Dendritations of Surfaces
In this paper we develop a generalization of foliated manifolds in the context of metric spaces. In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper semicontinuous local decompositions into dendrites. Applications are given in modeling stable and unstable sets of topological dynamical systems. For this purpose new forms of expansivity are defined.
In this paper we develop a Theory of Foliations from a viewpoint of Continuum Theory. We define and study what we call continuumwise foliations or simply cw-foliations and as a special case dendritations. These are generalizations of foliations of smooth manifolds, laminations, singular pseudo-Anosov foliations and the generalized foliations used by Hiraide to study expansive homeomorphisms. The idea is to consider monotone upper semicontinuous decompositions as local charts. We will not assume that the plaques are distributed as a product structure as in standard foliation theory. Moreover, two plaques in a common local chart have not even to be homeomorphic.
As we will show in Theorem 5.1.2, cw-foliations are a conceptual framework to understand the distribution of local stable and unstable continua in the dynamical systems that we will consider. A motivation is to classify cw-expansive surface homeomorphisms. We say that is cw-expansive (continuum-wise expansive) [Ka93] if there is such that if for all and is connected then , i.e., is a singleton. Important examples of cw-expansive dynamics are Anosov diffeomorphisms of a compact manifold of arbitrary dimension and pseudo-Anosov maps of compact surfaces with singular points and 1-prongs.
For an Anosov diffeomorphism local stable sets form foliated charts (at least , see [HPS]). If we consider an expansive homeomorphism of a compact surface, via [L, Hi], we know that local stable sets form a singular foliation. Recall that a homeomorphism is expansive if there is such that if for all then . In fact, Hiraide and Lewowicz independently proved that expansive homeomorphisms of compact surfaces are conjugate to pseudo-Anosov diffeomorphisms. If we consider cw-expansivity on a compact surface we have that local stable sets may be far from determining foliations in the standard sense. Moreover, a local stable set may not even be a finite union of arcs. For example, in [ArAnomalous], it is constructed a cw-expansive homeomorphism on a compact surface with a fixed point whose local stable set is connected but not locally connected, see §2.2.4. This example suggests that the goal of classifying all the cw-expansive surface homeomorphisms requires new technology.
In §2.1, we introduce some definitions located between expansivity and cw-expansivity that will be called cwN-expansivity. In a sense, a version of -expansivity from a viewpoint of continuum theory. Recall that, given , we say that is N-expansive [Mo12] if there is such that if (an arbitrary subset) and for all then (i.e., has at most points). We will say, see Definition 2.1.2, that a homeomorphism is cwN-expansive if there is such that if are continua, for all and for all then . Also, a homeomorphism is -expansive (where the means finite) if there is such that if are continua, for all and for all then is a finite set. These new forms of expansivity allow us to prove the local connection of stable sets and to conclude that they are dendrites. See Theorem 6.7.1. Recall that a dendrite is a Peano continuum containing no simple closed curve, a continuum is a compact connected metric space and a Peano continuum is a locally connected continuum. It is known, see Theorem 2.3.3, that such dendrites (the stable and unstable local sets mentioned above) have a uniform size, i.e., there is such that for all in the surface the stable and the unstable dendrites of meet the boundary of the disc centered at with radius . In this way we arrive naturally to the concept of dendritic decomposition, see §3.2.1.
The topic of topological decompositions has a long literature, see for example [Sm, Ro, Da, Nadler2, ChaCha]. For applications in dynamical systems the reader is referred to [ArAnomalous, CaPa, PX, BM, KTT]. A celebrated result proved by Moore in [Mo25] states that the quotient space of an upper semicontinuous decomposition of the two-dimensional sphere in non-separating continua is again the sphere (assuming that the decomposition has at least two elements, in other case the quotient is a point). In some sense, Moore’s decompositions are a generalization of zero-dimensional foliations, i.e., the decomposition in singletons. A standard foliated chart of by horizontal lines has two properties: each plaque is an arc and the quotient space is an arc. By Proposition 6.2.2, a local chart of a dendritation has the following properties: each plaque is a dendrite and the quotient space is a dendrite. The mentioned properties of a standard foliated chart do not characterize the foliated chart, see Example 6.4.6. In §6.4 we give necessary and sufficient conditions for a decomposition to be a standard foliated chart. In Corollary 6.6.3 we conclude that a continuous and -smooth dendritation is a standard foliation. The terms continuous and -smooth have a special meaning in this paper, see Definitions 6.6.1 and 3.2.14.
