# Continuum-wise hyperbolic homeomorphisms on surfaces

Rodrigo Arruda  ,  Bernardo Carvalho  and  Alberto Sarmiento
<EMAIL_ADDRESS>
###### Abstract.

This paper discusses the dynamics of continuum-wise hyperbolic surface
homeomorphisms. We prove that $cw_{F}$-hyperbolic surface homeomorphisms
containing only a finite set of spines are $cw_{2}$-hyperbolic. In the case of
$cw_{3}$-hyperbolic homeomorphisms we prove the finiteness of spines and,
hence, that $cw_{3}$-hyperbolicity implies $cw_{2}$-hyperbolicity. In the
proof, we adapt techniques of Hiraide [Hiraide1987ExpansiveHO] and Lewowicz
[Lewowicz1989ExpansiveHO] in the case of expansive surface homeomorphisms to
prove that local stable/unstable continua of $cw_{F}$-hyperbolic
homeomorphisms are continuous arcs. We also adapt techniques of Artigue,
Pacífico and Vieitez [Artigue2013NexpansiveHO] in the case of N-expansive
surface homeomorphisms to prove that the existence of spines is strongly
related to the existence of bi-asymptotic sectors and conclude that spines are
necessarily isolated from other spines.

###### Key words and phrases:

cw-hyperbolicity, classification, spines

2020 _Mathematics Subject Classification_ : Primary 37B45; Secondary 37D10.

##  1\. Introduction and statement of results

In the classification of expansive homeomorphisms on surfaces, Lewowicz and
Hiraide [Lewowicz1989ExpansiveHO, Hiraide1987ExpansiveHO] prove that
expansiveness implies the homeomorphism is pseudo-Anosov. The hypothesis in
the dynamics is only expansiveness, the space is a surface, and from these a
complete understanding of the structure of local stable/unstable sets is
obtained: they are subsets of $C^{0}$ transversal singular foliations with a
finite number of singularities. A classification of topologically hyperbolic
homeomorphisms on surfaces was also obtained [hiraide2toro]. Expansive surface
homeomorphisms satisfying the shadowing property are conjugate to an Anosov
diffeomorphism of the Torus $\mathbb{T}^{2}$. Expansiveness and shadowing
together define a local product structure that is continuous and rule out the
presence of singularities on local stable/unstable sets.

With the study of dynamics beyond topological hyperbolicity (see
[antunes2023firsttime, Artigue2019CountablyAE, ARTIGUE20203057,
artigue2020continuumwise, CARVALHO20163734, carvalho2019positively]), which is
being developed as the understanding of generalizations of hyperbolicity from
a topological perspective, we could try to extend the reach of these
techniques to distinct and more general scenarios. This idea was first
explored by Artigue, Pacífico and Vieitez [Artigue2013NexpansiveHO] in the
study of N-expansive homeomorphisms on surfaces. They exhibit an example of a
2-expansive homeomorphism on the bitorus that is not expansive, and prove that
non-wandering 2-expansive homeomorphisms on surfaces are, indeed, expansive.
The techniques of Lewowicz/Hiraide are applied to study the structure of bi-
assymptotic sectors, which are discs bounded by a local stable and a local
unstable arc. Bi-asymptotic sectors naturally appear in the dynamics of
2-expansive homeomorphisms, and it is proved in [Artigue2013NexpansiveHO] that
these sectors should be contained in the wandering part of the system. The
techniques seem to be restricted to the case of 2-expansive homeomorphisms and
extend them to 3-expansiveness seems complicated. In particular, use them to
understand the dynamics of more general continuum-wise expansive
homeomorphisms, introduced by Kato in [kato_1993], seems difficult.

However, in the study of cw-expansive homeomorphisms a recent work of Artigue,
Carvalho, Cordeiro and Vieitez [artigue2020continuumwise] discussed the
continuum-wise hyperbolicity, assuming that local stable and unstable continua
of sufficiently close points of the space intersect (see Definition 2.1). Cw-
hyperbolic systems share several important properties with the topologically
hyperbolic ones, such as the L-shadowing property [ARTIGUE20203057] and a
spectral decomposition theorem [artigue2020continuumwise], but a few important
differences are noted on the pseudo-Anosov diffeomorphism of $\mathbb{S}^{2}$:
the existence of stable/unstable spines, bi-asymptotic sectors, cantor sets in
arbitrarily small dynamical balls, and a cantor set of distinct arcs in local
stable/unstable sets.

In this paper we start to adapt the techniques of Lewowicz/Hiraide to the
study of cw-hyperbolic homeomorphisms on surfaces. One important hypothesis in
our results is cwF-expansiveness, that asks for a finite number of
intersections between any pair of local stable and local unstable continua
(see Definition 2.1). We do not know an example of a cw-hyperbolic surface
homeomorphism that is not cwF-expansive, so the study of cwF-hyperbolicity on
surfaces seems to be the perfect first step in the theory. In our first
result, we prove that local stable/unstable continua of cwF-hyperbolic surface
homeomorphisms are arcs (see Proposition 2.12). CwF-hyperbolicity is important
in this step since in [artigueanomalous] there is a cw-expansive surface
homeomorphism with non locally connected local stable continua. But even in
the case they are locally connected, they are only assured to be contained in
dendritations as proved in [artigue_2018]. Section 2 is devoted to prove that
they are arcs and this is done in a few important steps that are based in the
ideas of Lewowicz and Hiraide.

In our second result we relate bi-asymptotic sectors and spines in a way that
every regular sector contains a single spine and every spine is contained in a
regular bi-asymptotic sector. The notion of regular bi-asymptotic sector is
defined and pictures of non-regular sectors are presented. We give a complete
description of the structure of local stable and local unstable continua
inside a regular bi-asymptotic sector. We also prove that every bi-asymptotic
sector contains a regular bi-asymptotic sector and, hence, a spine, and that
non-regular sectors contain at least two distinct spines. All these results on
sectors and spines are proved in Section 3 and allow us to conclude that the
spines of a cwF-hyperbolic surface homeomorphism are isolated from other
spines and, hence, we obtain that there is at most a countable number of them.

In Section 4 we prove our main result using all the techniques developed in
the previous sections. The hypothesis of the existence of at most a finite
number of spines is important and will be discussed. We do not know examples
of cw-hyperbolic surface homeomorphisms with an infinite number of spines, so
this hypothesis also seems reasonable. The following is the main result of
this article:

######  Theorem 1.1.

If a cwF-hyperbolic surface homeomorphism has only a finite number of spines,
then it is cw2-hyperbolic. Cw3-hyperbolic surface homeomorphisms have at most
a finite number of spines, and are cw2-hyperbolic.

We note that in [artigue2020continuumwise] it is proved that the product of
$n$ copies of the pseudo-Anosov diffeomorphism of $\mathbb{S}^{2}$ is
cw${}_{2^{n}}$-hyperbolic but is not cw${}_{2^{n}-1}$-expansive. Thus, the
hypothesis on the space being a surface is important for our main result. We
state the following questions that follow naturally from our results:

######  Question 1.

Does there exist a $cw$-hyperbolic surface homeomorphism that is not
$cw_{2}$-hyperbolic?

######  Question 2.

Does $cw_{F}$-hyperbolicity on surfaces imply finiteness on the number of
spines and, hence, $cw_{2}$-hyperbolicity?

######  Question 3.

Does $cw$-hyperbolicity on surfaces imply local connectedness of
$C^{s}_{\varepsilon}(x)$?

######  Question 4.

Can we adapt the techniques of this paper to prove that 3-expansive surface
homeomorphisms are 2-expansive?

