topological disk , and as such the diameter of all its connected components is uniformly bounded by the diameter of .
Given a homeomorphism , we say that is inessential for if there exists an invariant neighborhood of which is inessential, otherwise is called essential. The set of essential and inessential points of are denoted by and , respectively. If is homotopic to the identity, and is a fixed lift of that commutes with deck transformations, we say that the orbit of is bounded if, for any , the set is bounded. We denote the set of nonwandering points of by .
Whenever is isotopic to the identity, we chose an isotopy satisfying and . We denote by the induced isotopy on the covering space, and by the corresponding lift of . We also denote by and we write just whenever there is no possibility for confusion.
Given curves we denote their respective images by , and their concatenation by . If is homeomorphic to , is a closed curve and does not belong to , we denote by the winding number of around . Whenever is a homeomorphism of , is a periodic point for with minimal period , and , we define the index of the orbit of with respect to as .
Given , we denote by , the projection in the direction. If and , we say that the orbit of is bounded in the direction if for any , the set is bounded.
2.1. Rotation set for homeomorphisms of the annulus
Let be the closed annulus, be a homeomorphism preserving orientation and boundary components, and a lift of to the universal covering , and let be the covering map . Given a point , we define, whenever the limit exists, the rotation number of as , where and denotes the first coordinate of . Unlike the rotation number of an orientationpreserving circle homeomorphism, the rotation number of a point does not need to exist. Furthermore, although all points have the same rotation number for a circle homeomorphism, for annulus homeomorphisms different points may have different rotation numbers. Therefore it is useful to define the pointwise rotation set of as , the set of all different rotation numbers.
The main result in [Ha90] shows the strong relation between rotation numbers and periodic orbits:
Theorem 6.
For every rational there exists in such that
2.2. Brouwer–Le Calvez foliation
The equivariant Brouwer theory developed in [LC05] is a very useful tool in dealing with homeomorphisms of surfaces homotopic to the identity, and we briefly describe it in the context we require.
For our purposes, let be a homeomorphism of isotopic to the identity, an isotopy between and , and the lifted isotopy between and , such that commutes with any . We assume that is totally disconnected, and that there exists a such that for every in . Let be an oriented topological foliation of , which we regard as a flow with singularities. We say that is dynamically transverse to if, for every , the curve is homotopic with fixed end points, in , to an arc positively transversal to , that is, that locally crosses every leaf from left to right. In this case, if and is the lifted foliation , then is dynamically transverse to . We call a Brouwer–Le Calvez foliation for . The result we use, a consequence of [LC05] and [Jau12], is:
Theorem 7.
If is homotopic to the identity and is totally disconnected, then there exists a compact set , an isotopy between and which pointwise fixes and a Brouwer–Le Calvez foliation for .
The following results follow from the ideas in [KT13d], and are included for completeness:
Proposition 8.
If is a Brouwer–Le Calvez foliation for , then for any there exists such that, if is the leaf of passing through , then:

For any , the curve is homotopic, with fixed endpoints, to an arc positively transversal to the foliation that crosses at least once,

If the restriction of to is equal to , then there exist such that is a Brouwer–Le Calvez foliation for .
Proof.
If and is its universal covering space and the lift of to , then has no fixed points. Let be the lift of to . Since is a oriented foliation without singularities of and is homeomorphic to the plane, every leaf of is a line, that is, the image of a proper and injective map from into , and naturally divides into two regions, the left of , denoted by and the right of , denoted by . Since is dynamically transverse to , the preimage of every leaf is contained in its right, and the image of any leaf must be contained in its left. Therefore every leaf of lifts to a Brouwer line in . In particular, if is a preimage of in and is leaf of that passes through , there exists a neighborhood of such that and are on opposite sides of . Both claims follow trivially by choosing such that ∎
Proposition 9.
Let be a topological flow, a closed curve positively transversal to the flow lines of with finitely many selfintersections, and a flow line intersecting the image of . If and are the and limits of the flow line, then at least one of them is nonempty, compact, and contained in a single bounded connected component of . Furthermore, there exists an integer such that either for every in , or for every in .
Proof.
Let and define inductively, for ,
and as the point in such that . Note that, by the definition of as minimal, is uniquely defined. Let be a positive reparametrization of the , and, provided a positive reparametrization of the concatenation . Note also that, since had finitely many self–intersections, then there exists such that and therefore . We remark that each is a simple closed curve. By construction and, if , then . In particular, if , then .
But if is a simple closed curve positively transversal to a flow, then intersects a flow line at most once, in which case either is a nonempty compact set contained in the interior of , is contained in the exterior of , and for every and , or is a nonempty compact and contained in the interior of , is contained in the exterior of , and for every and . Furthermore, if is disjoint from , and if and , then . This implies that, for , , with the equality holding only if is disjoint from . As is not disjoint from , there exists such that is not disjoint from and , and so ∎
2.3. Nielsen Thurston Theory
We need to make use of the Thurston’s classification theorem for homeomorphisms of surfaces. We describe only the elements needed for our result, the reader is referred to [FLP79] for a more comprehensive description. Let be a compact surface and a collection of a finite number of points such that the Euler characteristic of is negative. Let be a homeomorphism such that and let . Then is said to be:

reducible relative to if there exists a finite set of disjoint simple closed curves such that each is neither null homotopic in nor homotopic to a boundary component or to a topological end of , and such that leaves invariant the set .