In [Hi89] Hiraide considered generalized foliations for the study of expansivity from a topological viewpoint (see §4.4). However, this definition seems to be useful jointly with the pseudo orbit tracing property, and not in our case. For example, pseudo-Anosov singular foliations are not generalized foliations in the sense of Hiraide and they are our main examples of dendritations. Our generalization of a foliation is designed to accompany cw-expansivity.
The concept of cwN-expansivity appears naturally in the deep study of the articles [L, Hi]. In these papers expansive homeomorphisms of surfaces are classified. A careful reading reveals that several arguments can be done assuming cw1-expansivity instead of expansivity. The concept of cw1-expansivity was previously considered in [Samba] where some topological properties of stable sets were proved. In Theorem 6.8.5 we show that every cw1-expansive homeomorphism of a compact surface is expansive. The key of this proof is Theorem 5.3.4 where we give sufficient conditions for a cw1-expansive homeomorphism of a compact metric space to be expansive. In Example 5.3.2 we show that cw1-expansivity does not imply expansivity on arbitrary compact metric spaces (this space is not locally connected). In §2.2.1 we show that there is a cw2-expansive homeomorphism of the two-dimensional sphere that is not 2-expansive. It is a pseudo-Anosov with 1-prongs.
We obtained some general results on surface dendritations. In Theorem 6.2.7 we show that for every dendritation of a closed surface there is a residual set of points without ramifications. In particular, generic leaves are one-dimensional submanifolds. This is a consequence of another result by Moore [Mo28], saying that at most a countable number of disjoint triods can be embedded in a plane. In Theorem 6.7.1 we consider a -expansive homeomorphism of a compact surface. We show that stable and unstable continua form dendritations. We also prove that: no leaf is a Peano-continuum, generic leaves are non-compact one-dimensional manifolds and in a dense subset of the surface stable and unstable leaves are topologically transverse. One can feel that this result gives a nice generic picture of what is a -expansive homeomorphism of a compact surface. However, we think that the goal of classifying surface -expansive homeomorphism is far from the present paper, not to mention cw-expansivity.
A brief sketch of the paper is as follows. In §2 we introduce new forms of expansivity and we study the topology of stable and unstable sets. In §3 we recall the main results from continuum theory that will be needed and study decompositions, the local charts of our foliations. In §4 we define and study cw-foliations on metric spaces. They are defined via atlases of upper semicontinuous decompositions. We study the induced partition of the space into leaves. The contents of §2 and §4 are independent. In §5 the results of the previous sections are joined to study the stable and the unstable cw-foliations defined by a cw-expansive homeomorphism of a metric space. In §6 we study some special cases of dendritations of surfaces. We give sufficient conditions to prove that they are (singular) foliations. Also, some properties of the stable and the unstable dendritations of a homeomorphism with some kind of expansivity are derived. Throughout the paper several open problems are given.
The author thanks the referee for the careful reading and several suggestions that helped to improve the article.
2 Variations of expansivity
We start presenting general results of cw-expansive homeomorphisms on Peano continua. In §2.1 we introduce cwN-expansivity and we summarize the main variations of expansivity that will be considered in this paper. In §2.2 we present some examples that will be used to illustrate our results in subsequent sections. In particular, in §2.2.1 we give an example of a cw2-expansive homeomorphism on the two-dimensional sphere that is not 2-expansive. In §2.3 basic topological properties of stable sets are stated. We recall the Invariant Continuum Theorem for such dynamics. In §2.4 we introduce capacitors as a tool to understand what happens with unstable continua between two close stable plates. In §2.5 we give another form of expansivity, called partial expansivity, that generalizes partial hyperbolicity of diffeomorphisms. We do not develop this concept in the paper, however we think that it could be of interest. In §2.6 we consider the relationship between stable sets and stable continua. Also, we give sufficient conditions for a cwN-expansive homeomorphism to be N-expansive. For these purposes we introduce expansivity modulo an equivalence relation.