##  2\. Local stable/unstable continua are arcs

We begin this section with some precise definitions. Let $(X,d)$ be a compact
metric space and $f\colon X\to X$ be a homeomorphism. We consider the _c-
stable set_ of $x\in X$ as the set

$W^{s}_{c}(x):=\\{y\in X;\,\,d(f^{k}(y),f^{k}(x))\leq c\,\,\,\,\textrm{for
every}\,\,\,\,k\geq 0\\}$

and the _c-unstable set_ of $x$ as the set

$W^{u}_{c}(x):=\\{y\in X;\,\,d(f^{k}(y),f^{k}(x))\leq c\,\,\,\,\textrm{for
every}\,\,\,\,k\leq 0\\}.$

We consider the _stable set_ of $x\in X$ as the set

$W^{s}(x):=\\{y\in X;\,\,d(f^{k}(y),f^{k}(x))\to
0\,\,\,\,\textrm{when}\,\,\,\,k\to\infty\\}$

and the _unstable set_ of $x$ as the set

$W^{u}(x):=\\{y\in X;\,\,d(f^{k}(y),f^{k}(x))\to
0\,\,\,\,\textrm{when}\,\,\,\,k\to-\infty\\}.$

We denote by $C^{s}_{c}(x)$ the $c$-stable continuum of $x$, that is the
connected component of $x$ on $W^{s}_{c}(x)$, and denote by $C^{u}_{c}(x)$ the
$c$-unstable continuum of $x$, that is the connected component of $x$ on
$W^{u}_{c}(x)$.

######  Definition 2.1.

We say that $f$ is $cw$-expansive if there exists $c>0$ such that

$W^{s}_{c}(x)\cap W^{u}_{c}(x)\quad\text{ is totally disconnected }$

for every $x\in X$. A $cw$-expansive homeomorphism is said to be
$cw_{F}$-expansive if there exists $c>0$ such that

$\\#(C^{s}_{c}(x)\cap C^{u}_{c}(x))<\infty\quad\text{ for every }\quad x\in
X.$

Analogously, $f$ is said to be $cw_{N}$-expansive if there is $c>0$ such that

$\\#(C^{s}_{c}(x)\cap C^{u}_{c}(x))\leq N\quad\text{ for every }\quad x\in X.$

We say that $f$ satisfies the $cw$-local-product-structure if for each
$\varepsilon>0$ there exists $\delta>0$ such that

$C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(y)\neq\emptyset\quad\text{
whenever }\quad\operatorname{\textit{d}}(x,y)<\delta.$

The $cw$-expansive homeomorphisms (resp. $cw_{F}$, $cw_{N}$) satisfying the
$cw$-local-product-structure are called $cw$-hyperbolic (resp. $cw_{F}$,
$cw_{N}$).

The main examples of cw-hyperbolic surface homeomorphism are the Anosov
diffeomorphisms (or more generally the topologically hyperbolic
homeomorphisms), and the pseudo-Anosov diffeomorphism of $\mathbb{S}^{2}$. The
sphere $\mathbb{S}^{2}$ can be seen as the quotient of $\mathbb{T}^{2}$ by the
antipodal map, and thus any 2x2 hyperbolic matrix $A$ with integer
coefficients and determinant one induces diffeomorphisms $f_{A}$ on
$\mathbb{T}^{2}$ and $g_{A}$ on $\mathbb{S}^{2}$. The diffeomorphism $f_{A}$
is Anosov and, hence, cw1-hyperbolic, while $g_{A}$ is cw2-hyperbolic but not
cw1-expansive (see [Artigue2019CountablyAE, artigue2020continuumwise,
ARTIGUE20203057] for more details). We recall some known properties of local
stable/unstable continua for cwF-hyperbolic homeomorphisms. Most of them hold
assuming that $X$ is a Peano continuum, that is a compact, connected, and
locally connected metric space, but we assume from now on that $f\colon S\to
S$ is a cwF-hyperbolic homeomorphism of a closed surface $S$. We will use the
symbol $\sigma$ to denote both $s$ and $u$. Since this will appear several
times in what follows, we will not write in all of them that some statement
holds for $\sigma=s$ and $\sigma=u$, we will simply say that it holds for
$\sigma$.

######  Lemma 2.2 ([katoconcerning] Thm. 1.6).

For every $\varepsilon>0$, there exists $\delta>0$ such that

$\operatorname{diam}(C^{\sigma}_{\varepsilon}(x))>\delta\,\,\,\,\,\,\text{for
every}\,\,\,\,\,\,x\in S,$

where $\operatorname{diam}(A)=\sup\\{d(a,b);a,b\in A\\}$ denotes the diameter
of the set $A$.

######  Corollary 2.3.

$\operatorname{Int}C^{\sigma}_{\varepsilon}(x)=\emptyset$ for every $x\in S$
and $\varepsilon\in(0,\frac{c}{2})$.

###### Proof.

The following is the proof for $\sigma=s$, but for $\sigma=u$ the proof is
analogous. By contradiction, assume that
$y\in\operatorname{Int}C^{s}_{\varepsilon}(x)\neq\emptyset$ for some $x\in X$
and $\varepsilon<\frac{c}{2}$. Since
$\operatorname{diam}C^{u}_{\varepsilon}(y)>\delta$ for some $\delta>0$, then
$C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(y)$ contains a non-trivial
continuum. By the choice of $\varepsilon$, we have that
$C^{u}_{\varepsilon}(y)\subset C^{u}_{c}(y)$, then $C^{s}_{c}(x)\cap
C^{u}_{c}(x)$ contains a non-trivial continuum, which contradicts the
$cw$-expansiveness. ∎

Let $\mathcal{C}$ denote the space of all sub-continua of $S$ and
$\mathcal{C}_{\delta}$ denote the set of all sub-continua of $S$ with diameter
smaller than $\delta$. Let $\mathcal{C}^{s}$ and $\mathcal{C}^{u}$ denote the
set of all stable and unstable continua of $f$, respectively. More precisely,

$\mathcal{C}^{s}=\\{C\in\mathcal{C};\,\,\,\operatorname{diam}(f^{n}(C))\to
0,\,\,\,n\to\infty\\}$
$\mathcal{C}^{u}=\\{C\in\mathcal{C};\,\,\,\operatorname{diam}(f^{n}(C))\to
0,\,\,\,n\to-\infty\\}.$

The following lemma is actually a characterization of cw-expansiveness with
respect to local stable/unstable continua.

######  Lemma 2.4 ([artigue_2018] Prop. 2.3.1).

  1. (1)

There exists $\varepsilon^{*}>0$ such that
$C^{\sigma}_{\varepsilon^{*}}\subset C^{\sigma}$,

  2. (2)

For all $\varepsilon>0$ there exists $\delta>0$ such that $C^{\sigma}\cap
C_{\delta}\subset C^{\sigma}_{\varepsilon}$.

The following lemma is similar to Lemma 4.1 in [Hiraide1987ExpansiveHO] but
assuming cw-expansiveness. Let $B_{\delta}(x)$ denote the ball of radius
$\delta$ centered at $x$, that is the set of points whose distance to $x$ is
less than $\delta$.

######  Lemma 2.5.

For each $0<\varepsilon<\frac{\varepsilon^{*}}{4}$ there exists
$\delta\in(0,\varepsilon)$ such that

$C^{\sigma}_{\varepsilon}(x)\cap
B_{\delta}(x)=C^{\sigma}_{2\varepsilon}(x)\cap B_{\delta}(x)$

for every $x\in S$.