pseudo–Anosov relative to if there exists a pair of transverse measured foliations of with finitely many singularities, and a real such that preserves the foliations and such that if are the respective transverse measures, then and . Furthermore, every singularity of this pair of foliations is a pronged saddle, and if , then .
Thurston’s classification theorem states that, in this setting, for any given homeomorphism such that , there exists a homeomorphism and an isotopy from to leaving invariant and such that is either periodic, reducible relative to or pseudo–Anosov relative to .
Whenever is homotopic to a pseudo–Anosov homeomorphism , then each orbit must be globally shadowed by an orbit, and periodic orbits of are globally shadowed by periodic orbits of . In particular, and most relevant for us, we have the following result, a direct consequence of the main theorem of [Ha85]:
Theorem 10.
Let be pseudo–Anosov relative to , and let be homotopic to . Let the universal covering of and let and be homotopic lifts of and to , respectively. Then, if is a deck transformation and are such that , then there exists such that .
For homeomorphisms of the closed annulus , we use the following theorem from [BGH93] relating NielsenThurston theory and rotation sets :
Theorem 11.
Let be a homeomorphism of the annulus pseudo–Anosov relative to an invariant finite set , and let be its lift to the universal covering of . If, for every in , we have , then .
Finally,
Lemma 12.
Let be the open annulus, be a finite set, be a homeomorphisms such that and is homotopic in to , a homeomorphisms pseudo–Anosov relative to , and let be the lift of to the universal covering of . If for every in , then there exists such that, for any with and , there exists such that .
Proof.
First note that, from Brouwer theory, since has periodic points it must also have fixed points, and so the result follows when .
Let be the lift of to such that, if , then . If is the compactification of by adding two boundary circles at each end of the annulus, then there exists an extension of to such that is pseudo–Anosov relative to in . Furthermore, there exists a lift of to such that for all , if exists, then also exists and is equal to .
By Theorem 11, the origin lies in the interior of . Let be such that . Since the rotation number of all points in a boundary component of is the same, by eventually decreasing , we may assume that the rotation number of the boundary points do not belong to . By theorem 6, for every with and and , there exists such that . By the choice of must belong to the interior of and so, by the construction of it follows that . Finally, by the Handel’s shadowing theorem 10, for every there exists such that . ∎
2.4. Strictly toral homeomorphisms and irrotational homeomorphisms of
Following [KT13a] a homeomorphisms of homotopic to the identity is called annular if there exists and a lift of such that, for any . A homeomorphisms of is said to be strictly toral if, for every , is not annular and is inessential. The following result follows from Theorems C and D of [KT13a]:
Theorem 13.
If is strictly toral and , then is an inessential set, and for any open sets intersecting there exists such that .
A homeomorphisms isotopic to the identity of a surface is called sympletic if it preserves a Borel probability measure of full support and it is called hamiltonian if the rotation vector of is null (see [LC06] for a precise definition). It is called irrotational if, for every in . The main theorem of [KT13c] gives a description of homeomorphisms of which are area preserving and irrotational (and, in particular, hamiltonian):
Theorem 14.
If is irrotational and sympletic, then either is not strictly toral, or the orbit of any point is bounded.
3. Proof of the main results
Assume that neither (1) nor (3) holds, so that is inessential and that there exists a sequence of periodic points such that the diameter of the orbits is not uniformly bounded. We will show that condition (2) holds.
The following two propositions are direct consequences of being closed and inessential.
Proposition 15.
There exists a topological disk containing such that every connected component of is bounded.
Proposition 16.
There exists open neighborhood of such that, if is a connected component of , is the connected component of and , then .
Proposition 17.
We may assume is totally disconnected.
Proof.
Since is contained in and is a bounded topological disk, is connected. Let be the connected component of that contains , and let be ( is the filling of the set ). Note that, by a theorem of Brown–Kister ([BK84]), since , every connected component of is invariant. Since the connected components of are invariant and uniformly bounded, there exists a such that, if the orbit of has diameter larger than , then .
If is the partition of into sets of the form , if , and the connected components of , then is a upper semicontinuous decomposition of (see, for instance, proposition 1.6 of [KT13a]), and there exists a continuous surjection , and a homeomorphisms homotopic to the id such that:

is homotopic to the identity,

is a totally disconnected set,

and is a homeomorphisms between and .
Let be the lift of such that . Then there exists is such that if, and only if, there exists such that . Furthermore, if is a sequence of periodic points such that the diameter of their orbits is not uniformly bounded, then the same is true for the orbits of the corresponding sequence of –periodic points. Therefore conditions (1) and (3) also do not hold for , and if satisfies (2), then so does ∎
The previous proposition, together with Theorem 7, imply that there exists an isotopy , a closed subset and a Brouwer–Le Calvez foliation for .
Let now and be sets given by Propositions 15 and 16 and consider the compact set . For each , there exists , given by proposition 8, such that, if , then is homotopic with fixed points in to an arc transversal to the foliation crossing the leaf . Let and , be such that is a covering of and choose, for each .
Proposition 18.
There exists such that, if the orbit of has diameter greater then , then there exists and three distinct transformations in such that .
Proof.
Since is isotopic to the identity, there exists a constant such that, if the orbit of a point has diameter greater than , then there exists at least distinct integers such that whenever . Furthermore, by proposition 16, we can assume all of these points lie in . The result follows from taking ∎
Lemma 19.
There exists such that, if the orbit of has diameter greater then , then there exists and a singularity of the foliation such that the orbit of has nonzero index with both and
Proof.
By the previous proposition, if the orbit of has a sufficiently large diameter, then there exists and distinct deck transformations and such that , and we assume that where . If is the concatenation , then is a closed loop passing through the whole orbit of , and is homotopic to a closed loop positively transversal to the foliation , and crossing the leafs for . By the choice of , for each , has a nonzero winding number with either any point in or with any point in . We assume, with no loss in generality, that for any . The result follows by taking and either a singularity in or, in case is a simple closed curve, a singularity in its interior ∎
Let now be a –periodic point whose orbit has diameter greater than and consider given by the previous lemma. Let and be the isotopy given by concatenating times. Then is fixed, but is not homotopically trivial in , and the orbit of by has nonzero index with both and . Let which is homeomorphic to the open annulus , the projection, and be the induced dynamics satisfying . Let and .
Lemma 20.
is homotopic in to a homeomorphism pseudoAnosov relative to
Proof.
Let be a simple closed curve in which is neither homotopic to a topological end of , nor homotopically trivial. Note that if is another curve which is also neither homotopic to a topological end of , nor homotopically trivial, and such that is disjoint from , then is homotopic to in . Therefore, if is a reducible homeomorphism and is one of the reducing curves, must leave invariant.
Let be homotopic to relative to . We show cannot leave invariant which in turn implies, by Thurston’s classification, the stated lemma. Let be an isotopy between and leaving fixed and let be the isotopy between the identity and , and let be the lifted isotopy to . Then both and are in and furthermore Note that, since is the universal covering of commutes with , but not necessarily with every transformation in Deck(). Finally, note that the closed loops and are homotopic.
There are two possibilities to consider

is homotopically trivial in , in which case has two connected components, one which contains and and another which contains the ends of ,

is not homotopically trivial, in which case it separates the ends of , leaving in one side and in the other,
see figure 1.
First assume (1). Let be the connected component of such that is in its interior. Since is homotopically trivial in , is also homotopically trivial in , and its interior can contain either or , but not both. Since and since we now that has nonzero winding number around both and , cannot keep invariant.
Now assume (2). In this case, and are in different unbounded connected components of , and again, since the winding number of around is not zero, also cannot leave invariant. ∎
Now, since both and are fixed by , and is the lift of to the universal covering of , then both and are null. Therefore, Lemmas 12 and Lemma 20 imply that, if is sufficiently small, then there exists such that Therefore, as and commute, is a periodic point for and by Brouwer theory there exists fixed by this same map, which ends the proof of Theorem 1.
3.1. Proof of Theorem 3
Assume that is inessential. Let be a Brouwer  Le Calvez foliation for , and let be as in proposition 18.
Assume that there exists be such that , and such that the diameter of the orbit of is larger than , otherwise we are done. Since is an inessential nonwandering point, there exists a topological disk and a integer such that , and . Since has a nonwandering point, there must also exists a –periodic point in .
Let be a connected component of . Then there exists a deck transformation such that , and in particular, . If is not the identity, we are done, so assume that . Since and is nonwandering, itself is nonwandering. Let be sufficiently small such that, for every the orbit of has diameter greater than , and that . We also assume, by proposition 8, that is sufficiently small such that, if is a homeomorphism whose restriction to