Let be a homeomorphism of a compact metric space . We say that is a subcontinuum if is compact and connected. Denote by the space of subcontinua of
In we consider the Hausdorff metric. It is usually called hyperspace of and has the remarkable properties of being compact (if is compact) and arc-connected (provided that is connected), see [Nadler, Nadler2]. For define the sets
The continua in are called -stable and those in are -unstable. We also consider the sets
The continua in are called stable and those in are unstable.
The sets and are closed in .
Given we say that is cwN-expansive if there is such that if and then . In this case is a cwN-expansivity constant. We say that is -expansive if there is such that if and then is a finite set.
In Table 1 the main variations of expansivity considered in this paper are summarized.
In this section we explain the examples that we had in mind while developing this paper. We start with three classical families for which there are well established theories for modeling stable and unstable sets:
Anosov diffeomorphisms. These diffeomorphisms are defined on smooth manifolds and are characterized by the uniform hyperbolicity on the tangent bundle. Considering the distribution of stable and unstable sets, they are the most regular kind of expansive homeomorphisms because they form continuous foliations (see [HPS]).
Smale spaces. These are expansive homeomorphisms of compact metric spaces with local product structure (equivalently, canonical coordinates or pseudo-orbit tracing property). The category includes Smale’s basic sets of Axiom A diffeomorphisms. Stable and unstable sets can be modeled with Hiraide’s generalized foliations (see §4.4).
Pseudo-Anosov diffeomorphisms. These diffeomorphisms are defined on compact surfaces. They have local product structure except for a finite number of points called singular. Stable and unstable sets form singular foliations (see for example [Hi]).
Next we describe other examples that will be essential in the development of the paper. Except for the first one, we assume that the reader is familiar with Smale’s derived from Anosov diffeomorphisms (which in particular is an interesting example of a Smale space).
2.2.1 A pseudo-Anosov with 1-prongs
The dynamics of pseudo-Anosov diffeomorphisms is not simple, at least from author’s viewpoint. In this section we wish to discuss in detail some properties of a special example, a pseudo-Anosov with 1-prongs of the sphere. \textcolorblackThis example has been considered several times in the literature. It has some properties that may not be easy to predict at first sight. In [WaltersET]*Example 1, p. 140 Walters considered it to show that a factor of an expansive homeomorphism may not be expansive. In [PaPuVi]*§2.4 it is proved that the local stable set of some points is not locally connected. In [PaVi] it is shown that it is not entropy expansive, in fact they show that there are arbitrarily small horseshoes. We will show that it is cw2-expansive but not -expansive, for all .
blackThe example is as follows. Let be the two dimensional torus where . Consider the equivalence relation for . The quotient space is a two-dimensional sphere . Denote by the canonical projection. On the torus consider the Anosov diffeomorphism defined by . Define by for all . \textcolorblackFor a more detailed construction the reader is referred to the works mentioned above.
blackThe homeomorphism is cw2-expansive.
Denote by and the stable and the unstable singular foliations of , respectively. These are transverse foliations except at the singularities. Singular points are 1-prongs and the foliations looks as in Figure 1. Then, a small arc of the stable foliation intersects in at most two points an unstable arc. Thus, the proof is reduced to show that every stable continuum is contained in a stable leaf. Arguing by contradiction, let be a stable continuum that it is not contained in a stable leaf. Then there is a hyperbolic periodic point such that , \textcolorblackwhere denotes the stable leaf trough . Since is hyperbolic we see that cannot go to zero as because is not contained in . This contradiction proves that every stable continuum is contained in a stable leaf. ∎
As usual we define local stable and unstable sets as
respectively. The next result can be derived from [PaVi], however, since we think that more details can be given, a proof is included. The author learned this proof from J. Vieitez and J. Lewowicz. We will use the notation for the closure of .
For all there is a Cantor set such that for all , in particular is not -expansive for all .