###### Proof.

For each $0<\varepsilon<\frac{\varepsilon^{*}}{4}$ let
$\delta^{*}\in(0,\varepsilon)$ be given by Lemma 2.4 and
$\delta=\frac{\delta^{*}}{2}$. Since $\varepsilon<\frac{\varepsilon^{*}}{4}$,
it follows that

$C^{\sigma}_{\varepsilon}(x)\subset C^{\sigma}_{2\varepsilon}(x)\in
C^{\sigma}.$

The choice of $\delta$ ensures that $C^{\sigma}_{2\varepsilon}(x)\cap
B_{\delta}(x)\subset C^{\sigma}_{\varepsilon}(x)$, and, hence,

$C^{\sigma}_{2\varepsilon}(x)\cap
B_{\delta}(x)=C^{\sigma}_{\varepsilon}(x)\cap B_{\delta}(x).$

∎

In the following lemma, the hypothesis of cwF-expansiveness will be necessary.
It was first proved in [artigue_2018] for cwF-expansive homeomorphisms, but we
include a different proof using cwF-hyperbolicity. In this article, an arc is
a subset of $S$ homeomorphic to $[0,1]$.

######  Lemma 2.6 ([artigue_2018] Thm. 6.7.1).

There exists $\varepsilon>0$ such that $C^{\sigma}_{\varepsilon}(x)$ is
locally connected for every $x\in S$.

###### Proof.

We prove for $\sigma=s$ but the proof for $\sigma=u$ is similar. Let
$0<\varepsilon<\frac{c}{4}$ and by contradiction assume that
$C^{s}_{\varepsilon}(x)$ is not locally connected for some $x\in S$. Then we
can consider a sequence of arcs $(P_{n})_{n\in\mathbb{N}}\subset
C^{s}_{\varepsilon}(x)$ (as in the proof of Proposition 3.1 in
[Hiraide1987ExpansiveHO]) such that $P_{i}\cap P_{j}=\emptyset$ if $i\neq j$
and $\operatorname{\textit{d}}(P_{n},P^{*})\to 0$ for some non-trivial arc
$P^{*}\subset C^{s}_{\varepsilon}(x)$ (here we also denote by $d$ the
Hausdorff distance on the space of continua). Let $y\in P^{*}$ be an interior
point of $P^{*}$ and consider $r\in(0,\varepsilon)$ such that

  1. (1)

$P^{*}$ separates $B_{r}(y)$, and

  2. (2)

$P_{n}$ separates $B_{r}(y)$ for every $n>n_{0}$ and for some
$n_{0}\in\mathbb{N}$.

We can assume that every $P_{n}$ is contained in the same component $A$ of
$B_{r}(y)\setminus P^{*}$ taking a sub-sequence if necessary. The cw-local-
product-structure ensures the existence of $\delta\in(0,\frac{r}{4})$ such
that

$C^{s}_{\frac{r}{4}}(a)\cap C^{u}_{\frac{r}{4}}(b)\neq\emptyset\quad\text{
whenever }\quad\operatorname{\textit{d}}(a,b)<\delta.$

Since $\operatorname{Int}C^{s}_{2\varepsilon}(y)=\emptyset$, there exists
$z\in A\cap B_{\delta}(y)$ such that $z\notin C^{s}_{2\varepsilon}(y)$, and,
hence,

$C^{s}_{r}(z)\cap C^{s}_{\varepsilon}(y)=\emptyset.$

In particular, $C^{s}_{r}(z)\cap P_{n}=\emptyset$ for every $n\in\mathbb{N}$.
Choose $n_{1}>n_{0}$ such that $P_{n_{1}}\cap B_{r}(y)$ separates $P^{*}\cap
B_{r}(y)$ and $C^{s}_{\frac{r}{4}}(z)$. The choice of $\delta$ ensures that
$C^{u}_{\frac{r}{4}}(y)\cap C^{s}_{\frac{r}{4}}(z)\neq\emptyset$. In
particular, $C^{u}_{\frac{r}{4}}(y)\cap P_{n_{1}}\neq\emptyset$, and, hence,
$C^{u}_{\varepsilon}(y)$ intersects an infinite number of distinct
$P_{n}^{\prime}s$ (see Figure 1). This contradicts cwF-expansiveness and
finishes the proof.

Figure 1.

∎

The following corollary is a consequence of this result. A subset $A\subset S$
is arcwise connected if for each pair of distinct points $x,y\in A$ there
exists an arc $h\colon[0,1]\to A$ such that $h(0)=x$ and $h(1)=y$.

######  Corollary 2.7.

There exists $\varepsilon>0$ such that $C^{\sigma}_{\varepsilon}(x)$ is
arcwise connected and locally arcwise connected for every $x\in S$. Moreover,
for each pair of distinct points $y,z\in C^{\sigma}_{\varepsilon}(x)$, there
is a unique arc $\sigma(y,z;x)$ in $C^{\sigma}_{\varepsilon}(x)$ connecting
$y$ and $z$.

###### Proof.

From Lemma 2.6 and Theorem 5.9 of [hall1955elementary], it follows that
$C^{\sigma}_{\varepsilon}(x)$ is a Peano space. Hence, Theorem 6.29 of
[hall1955elementary] ensures that $C^{\sigma}_{\varepsilon}(x)$ is arcwise
connected and locally arcwise connected. The uniqueness comes from the
observation that two distinct arcs connecting $y$ and $z$ would create an open
set bounded by a local stable (in the case $\sigma=s$) or local unstable (in
the case $\sigma=u$) curve, which contradicts cw-expansiveness on surfaces. ∎

From now on we choose $\varepsilon>0$ given by Corollary 2.7 and
$\delta\in(0,\varepsilon)$ satisfying the Lemmas 2.2, 2.4, and 2.5. Following
the steps of Hiraide in [Hiraide1987ExpansiveHO], we define an equivalence
relation in the set of arcs starting on $x$ and contained in
$C^{\sigma}_{\varepsilon}(x)$.

######  Definition 2.8.

Let $x\in S$, and $y,z\in C^{\sigma}_{\varepsilon}(x)$. We write $y\sim z$ if

$\sigma(x,y;x)\cap\sigma(x,z;x)\supsetneq\\{x\\}.$

We define the number of stable/unstable separatrices at $x$ as

$P^{\sigma}(x)=\\#(C^{\sigma}_{\varepsilon}(x)/\sim).$

Lemma 2.5 ensures that the number of separatrices at $x$ does not depend on
the choice of $\varepsilon<\frac{\varepsilon^{*}}{4}$. This explains the
notation $P^{\sigma}(x)$ without $\varepsilon$ being mentioned.

The following lemma follows an idea present in the works of Lewowicz/Hiraide:
stable and unstable separatrices must be alternated as in Figure 2. In this
step, the cwF-hyperbolicity will also be necessary. Let $\partial
B_{\delta}(x)$ denote the boundary of the ball $B_{\delta}(x)$, that is the
set of points whose distance to $x$ equals $\delta$.

Figure 2. Lemma 2.9

######  Lemma 2.9.

For each $x\in S$, there exists $r_{0}>0$ such that if $r\in(0,r_{0})$,
$y,z\in\partial B_{r}(x)$ are in different classes of $\sim$, $s(x,y;x)$ and
$s(x,z;x)$ are arcs intersecting $\partial B_{r}(x)$ at one point and $A$ is a
component of $B_{r}(x)\setminus(s(x,y;x)\cup s(x,z;x))$, then there is $a\in
C^{u}_{\varepsilon}(x)\cap A$ such that $u(x,a;x)\subset A$.

###### Proof.

If $C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(x)=\\{x\\}$, let
$r_{0}=\varepsilon$, otherwise,

$C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(x)\supsetneq\\{x\\},$

and we let

$r_{0}=\operatorname{\textit{d}}(x,(C^{s}_{\varepsilon}(x)\cap
C^{u}_{\varepsilon}(x))\setminus\\{x\\}),$

which is a positive number by $cw_{F}$-expansiveness. Let $x$, $r$, $y$, $z$,
and $A$ as above, and suppose there is no unstable arc $u(x,a;x)\subset A$.
Choose $\delta_{r}\in(0,\frac{r}{4})$, given by the cw-local-product-
structure, such that

$C_{\frac{r}{4}}(x)\cap C_{\frac{r}{4}}(y)\neq\emptyset\,\,\,\textnormal{
whenever }\,\,\,d(x,y)<\delta_{r}.$

Lemma 2.3 assures the existence of

$b\in A\setminus
C^{s}_{\varepsilon}(x)\,\,\,\,\,\,\text{with}\,\,\,\,\,\,\operatorname{\textit{d}}(b,x)<\frac{\delta_{r}}{2}.$

It follows that

$C^{s}_{\frac{r}{4}}(b)\subset A.$

If $C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(x)=\\{x\\}$, then
$C^{u}_{\frac{r}{4}}(x)\cap A=\emptyset$ since there is no unstable arc
$u(x,a;x)\subset A$; otherwise, $C^{s}_{r}(x)\cap C^{u}_{r}(x)=\\{x\\}$ since
$r<r_{0}$, and, hence, $C^{u}_{\frac{r}{4}}(x)\cap A=\emptyset$.

Figure 3.

In both cases, we obtain

$C^{s}_{\frac{r}{4}}(b)\cap C^{u}_{\frac{r}{4}}(x)=\emptyset.$

This contradicts the choice of $\delta_{r}$ since
$\operatorname{\textit{d}}(b,x)<\frac{\delta_{r}}{2}$ (see Figure 3) and
proves the existence of an unstable arc $u(x,a;x)\subset A$ and finishes the
proof. ∎

The following corollary is a direct consequence of the previous Lemma.

######  Corollary 2.10.

$P^{s}(x)=P^{u}(x)$ for every $x\in S$.

We recall that $C^{\sigma}_{\varepsilon}(x)$ is locally connected but the case
$P^{\sigma}(x)=\infty$ is still not ruled out since we could have an infinite
number of separatrices with diameter converging to zero. Also, in the
expansive case, examples of pseudo-Anosov homeomorphisms with singularities
containing a number of stable/unstable separatrices greater than two can be
constructed. In the following lemma, we observe that these two scenarios do
not occur in the case of cwF-hyperbolic homeomorphisms. Indeed, the existence
of bifurcation points contradicts cwF-hyperbolicity.

######  Lemma 2.11.

$P^{\sigma}(x)\leq 2$ for every $x\in S$.

###### Proof.

Suppose, by contradiction, that there exists $x\in S$ with $P^{\sigma}(x)\geq
3$. Let $r>0$ and $x_{1},x_{2},x_{3}$ be in different classes of
$C_{\varepsilon}^{s}(x)$ such that $s(x,x_{i};x)\subset B_{r}(x)$ and
intersects $\partial B_{r}(x)$ only at $x_{i}$ for every $i=1,2,3$. Then
$\bigcup_{i=1}^{3}s(x,x_{i};x)$ separates $B_{r}(x)$ in exactly three
components, $A_{1},A_{2},A_{3}$ as in Figure 4.

Figure 4.

Using Lemma 2.9, we can find $y_{i}\in A_{i}$ such that $u(x,y_{i};x)\subset
A_{i}$. If $C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(x)=\\{x\\}$, let

$r_{0}=\min(\operatorname{diam}u(x,y_{i};x))$

otherwise, let

$r_{0}=\min(\operatorname{\textit{d}}(x,(C^{s}_{\varepsilon}(x)\cap
C^{u}_{\varepsilon}(x))\setminus\\{x\\}),\min(\operatorname{diam}u(x,y_{i};x))),$

which is a positive number by $cw_{F}$-expansiveness. It follows that
$\bigcup_{i=1}^{3}u(x,z_{i};x)$ divides $B_{r_{0}}(x)$ in three components,
$B_{1},B_{2},B_{3}$, for some $z_{i}\in u(x,y_{i};x)$ (see Figure 5).

Figure 5.

Since the stable and unstable arcs are alternating, then

$A_{i}\cap B_{j}=\emptyset.$

for some $i,j\in\\{1,2,3\\}$ (see Figure 6).

Figure 6.

Choose $\delta_{r_{0}}\in(0,\frac{r_{0}}{4})$, given by cw-local-product-
structure, such that

$C^{s}_{\frac{r_{0}}{4}}(x)\cap
C^{u}_{\frac{r_{0}}{4}}(y)\neq\emptyset\,\,\,\textnormal{ whenever
}\,\,\,d(x,y)<\delta_{r_{0}}.$

Lemma 2.3 assures the existence of

$a\in A_{i}\setminus
C^{s}_{\varepsilon}(x)\,\,\,\,\,\,\text{with}\,\,\,\,\,\,\operatorname{\textit{d}}(a,x)<\frac{\delta_{r_{0}}}{2}$

and

$b\in B_{j}\setminus
C^{u}_{\varepsilon}(x)\,\,\,\,\,\,\text{with}\,\,\,\,\,\,\operatorname{\textit{d}}(b,x)<\frac{\delta_{r_{0}}}{2}.$

It follows that

$C^{s}_{\frac{r_{0}}{4}}(a)\subset
A_{i}\,\,\,\,\,\text{and}\,\,\,\,\,C^{u}_{\frac{r_{0}}{4}}(b)\subset B_{j},$

and, hence, $C^{s}_{\frac{r_{0}}{4}}(a)\cap
C^{u}_{\frac{r_{0}}{4}}(b)=\emptyset$ (see Figure 7).

Figure 7.

This contradicts the choice of $\delta_{r_{0}}$ and finishes the proof. ∎

There are two possible cases: either $P^{\sigma}(x)=1$ and $x$ is said to be a
spine, or $P^{\sigma}(x)=2$ and $x$ is said to be a regular point. Let
$\operatorname{Spin}(f)$ denote the set of all spines of $f$. The following
proposition gathers all results we obtained so far. We prove that
$C^{\sigma}_{\varepsilon}(x)$ is an arc for every $x\in S$.

######  Proposition 2.12.

If $x\in\operatorname{Spin}(f)$, then there is a homeomorphism
$h^{\sigma}:[0,1]\to C^{\sigma}_{\varepsilon}(x)$ with $h^{\sigma}(0)=x$.
Otherwise, there is a homeomorphism $h^{\sigma}:[-1,1]\to
C^{\sigma}_{\varepsilon}(x)$ with $h^{\sigma}(0)=x$.