Let be a 1-prong of . Take such that the orbit is dense in . A point with this property will be called transitive. Consider . We will show that contains a Cantor set contained in the unstable arc of . Consider such that there is with . We have that . Also we can assume that is in the unstable arc of , see Figure 1. Since and are in a stable arc and is transitive we have that is transitive too.
Denote by . For each take a transitive point in the unstable arc of with . Define as the set containing and these two new points. Again, for each take a transitive point in the unstable arc of with . Inductively we define a sequence of sets with . By construction each is contained in . Denote by . Since is closed we have that . Also, has no isolated point and it cannot contain an arc because is cw-expansive (in fact cw2-expansive as we proved). Then, is a Cantor set contained in the unstable arc of and in . ∎
2.2.2 \textcolorblackQuasi-Anosov diffeomorphisms
By definition, a quasi-Anosov diffeomorphism is a diffeomorphism of a compact manifold such that is unbounded for every non-vanishing tangent vector , where is the differential of and is the norm induced by a Riemannian metric on . They were characterized by Mañé [Ma75] as Axiom A diffeomorphisms with quasi-transverse stable and unstable spaces. Recall that Axiom A means that the non-wandering set is hyperbolic and periodic points are dense in the non-wandering set. For an Axiom A diffeomorphism, at every point the tangent space contains a contracting subspace and an expanding subspace . The quasi-transversality condition means that for all .
We proceed to sketch the construction of a particular quasi-Anosov diffeomorphism that is not Anosov [FR]. Consider two derived from Anosov diffeomorphisms , , where is an -torus, is conjugate to and presents a codimension one shrinking repeller and a sink fixed point . Consequently, presents a codimension one expanding attractor and a source fixed point . Let , , be an open ball around such that and . Consider the manifolds with boundary and a diffeomorphism such that is a closed manifold and there is a diffeomorphism extending the dynamics of and .
In [FR] it is proved that for , there is a diffeomorphism making quasi-Anosov. As we said, the non-wandering set is hyperbolic. At wandering points stable and unstable manifolds are one-dimensional and its tangent lines are quasi-transverse. Since is 3-dimensional, they are not transverse and is not Anosov.
2.2.3 \textcolorblackQ-Anosov diffeomorphisms
In [ArRobNexp] the construction of §2.2.2 was considered for the simpler case . On a surface there is not enough space to construct a quasi-Anosov diffeomorphism (not being Anosov). However, the construction can be performed. In this case the map will introduce tangencies between stable and unstable manifolds at wandering points. Assuming that is of class , if these tangencies are of order at most we say that is Q-Anosov. In [ArRobNexp] it is shown that the set of Q-Anosov diffeomorphisms of a closed surface is an open set (in the topology) of -expansive diffeomorphisms, where -expansive means -expansive with . These examples are Axiom A and show that -expansivity does not imply -expansivity.
2.2.4 \textcolorblackAnomalous cw-expansivity
In [ArAnomalous] it was considered another variation of the quasi-Anosov. On a surface, as in §2.2.3, it is defined a cw-expansive homeomorphism with what we called an anomalous saddle. It is a fixed point whose stable set is connected but not locally connected. Let us briefly give a description. Let be the set . It is a countable union of vertical segments. Consider the auxiliary map defined as . Define and . The anomalous saddle in this case is the origin. In [ArAnomalous] it is constructed a homeomorphism around such that its local stable set is and the unstable set is (intersected with a neighborhood of the origin). This anomalous saddle is inserted in a -Anosov diffeomorphism of the previous section. In the example, the intersection of local stable and unstable sets is totally disconnected, thus implying cw-expansivity.
2.3 Invariant continua
Our interest is centered at dynamical systems on surfaces, but some fundamental results on cw-expansivity hold for homeomorphisms on Peano continua, i.e., a connected, locally connected and compact metric space. In this setting, Theorem 2.3.3 plays the role of the Invariant Manifold Theorem [HPS] for hyperbolic diffeomorphisms on smooth manifolds.
The following result is a characterization of cw-expansivity in terms of stable and unstable continua. \textcolorblackThe direct part is known [Ka93]*§2. The converse may be new, however it is quite direct from the definitions.
A homeomorphism of a compact metric space is cw-expansive if and only if there is such that
for all there is such that
for and .