###### Proof.

Let $IC^{\sigma}(x)$ denote the union of all open arcs in
$C^{\sigma}_{\varepsilon}(x)$ and

$BC^{\sigma}(x)=C^{\sigma}_{\varepsilon}(x)\setminus IC^{\sigma}(x).$

Note that $BC^{\sigma}(x)$ is always formed by two distinct points $x_{1}$ and
$x_{2}$, since local connectedness of $C^{\sigma}_{\varepsilon}(x)$ ensures
the existence of at least two points in $BC^{\sigma}(x)$ (as in Lemma 4.5 of
[Hiraide1987ExpansiveHO]), and the existence of three distinct points in
$BC^{\sigma}(x)$ would imply the existence of $y\in
C^{\sigma}_{\varepsilon}(x)$ with $P^{\sigma}(y)\geq 3$, contradicting Lemma
2.11. If $x\in\operatorname{Spin}(f)$, then either $x=x_{1}$ or $x=x_{2}$, and
the arc connecting $x_{1}$ to $x_{2}$ gives us a homeomorphism
$h^{\sigma}\colon[0,1]\to C^{\sigma}_{\varepsilon}(x)$ such that
$h^{\sigma}(0)=x$. If $x\notin\operatorname{Spin}(f)$, then $x\notin
BC^{\sigma}(x)$ and the arc connecting $x_{1}$ to $x_{2}$ gives us a
homeomorphism $h^{\sigma}:[-1,1]\to C^{\sigma}_{\varepsilon}(x)$ with
$h^{\sigma}(0)=x$. ∎

Now we exhibit two important consequences of above results that will be
important in the proofs of Section 3. In the first lemma, we prove that either
a local stable/unstable continuum separates a small ball, or it contains a
spine in this ball. The notation $c.c_{x}(A)$ is used to denote the connected
component of $x$ in the set $A$.

######  Lemma 2.13.

For each $0<\varepsilon<\frac{\varepsilon^{*}}{4}$, there exists
$\delta\in(0,\varepsilon)$ such that for each $x\in S$ one of the following
holds:

  1. (1)

$c.c_{x}(C^{\sigma}_{\varepsilon}(x)\cap B_{\delta}(x))$ separates
$B_{\delta}(x)$,

  2. (2)

$c.c_{x}(C^{\sigma}_{\varepsilon}(x)\cap B_{\delta}(x))$ contains a spine.

###### Proof.

If $x\in\operatorname{Spin}$, we are in case (2). Then we assume that $x\in
S\setminus\operatorname{Spin}(f)$. For each
$0<\varepsilon<\frac{\varepsilon^{*}}{4}$, let $\delta\in(0,\varepsilon)$ be
given by Lemma 2.5 such that

$C^{\sigma}_{\varepsilon}(x)\cap
B_{\delta}(x)=C^{\sigma}_{2\varepsilon}(x)\cap B_{\delta}(x).$

Let $h^{\sigma}:[-1,1]\to C^{\sigma}_{\varepsilon}(x)$ be a homeomorphism as
in Proposition 2.12 with $h^{\sigma}(0)=x$. Lemma 2.2 ensures the existence of
$z\in C^{\sigma}_{\varepsilon}(x)\cap\partial B_{\delta}(x)$. Without loss of
generality, we can assume that $z\sim h(-1)$. We assume item (1) is false and
prove item (2). Suppose that $c.c_{x}(C^{\sigma}_{\varepsilon}(x)\cap
B_{\delta}(x))$ does not separate $B_{\delta}(x)$. Since $z\in
C^{\sigma}_{\varepsilon}(x)\cap\partial B_{\delta}(x)$ and $z\sim h(-1)$, then
$\sigma(x,y;x)\subset\operatorname{Int}B_{\delta}(x)$, where $y=h(1)$. Since
$y\in C^{\sigma}_{\varepsilon}(x)$, then $C^{\sigma}_{\varepsilon}(y)\subset
C^{\sigma}_{2\varepsilon}(x)$ and, hence,

$C^{\sigma}_{\varepsilon}(y)\cap B_{\delta}(x)\subset
C^{\sigma}_{2\varepsilon}(x)\cap B_{\delta}(x)=C^{\sigma}_{\varepsilon}(x)\cap
B_{\delta}(x).$

Thus, $y\in c.c_{x}(C^{\sigma}_{\varepsilon}(x)\cap B_{\delta}(x))$ and
$P(y)=1$, that is, $y\in\operatorname{Spin}(f)$.

∎

In the last result of this section, we observe that local stable/unstable sets
intersect transversely. First, we state a precise definition for topological
transversality.

######  Definition 2.14.

Let $\alpha,\beta$ be arcs in $S$ meeting at $x$. We say that $\alpha$ is
topologically transversal to $\beta$ at $x$ if there exists a disk $D$ such
that

  1. (1)

$\alpha\cap\beta\cap D=\\{x\\}$,

  2. (2)

$\beta$ separates $D$, and

  3. (3)

the connected components of $(\alpha\setminus\beta)\cap D$ are in different
components of $D\setminus\beta$ (See Fig. 8).

Figure 8. Two arcs topologically transverse at the point $x$.

######  Lemma 2.15.

If $x,y\in S$ and $z\in C^{s}_{\varepsilon}(x)\cap
C^{u}_{\varepsilon}(y)\setminus(Spin(f))$, then $C^{s}_{\varepsilon}(x)$
intersects $C^{u}_{\varepsilon}(y)$ transversely at $z$.

###### Proof.

Since $z\notin Spin(f)$, then $P^{s}(z)=P^{u}(z)=2$. If the intersection is
not transversal we find $a_{1},a_{2}\in C^{s}_{2\varepsilon}(z)$ and a disk
$D$ around $z$ such that $s(a_{1},a_{2};z)$ separates $D$ and there is a
component of $D\setminus s(a_{1},a_{2};z)$ containing $b_{1}\not\sim b_{2}\in
C^{u}_{\varepsilon}(z)$. This is a contradiction with 2.9 because
$P_{\varepsilon}(z)=2$. ∎

##  3\. Bi-asymptotic sectors and spines

Bi-asymptotic sectors were introduced in [Artigue2013NexpansiveHO] for
$N$-expansive homeomorphisms on surfaces. These sectors were defined as being
a disk bounded by the union of a local stable and a local unstable arc (see
Figure 9(a)). In the case of 2-expansive homeomorphisms, a consequence of the
arguments of [Artigue2013NexpansiveHO] is that both intersections
$a_{1},a_{2}$ of a bi-asymptotic sector are not only transversal, but point
outside the disk (see Figure 9(a)). Indeed, 2-expansiveness and non-existence
of wandering points imply the non-existence of spines inside by-asymptotic
sectors (see Proposition 3.5 in [Artigue2013NexpansiveHO]), and this ensure
the existence of a third intersection between the stable and unstable arcs
bounding the sector if the intersection points inward.

(a)

(b)

Figure 9.

For cwF-hyperbolic homeomorphisms, it is not possible to ensure the
intersections points outward the disk. First, because there will be spines
inside the sector, but also because this case allows more intersections
between local stable/unstable arcs. The goal is to prove that every bi-
asymptotic sector contains a spine and that every spine is contained in a bi-
asymptotic sector. To prove this, we first understand the case of bi-
asymptotic sectors with intersections pointing outward the sector (these
sectors will be called regular). We will characterize the structure of
stable/unstable arcs inside a regular bi-asymptotic sector obtaining a single
spine inside it.

######  Definition 3.1.

We say that $C^{s}_{\varepsilon}(x)$ and $C^{u}_{\varepsilon}(x)$ form a bi-
asymptotic sector if there exists a pair of sub-arcs $a^{s}$, $a^{u}$
contained in $C^{s}_{\varepsilon}(x)$ and $C^{u}_{\varepsilon}(x)$,
respectively, such that $a^{s}\cup a^{u}$ bounds a disk $D$. In this case, $D$
is called the bi-asymptotic sector. Let $a_{1}$ and $a_{2}$ be the end points
of $a^{s}$ and $a^{u}$. A bi-asymptotic sector $D$ is said to be regular if it
satisfies the following:

(Regularity condition) There exists neighborhoods $V_{a_{1}}$ of $a_{1}$ and
$V_{a_{2}}$ of $a_{2}$ such that $C^{\sigma}_{\varepsilon}(x)\cap
V_{a_{1}}\cap\operatorname{Int}D=\emptyset$ and
$C^{\sigma}_{\varepsilon}(x)\cap V_{a_{2}}\cap\operatorname{Int}D=\emptyset$.

Without the regularity condition, the bi-asymptotic sectors can contain more
than one spine, and, hence, a more complicated structure of stable/unstable
arcs, as in Figure 10(a), 10(b) and 10(c).

(a)

(b)

(c)

(d)

Figure 10.

Note that both the stable and unstable continua enter the disk passing through
$a_{2}$, that is, for every neighborhood $V$ of $a_{2}$ we have
$C^{\sigma}_{\varepsilon}(a_{2})\cap V\cap\operatorname{Int}D\neq\emptyset$.
Thus, the sector in Figure 10(b) formed considering the stable arc from
$a_{1}$ and $a_{2}$ does not satisfy the regularity condition. The same
happens with the sector bounded by the stable and unstable arcs connecting
$a_{3}$ and $a_{4}$. Also, note that the sectors formed by the stable arc from
$a_{1}$ to $a_{4}$ and from $a_{2}$ to $a_{3}$ satisfy the regularity
condition. Inside these sectors, the structure of stable/unstable arcs is the
same: there is a single spine and all stable/unstable arcs turn around this
spine. We prove in this section that this is exactly the structure of
stable/unstable arcs inside any regular bi-asymptotic sector. Let $D$ be a
regular bi-asymptotic sector bounded by $a^{s}$ and $a^{u}$ with
$\operatorname{diam}D<\delta$ (given by Lemma 2.13), and let $a_{1}$ and
$a_{2}$ be the end points of $a^{s}$ and $a^{u}$. For $p\in D$, define
$C^{u}_{D}(p)$ and $C^{s}_{D}(p)$ as the connected component of $C^{u}(p)\cap
D$ and $C^{s}(p)\cap D$ containing $p$ respectively. We remark that Lemma 2.13
also holds changing the ball $B_{\delta}(x)$ for the sector $D$, that is, for
each $p\in D$, either $C^{\sigma}_{D}(p)$ separates $D$ or $C^{\sigma}_{D}(p)$
contains a spine. The hypothesis of regularity is important to ensure the
following result.

######  Lemma 3.2.

$C^{\sigma}_{D}(p)=a^{\sigma}$ for every $p\in a^{\sigma}$.

###### Proof.

It is clear that $C^{\sigma}_{D}(p)\supset a^{\sigma}$. By contradiction,
assume that there exist $p\in a^{\sigma}$ and $y\in C^{\sigma}_{D}(p)\setminus
a^{\sigma}$. This means that either $\sigma(a_{1},y;x)$ or $\sigma(a_{2},y;x)$
is contained in $\operatorname{Int}D$ and contradicts the regularity
condition. ∎

Note that in Figure 10(b), the arc $a^{u}$ connects $a_{1}$ to $a_{2}$, while
$C^{u}_{D}(a_{1})$ contains $a^{u}$ and also an arc from $a_{2}$ to a spine in
the interior of $D$. Also, $a^{s}$ is the stable arc from $a_{1}$ to $a_{2}$,
while $C^{s}_{D}(a_{1})$ contains the arc connecting $a_{2}$ to $a_{4}$ (see
Figure 10(a)). The following lemma is a consequence of the previous lemma and
the transversality explained in Lemma 2.15. The following results also hold
changing the roles of $s$ and $u$ but we will not use $\sigma$ as before since
it would make the presentation more complicated.

######  Lemma 3.3.

$1\leq\\#(C^{s}_{D}(p)\cap a^{u})\leq 2$ for every $p\in D$. If
$\\#(C^{s}_{D}(p)\cap a^{u})=1$, then $C^{s}_{D}(p)$ contains a spine.

###### Proof.

If $p\in a^{s}$, then $C^{s}_{D}(p)=a^{s}$ since $D$ is regular (see Lemma
3.2). This implies that $\\#(C^{s}_{D}(p)\cap a^{u})=\\#(a^{s}\cap a^{u})=2$
(See Fig. 9(a)). If $p\in D\setminus a^{s}$, then $C^{s}_{D}(p)\cap
a^{s}=\emptyset$, by Lemma 2.11, which implies that $C^{s}_{D}(p)\cap
a^{u}\neq\emptyset$ since Lemma 2.2 ensures that $C^{s}_{D}(p)\cap(a^{s}\cup
a^{u})\neq\emptyset$. Therefore $\\#(C^{s}_{D}(p)\cap a^{u})\geq 1$. Note that
every intersection between $C^{s}_{D}(p)$ and $a^{u}$ is transversal and this
together with Lemma 2.11 ensure that $\\#(C^{s}_{D}(p)\cap a^{u})\leq 2$. The
last part of the statement is obtained using Lemma 2.13, since
$\\#(C^{s}_{D}(p)\cap a^{u})=1$ implies that $C^{s}_{D}(p)$ does not separate
$D$, and, hence, contains a spine. ∎

Note that in the non-regular sectors of Figure 10(a), while $a^{s}\cap
a^{u}=\\{a_{1},a_{2}\\}$, we have $C^{s}_{D}(p)\cap
a^{u}=\\{a_{1},a_{2},a_{3}\\}$. Following [Artigue2013NexpansiveHO], we define
an order in the set $\mathcal{{F}}^{s}=\\{C^{s}_{D}(x):x\in D\\}$ as follows:
$C^{s}_{D}(x)<C^{s}_{D}(y)$ if $a^{s}$ and $C^{s}_{D}(y)$ are separated by
$C^{s}_{D}(x)$, i.e. , $a^{s}$ and $C^{s}_{D}(y)$ are in different components
of $D\setminus C^{s}_{D}(x)$. Note that $a^{s}$ is a minimal element for the
order and if $y\in\operatorname{Int}D\cap\operatorname{Spin}$, then
$C^{s}_{D}(y)$ does not separate $D$ and $C^{s}_{D}(y)$ cannot be smaller than
$C^{s}_{D}(z)$ for any $z\neq y$ in $D$. The following lemma is based on Lemma
3.2 in [Artigue2013NexpansiveHO]. The regularity condition on the sector
allows us to basically follow the original proof.

######  Lemma 3.4.

The order $<$ in $\mathcal{{F}}^{s}$ is total.

###### Proof.

Let $C^{s}_{D}(x)$ and $C^{s}_{D}(y)$ be different elements of
$\mathcal{{F}}^{s}$ and suppose by contradiction that neither
$C^{s}_{D}(x)<C^{s}_{D}(y)$ nor $C^{s}_{D}(y)<C^{s}_{D}(x)$. We assume that
$x$ and $y$ are spines since in the other cases we obtain the result
similarly. Consider $\gamma_{1},\gamma_{2},\gamma_{3}\subset a^{u}$ sub-arcs
as in Figure 11.

Figure 11. Non-comparable stable arcs.

Since $x$ and $y$ are spines, it follows that $E=D\setminus(C^{s}_{D}(x)\cup
C^{s}_{D}(y))$ is connected. For $1\leq i<j\leq 3$, define

$A_{ij}=\\{x\in
E:C^{s}_{D}(x)\cap\gamma_{i}\neq\emptyset,C^{s}_{D}(x)\cap\gamma_{j}\neq\emptyset\\}.$

By the definition of the subarcs, we have that $A_{ij}$ is non-empty for all
$1\leq i<j\leq 3$. In addition, these sets are closed and cover $E$. Since $E$
is connected, we can find $z$ that belongs to all of them. Hence
$\\#(C^{s}_{D}(z)\cap a^{u})\geq 3$ and this contradicts Lemma 3.3. In the
other cases we just need to choose the appropriate arcs $\gamma_{i}$ and
change the definition of the set $E$ accordingly. Figure 12 illustrates these
choices.