Direct. Let be such that if is connected and for all then is a singleton. Suppose that and take , that is, for all . If this diameter does not converge to 0 then there are and such that for all . Since is compact we can assume that in the Hausdorff metric. We will show that
Since , given there is such that for all . Then, for all . Since and we conclude (2). Since we have that is not a singleton. This contradicts the cw-expansivity of and proves that .
To prove the next part we argue by contradiction. Suppose that there is and a sequence such that and for all where . Note that and that we can assume that . Since there is such that for all . We can take a subcontinuum and another divergent sequence such that and for all . \textcolorblackThen, arguing as in the proof of (2), we conclude that a limit continuum of contradicts the cw-expansivity.
Converse. Let us show that is a cw-expansivity constant. Suppose that for all . Assume, by contradiction, that there is For this value of there is such that . Since and there is such that . Then . As we have that . This is a contradiction that finishes the proof. ∎
blackThe next theorem states the existence of non-trivial stable and unstable continua through each point of the space. Moreover, these continua have diameter bounded away from zero. Their invariance is in fact given by Proposition 2.3.1. The result was first proved independently by Hiraide and Lewowicz in [Hi, L] for the classification of expansive homeomorphisms on compact surfaces. It was generalized by Kato in [Ka2]*Theorem 1.6 for cw-expansivity. A partial result was previously given by Mañé in [Ma] to prove that minimal expansive homeomorphisms can only exist on totally disconnected spaces.
For the study of cw-expansive homeomorphisms it is essential to assume some kind of connection of the space. To illustrate this point consider that every homeomorphism of a Cantor set is cw-expansive. It turns that local connection is a good property to exploit the cw-expansivity. There is a minor loss of generality if, in addition, we assume that the space is a Peano continuum. Indeed, if is a locally connected compact metric space then it has a finite number of components. Therefore, a homeomorphism of will permute this components and taking a power of we will have a finite number of homeomorphisms of Peano continua.
For a set , as and , define .
Theorem 2.3.3 (Invariant Continuum Theorem).
If is a cw-expansive homeomorphism of a Peano continuum then for all there is such that
for all and .
This theorem has the following direct consequences.
Remark 2.3.4 (Uniform size of stable continua).
Note that (3) implies that for all there are stable and unstable continua through \textcolorblackof diameter greater than . Consequently, these continua meet the boundary of the ball .
Remark 2.3.5 (No stable points).
From the Invariant Continuum Theorem we see that if is a cw-expansive homeomorphism of a Peano continuum then neither stable nor unstable continua have interior points. This is because if is a stable set with an interior point then we can take an unstable continuum contained in that contradicts the cw-expansivity. In particular there are no Lyapunov stable trajectories.
Remark 2.3.6 (Surfaces with boundary).
Let us explain why we do not consider surfaces with boundary. Suppose that is a cw-expansive homeomorphism of a compact surface with boundary. By Brouwer’s Theorem on the Invariance of Domain [HW] we know that is invariant by . Then, the restriction is cw-expansive. This gives a contradiction because, on one hand there are non-trivial stable continua in , and on the other hand every non-trivial continuum of has interior points (relative to ). This is the argument to prove that the circle admits no expansive homeomorphisms that the author learned from Lewowicz.
Do cw-expansive homeomorphisms of compact manifolds with non-empty boundary exist? For example, does there exist a cw-expansive homeomorphism of the 3-ball? Does the 3-sphere admit expansive or cw-expansive homeomorphisms?
We say that \textcolorblackseparates if is not connected. We will show in Theorem 2.3.9, under a natural assumption on the Peano space , that no stable set separates. Previously we prove a topological lemma.
If is a continuum and every point has arbitrarily small neighborhoods with connected boundary then for all there is such that if is a closed set that separates and then there is a component of with .
Given consider a finite open cover of such that for all and is connected. Take such that if then there is such that . Suppose that is closed, separates and . Take such that .
Let us show that is connected. By contradiction, suppose that a union of disjoint, closed, non-empty sets. Since is connected, we can assume that . Then, and disconnects , a contradiction that proves that is connected.
Let be the component of containing . Since separates , there is at least another component. This component is contained in and has diameter smaller than . ∎
If is a cw-expansive homeomorphism of the Peano continuum and every point of has arbitrarily small neighborhoods with connected boundary then no stable closed set separates .
Arguing by contradiction assume that is a closed stable set separating . \textcolorblackAs is a homeomorphism, every iterate of separates . \textcolorblackSince it is stable, taking a positive iterate, we can suppose that it is as small as we want. By Lemma 2.3.8 there is a small component of its complement. By Theorem 2.3.3 each point of has a stable continuum meeting . \textcolorblackSince we have that is stable, therefore, is a stable set. Since has interior points we have a contradiction with Remark 2.3.5. ∎
If a Peano continuum has no locally separating points then every point has arbitrarily small neighborhoods with connected boundary. See [Jo] for a proof.
Theorem 2.3.9 holds if is a compact manifold with or without boundary. In the one-dimensional case, intervals \textcolorblackdo not have connected boundary, but since there are not cw-expansive homeomorphisms on one-dimensional manifolds the theorem holds true.
The following example shows that in Theorem 2.3.9 we need to assume that every point has arbitrarily small neighborhoods with connected boundary.
Consider two copies of an Anosov diffeomorphism of the two-dimensional torus identifying two fixed points. The gluing point has not arbitrarily small neighborhoods with connected boundary. Also, this point forms a stable set and it (locally and globally) separates.
The following technical definition is based on the arguments of the proof of [L]*Lemma 2.3.
Given , , an -capacitor is a triple such that:
, and are disjoint continua, , is open,
there is a continuum meeting and ,
In this case and are the plates of the capacitor, is the separation of the plates and is the radius. See Figure 2.
We say that a capacitor has stable plates if are stable sets for the homeomorphism .
In the next result we study unstable continua between two close stable plates. It generalizes [L]*Lemma 2.3 and it will be applied in Theorem 6.7.1 for the study of -expansivity on surfaces.
Assume that is a cw-expansive homeomorphism of a Peano continuum . Then, for all small there is such that if is an -capacitor \textcolorblackwith stable plates then:
for all there is an unstable continuum from to contained in ,
there is an unstable continuum meeting and .
Arguing by contradiction assume that there are , a sequence of -capacitors with stable plates and with no unstable continuum from to contained in . Since is a Peano continuum, by Theorem 2.3.3 for each there is an unstable continuum containing and intersecting the boundary of . Since we conclude that . Taking subsequences, we can assume that and in the Hausdorff metric. By definition of capacitor we have that which implies that . Therefore is a continuum that is stable and unstable. Since we have a contradiction with the cw-expansivity of .
Given consider satisfying the first item. Let be an -capacitor. By definition, there is a continuum \textcolorblackand points with and . Define
The sets and are non-empty because they contain and respectively. They are closed sets and, as we have shown, they cover . Since is connected, there is a point . ∎
Let be an expansive homeomorphism of a compact three-manifold. In [Vi2002] it is shown that local stable sets are locally connected if is smooth and without wandering points. By Theorem 2.4.2 we know that a stable set of cannot be homeomorphic to the closure of the set
2.5 \textcolorblackPartial expansivity
Consider a homeomorphism of a compact metric space. We will use some concepts of topological dimension. We refer the reader to [HW] for the definitions and basic properties.
Given an integer , we say that is partially expansive with central dimension and expansivity constant if for every non-trivial compact set with there is such that .
As usual, we say that is sensitive to initial conditions if there is such that for all and for all there are and such that .
For a homeomorphism the following hold:
is expansive if and only if it is partially expansive with ,
is cw-expansive if and only if it is partially expansive with ,
if in addition is a compact manifold of dimension then is sensitive to initial conditions if and only if is partially expansive with central dimension .
Sketch of the proof.
Since the arguments are quite direct we only give the details that we consider more relevant. To prove the first part, note that by definition (see [HW]) the condition means . Then, with is non-trivial if and only if it has at least two points. To conclude the stated equivalence one has to note that .
The statement related to cw-expansivity follows because positive dimension is equivalent to contain a non-trivial continuum.