Figure 12.

∎

Note that the regularity condition is important to conclude the order is total
since non-regular sectors can contain points $z\in\operatorname{Int}D$ such
that $\\#(C^{s}_{D}(z)\cap a^{u})\geq 3$, so the existence of a point in the
intersection of $A_{ij}$ would not imply a contradiction.

Lemma 3.4 ensures that inside a regular bi-asymptotic sector there is at most
one spine, but does not necessarily prove the existence of a spine. This will
be consequence of the following result that proves continuity for the
variation of the arcs inside a regular bi-asymptotic sector. It is based on
Lemma 3.3 in [Artigue2013NexpansiveHO]. Lemma 3.3 ensures that
$\\#(C^{s}_{D}(x)\cap a^{u})\leq 2$ for every $x\in a^{u}$. Then we consider a
map $g:a^{u}\to a^{u}$ (see Figure 13) defined as

$C^{s}_{D}(x)\cap a^{u}=\\{x,g(x)\\}.$

Note that if $C^{s}_{D}(x)\cap a^{u}=\\{x\\}$, then $g(x)=x$ and Lemma 3.3
ensures that $C^{s}_{D}(x)$ contains a spine.

Figure 13.

Note that we could have problems to define $g$ in sectors that are not regular
since in these cases $C^{s}_{D}(a_{1})$ may not coincide with $a^{s}$ and
intersect $a^{u}$ in three different points.

######  Lemma 3.5.

The map $g:a^{u}\to a^{u}$ is continuous.

###### Proof.

As $a^{u}$ is homeomorphic to $[0,1]$, we can induce an ordering in $a^{u}$
such that $a_{1}<a_{2}$. We prove that $g$ is decreasing with this order, and
since $g$ is bijective, we conclude continuity. Suppose, by contradiction,
that $g$ is not decreasing, so there exist $x<y$ such that $g(x)<g(y)$. Note
that $g(x)\neq x$ since $x<y$ and $x=g(x)<g(y)$ ensure the arcs $s(x,g(x);x)$
and $s(y,g(y);y)$ are not comparable, contradicting Lemma 3.4. The same reason
ensures that $g(y)\neq y$. If $x<y<g(x)<g(y)$, then there is an intersection
between $s(x,g(x);x)$ and $s(y,g(y);y)$, contradicting Lemma 2.11. If
$x<g(x)<y<g(y)$ the arcs $s(x,g(x);x)$ and $s(y,g(y);y)$ are not comparable,
contradicting Lemma 3.4. Other cases are obtained from these cases
interchanging $x$ and $g(x)$ or $y$ and $g(y)$, leading to the same
contradictions. ∎

The following proposition gathers all results we obtained so far about bi-
asymptotic sectors. It is one of the directions of the equivalence between the
existence of sectors and spines that we want to prove.

######  Proposition 3.6.

If $D$ is a bi-asymptotic sector with diameter less than $\delta$, then
$\operatorname{Int}D\cap\operatorname{Spin}(f)\neq\emptyset$. Moreover, if $D$
is regular, then $\\#(\operatorname{Int}D\cap\operatorname{Spin}(f))=1$.

###### Proof.

First, we note that every regular bi-asymptotic sector $D$ contains a unique
spine in its interior. Since $a^{u}$ is homeomorphic to $[0,1]$ and $g\colon
a^{u}\to a^{u}$ is continuous, it follows that $g$ has a fixed point, and
since $g$ is decreasing, this fixed point is unique. Clearly, $a_{1}$ and
$a_{2}$ are not fixed points of $g$. This and the transversality proved in
Lemma 2.15 ensure that this single spine is contained in
$\operatorname{Int}D$.

Now let $D$ be a non-regular bi-asymptotic sector. Without loss of generality,
let us assume that $D$ does not satisfy the regularity condition at $a_{2}$,
that is, $C^{s}_{\varepsilon}(x)$ enters $D$ through $a_{2}$. If
$C^{s}_{\varepsilon}(x)$ does not intersect the open arc
$a^{u}\setminus\\{a_{1},a_{2}\\}$, then the connected component of
$C^{s}_{\varepsilon}(x)$ containing $a_{2}$ does not separate $D$, and by
Lemma 2.13 we find a spine in $\operatorname{Int}D$. If
$C^{s}_{\varepsilon}(x)$ intersects $a^{u}\setminus\\{a_{1},a_{2}\\}$ at a
point $y$, then the transversality of the intersection ensures that

$s(a_{2},y;x)\cup u(a_{2},y;x)$

bounds a regular bi-asymptotic sector contained in $D$ (see Figure 14). Thus,
there exists a spine in $\operatorname{Int}D$.

Figure 14.

∎

Now we prove the other direction of the equivalence.

######  Lemma 3.7.

If $x\in\operatorname{Spin}(f)$, then there exists $D_{x}$ a regular bi-
asymptotic sector such that $x\in\operatorname{Int}D_{x}$.

###### Proof.

We begin choosing $y\in C^{s}_{\varepsilon}(x)\cap B_{\frac{\delta}{2}}(x)$
such that $s(x,y;x)\subset B_{\frac{\delta}{2}}(x)$. Since
$C^{u}_{\frac{\varepsilon}{2}}(y)$ is an arc transversal to
$C^{s}_{\varepsilon}(x)$ at $y$ and $y\notin\operatorname{Spin}(f)$, we can
choose a small neighborhood $U$ of $C^{s}_{\varepsilon}(x)$ and
$t_{1},t_{2}\in C^{u}_{\frac{\varepsilon}{2}}(y)\cap\partial U$ satisfying:

  1. (1)

$t_{1}\not\sim t_{2}$,

  2. (2)

$u(t_{1},t_{2};y)$ intersects $\partial U$ only at $t_{1}$ and $t_{2}$,

  3. (3)

$u(t_{1},t_{2};y)\cap C^{s}_{\varepsilon}(x)=\\{y\\}$.

Since $u(t_{1},t_{2};y)$ is transversal to $C^{s}_{\varepsilon}(x)$ at $y$,
and $u(t_{1},t_{2};y)\cap C^{s}_{\varepsilon}(x)=\\{y\\}$, then
$u(t_{1},t_{2};y)$ divides both $U$ and $C^{s}_{\varepsilon}(x)$ in exactly
two components (see Figure 15).

Figure 15.

The semi-continuity of the map $a\to C^{s}_{\frac{\varepsilon}{2}}(a)$ (see
page 15 and Theorem 6.7.1 of [artigue_2018]) allows us to choose a disk $V$
centered at $x$ such that

$C^{s}_{\frac{\varepsilon}{2}}(z)\subset U\,\,\,\,\,\,\text{for
every}\,\,\,\,\,\,z\in V$

and $C^{s}_{\frac{\varepsilon}{2}}(x)\cap\partial V=\\{\tau\\}$. In
particular, $\partial V\setminus\\{\tau\\}$ is connected. For $i=1,2$, let
$c_{i}=u(y,t_{i},y)$ and

${\mathcal{E}}_{i}=\\{z\in\partial
V\setminus\\{\tau\\}:C^{s}_{\frac{\varepsilon}{2}}(z)\cap
c_{i}\neq\emptyset\\}.$

Since ${\mathcal{E}}_{i}$ is closed and $\partial V\setminus\\{\tau\\}$ is
connected, if ${\mathcal{E}}_{1}$ and ${\mathcal{E}}_{2}$ are not empty, then
${\mathcal{E}}_{1}\cap{\mathcal{E}}_{2}\neq\emptyset$. Hence, by Lemma 2.11
there is $z\in\partial V\setminus\\{\tau\\}$ such that
$C^{s}_{\frac{\varepsilon}{2}}(z)$ intersects $c_{1}\setminus\\{y\\}$ and
$c_{2}\setminus\\{y\\}$. Choose intersections $a_{1},a_{2}$, respectively,
such that $s(z,a_{1};z)\setminus\\{a_{1}\\}$ and
$s(z,a_{2};z)\setminus\\{a_{2}\\}$ are contained in the component of $x$ in
$U\setminus u(t_{1},t_{2};y)$. Hence these arcs bound a disk $D$ with

$\partial D=s(z,a_{1};z)\cup s(z,a_{2};z)\cup u(a_{1},a_{2};y).$

Figure 16.