For the last part we recall [HW]*Corollary 1, p. 46 that if is a compact -dimensional manifold and then if and only if has non-empty interior. ∎
Let be a continuous flow. We consider the following weak form of expansivity. We say that a flow is separating [ArKinExp] if there is (a separating constant) such that if for all then for some . Examples of separating flows are expansive flows in the sense of Bowen-Walters [BW] and -expansive flows as defined by Komuro [Ko84].
If is a separating flow then for all the homeomorphism is partially expansive with .
Let be a separating constant and take such that if is a continuum and then for all . Then, if for all we have that for all . Therefore, is an orbit segment and consequently . ∎
2.6 Relative expansivity
Given a homeomorphism it is usual to define the stable set of as
and the unstable set
Consider the equivalence relation , , if there is a continuum such that and as . Similarly we define (taking ). The equivalence class of will be denoted as (and ). We will give conditions that allow us to prove that and .
Let be a partition of . We say that is separating mod if there is such that if for all then
If is separating mod then .
Assume that is separating mod with as in the definition. Suppose that as . Take such that for all it holds that . Then there is a stable continuum containing . This implies that are in the stable continuum . Then . ∎
Assuming that is cw-expansive does the condition imply that it is separating mod ?
The pseudo-Anosov with 1-prongs on the two-dimensional sphere given in §2.2.1 is not separating mod .
blackBy Proposition 2.2.2, for all there is a Cantor set contained in an unstable arc and contained in for some in this unstable arc. Since cuts the unstable arc in a countable set, there is that is not in . This proves that is not separating mod . ∎
Does the example in §2.2.1 satisfy ? The solution could be simple but we were not able to solve it.
We say that is expansive mod if for all there is such that if for all then and are in a common -stable continuum.
Note that every homeomorphism expansive mod is separating mod .
If is a cwN-expansive homeomorphism of a Peano continuum and is expansive mod and is expansive mod then is N-expansive.
Let be a cwN-expansivity constant for and take . Consider from Definition 2.6.6 and suppose that for all . For we have that for all . Then, there is an -stable continuum containing . We have that there is a -stable continuum containing . Similarly, there is a -unstable continuum containing . Since is a cwN-expansivity constant for we have that . This proves that is an N-expansivity constant for . ∎
It seems that pseudo-Anosov diffeomorphisms of surfaces without 1-prongs are expansive mod . We know that the pseudo-Anosov with 1-prongs of §2.2.1 is not expansive mod . Are the examples in §2.2.4 and §2.2.3 expansive mod ? It would be interesting to understand which cw-expansive surface homeomorphisms are expansive mod .
3 Continuum theory and decompositions
In this section we review some results from continuum theory that will be used throughout the article. Also, we study decompositions, that will play the role of foliated charts in the next section.
3.1 \textcolorblackBackground on Continuum Theory
We recall that a continuum is a compact connected metric space. General references for continuum theory are [Nadler2, Kur1, Kur].
3.1.1 Partitions and monotone restrictions
Let be a compact metric space and denote by the set of subsets of . A partition of is a function such that:
for all ,
if and only if and
and implies .
A partition is monotone if each is connected. Given and denote by the component of containing . For a partition and define the monotone restriction as
for all .
If is a partition of and then
From the definition (4) we see that we have to show that
for all . To prove the inclusion consider a connected set such that . Since we have that . Then, because . Given that we conclude that . Then . The converse inclusion is easier to prove. It follows from the fact that . ∎
We say that a partition is upper semicontinuous if for all and every open set containing there is a neighborhood of such that if then .
is upper semicontinuous if and only if given such that in the Hausdorff metric then . The upper semicontinuity of implies that each is a closed subset of .
We say that a partition is continuous at if for every we have that in the Hausdorff metric. We say that is continuous if it is continuous at every . A set is residual if it is a countable intersection of open and dense subsets of .
If is an upper semicontinuous partition of then:
[Kur]*p. 70-71 there is a residual subset such that is continuous at every ,
[Nadler2]*Theorem 3.9 if in addition is a continuum then with its quotient topology is a continuum.111It is clear that is compact and connected since it is the quotient of the continuum . In [Nadler2]*Theorem 3.9 it is shown that is metrizable.
3.1.2 Local connection
A continuum is hereditarily locally connected if every subcontinuum is locally connected. A convergence continuum of a compact metric space is a non-trivial subcontinuum of for which there is a sequence of continua such that in the Hausdorff metric, and for all .
Theorem 3.1.4 ([Nadler2]*Theorem 10.4).
A continuum is hereditarily locally connected if and only if contains no convergence continuum.
Theorem 3.1.5 ([Hy]*Theorem 3-17).
Every connected, locally connected, complete metric space is arc-connected.
Theorem 3.1.6 (Sierpiński’s Theorem [Kur1]*p. 218).
A continuum is Peano if and only if for all there is a finite cover of by connected sets of diameter less than .
A continuum is unicoherent if given subcontinua such that then is connected. A continuum is hereditarily unicoherent if every subcontinuum is unicoherent. To show the difference between these concepts and for future reference let us recall the following result.
Theorem 3.1.7 (Janisewski’s Theorem [Kur]*p. 506).
The union of two subcontinua of the 2-sphere disconnects the sphere if and only if is disconnected.
By Janisewski’s Theorem we see that the 2-sphere is a unicoherent continuum. It is not hereditarily unicoherent because it contains a circle, which is not unicoherent.
A dendrite is a Peano continuum containing no simple closed curve. The points of a dendrite are classified as: end point if its complement is connected, ramification or branch point if its complement has at least three components, regular point if its complement has two components. The following results from [Nadler2]*§10 summarizes several properties of dendrites.
Every subcontinuum of a dendrite is a dendrite.
The set of all the ramification points of a dendrite is countable.
If is a compact metric space then the following statements are equivalent:
is a dendrite,
is a hereditarily unicoherent Peano continuum,
is connected and any two points of the continuum are separated by a third point.
Every dendrite can be embedded in the plane. Moreover, the Wazewski’s universal dendrite is a dendrite in which contains a topological copy of any dendrite.
The standard theory of foliations is based on a special kind of local partitions in plaques, i.e., foliated charts. On a surface , a foliated chart is a homeomorphism where is an open subset. If then the plaques of are for all . We can define a map as . This map is a continuous monotone partition of .
For a pseudo-Anosov diffeomorphism, the stable and unstable sets form singular foliations. At a singular point, an -prong with , it is not possible to define a local chart as above where the plaques are the local stable sets. The partition in local stable sets in a neighborhood of a singularity is monotone and upper semicontinuous. The (full) continuity is lost at the singularity. This is the idea we have in mind for the next definition of decomposition, which is a standard concept in continuum theory.
Given a compact metric space , we will consider a compact subset . We can think that is the closure of the open set in the surface considered above. However, the theory is developed in such a way that may not be connected and may have empty interior.
A decomposition of is a monotone upper semicontinuous partition . The sets are the plaques of the decomposition.
Example 3.2.2 (Extremal examples).
For a compact metric space define and . It holds that and are decompositions of , a proof can be found in [Mo25]*Theorem 24. For every decomposition of we have that for all .
For standard foliations, the restriction of a foliated chart to an arbitrary subset may not be a foliated chart. This is because the product structure may be lost. Next we show that the restriction of a decomposition is a decomposition.
If is a decomposition of the compact set and is compact then is a decomposition.
By definition (4) is monotone. Suppose that and . Taking a subsequence we can also assume that for some subcontinuum . Since and is upper semicontinuous we have that and, as is a continuum contained in , we conclude that . This proves that is upper semicontinuous. ∎
Let be a continuum and consider a compact metric space . Define and as . If we consider the product topology on then is a continuous decomposition of .
Given two decompositions , , we say that and are equivalent if there is a homeomorphism such that for all . We say that a decomposition is a product structure if it is equivalent to a decomposition as in Example 3.2.4.
A decomposition of a continuum is a product structure if and only if there is a decomposition of such that
for all .
If is a product structure, we can assume that . Since is a continuum we have that and are continua. If we can define . Then
for all .
To show the converse note that by Proposition 3.1.3 we know that and are continua. Consider defined by