Since $a_{1},a_{2}\in C^{s}_{\frac{\varepsilon}{2}}(z)$, it follows that
$a_{2}\in C^{s}_{\varepsilon}(a_{1})$. Also, $a_{1},a_{2}\in
C^{u}_{\frac{\varepsilon}{2}}(y)$ ensures that $a_{2}\in
C^{u}_{\varepsilon}(a_{1})$. Thus,

$a_{2}\in C^{s}_{\varepsilon}(a_{1})\cap C^{u}_{\varepsilon}(a_{1}).$

The regularity condition follows from the fact that $\operatorname{Int}D$ is
contained in one connected component of $U\setminus u(t_{1},t_{2};y)$ and the
intersections in $a_{1}$ and $a_{2}$ being transversal ensure that
$C^{s}_{\varepsilon}(z)$ enters the other component of $U\setminus
u(t_{1},t_{2};y)$ through $a_{1}$ and $a_{2}$. This proves that $D$ is a
regular bi-asymptotic sector and $x\in\operatorname{Int}D$.

Now assume that ${\mathcal{E}}_{1}\neq\emptyset$ and
${\mathcal{E}}_{2}=\emptyset$. In this case, $\partial
V\setminus\\{\tau\\}\subset{\mathcal{E}}_{1}$. Choose

$y^{\prime}\in C^{s}_{\varepsilon}(x)\cap\operatorname{Int}V$

and $u(t_{1}^{\prime},t_{2}^{\prime};y^{\prime})$ a sub-arc of
$C^{u}_{\frac{\varepsilon}{2}}(y^{\prime})$ such that $t_{1}^{\prime}\not\sim
t_{2}^{\prime}$ and $u(t_{1}^{\prime},t_{2}^{\prime};y^{\prime})$ is contained
in $\operatorname{Int}U$ except, possibly, at $t_{1}^{\prime}$ and
$t_{2}^{\prime}$. Since
$\operatorname{diam}(C^{u}_{\frac{\varepsilon}{2}}(y^{\prime}))\geq\delta$, we
can assume that either $t_{1}^{\prime}$ or $t_{2}^{\prime}$ belongs to
$\partial U$. Let us assume that $t_{1}^{\prime}\in\partial U$. Then we choose
a sequence $(z_{n})_{n\in\mathbb{N}}\subset\partial V\setminus\\{\tau\\}$ such
that

  1. (1)

$z_{n}\to\tau$ as $n\to\infty$,

  2. (2)

$z_{n}$ and $c_{1}$ are in different components of

$U\setminus(u(y^{\prime},t_{1}^{\prime};y^{\prime})\cup s(y^{\prime},y;x)\cup
u(y,t_{2};y))$

for every $n\in\mathbb{N}$, and

  3. (3)

$C^{s}_{\frac{\varepsilon}{2}}(z_{n})\cap c_{1}\neq\emptyset$ for every
$n\in\mathbb{N}$.

Figure 17.

In particular, (3) implies that

$C^{s}_{\frac{\varepsilon}{2}}(z_{n})\cap
u(y^{\prime},t_{1}^{\prime};y^{\prime})\neq\emptyset\,\,\,\,\,\,\text{for
every}\,\,\,\,\,\,n\in\mathbb{N}.$

This ensures the existence of $n_{0}\in\mathbb{N}$ such that

$C^{s}_{\frac{\varepsilon}{2}}(z_{n})\cap
u(y^{\prime},t_{2}^{\prime};y^{\prime})\neq\emptyset\,\,\,\,\,\,\text{for
every}\,\,\,\,\,\,n\geq n_{0},$

since the intersection between $C^{s}_{\varepsilon}(x)$ and
$u(t_{1}^{\prime},t_{2}^{\prime};y)$ is transversal at $y^{\prime}$, and the
semi-continuity again ensure that $C^{s}_{\frac{\varepsilon}{2}}(z_{n})$
converge to a subset of $C^{s}_{\varepsilon}(x)$. As in the previous case, the
regularity condition is assured by the transversality of these intersections
and we obtain a regular bi-asymptotic sector $D$ with
$x\in\operatorname{Int}D$.

Figure 18.

∎

A direct consequence of the previous lemma and Proposition 3.6 is the
following:

######  Corollary 3.8.

Every spine is isolated from other spines, and $\operatorname{Spin}(f)$ is at
most countable.

In the proof of Lemma 3.7, the diameter of the sector $D$ containing the spine
$x$ depends on the spine and is not necessarily uniform over all spines, i.e.,
for a sequence of distinct spines the diameters of the respective sectors
could become arbitrarily small. The following Lemma ensures the existence of
bi-asymptotic sectors with uniform diameter close to the local stable/unstable
continua of every spine. However, these sectors may not contain the associated
spines in its interior.

######  Lemma 3.9.

Let $x\in\operatorname{Spin}(f)$ and $y\in C^{s}_{\frac{\varepsilon}{2}}(x)$
with $0<\operatorname{diam}(s(x,y;x))<\frac{\delta}{4}$. If $u(a_{1},a_{2};y)$
is a sub-arc of $C^{u}_{\varepsilon}(y)$ containing $y$ in its interior, then
there exists a neighborhood $V$ of $x$ and $z\in\partial V$ such that
$\\#(C^{s}_{\varepsilon}(z)\cap u(a_{1},a_{2};y))\geq 2$.

###### Proof.

Let $U$ be a small neighborhood of $C^{s}_{\varepsilon}(x)$ and
$u(a^{\prime}_{1},a^{\prime}_{2};y)$ be a sub-arc of $u(a_{1},a_{2};y)$
contained in $U$ such that

  1. (1)

$u(a^{\prime}_{1},a^{\prime}_{2};y)\cap C^{s}_{\varepsilon}(x)=\\{y\\}$,

  2. (2)

$u(a^{\prime}_{1},a^{\prime}_{2};y)\cap\partial
U=\\{a_{1}^{\prime},a_{2}^{\prime}\\}$,

  3. (3)

$\operatorname{\textit{d}}(x,u(a_{1}^{\prime},a_{2}^{\prime};y))<\frac{\delta}{2}$.

The existence of this sub-arc is ensured by $cw_{F}$-expansiveness. Thus,
$u(a^{\prime}_{1},a^{\prime}_{2};y)$ separates $U$ in two components and the
diameter of the component $W$ of $x$ is less than $\delta$. As in the proof of
Lemma 3.7, let $V$ be an open disk centered at $x$ such that
$C^{s}_{\varepsilon}(z)\subset U$ for all $z\in V$, and let being $\tau$ be
the first intersection of $\partial V$ with $C^{s}_{\varepsilon}(x)$. There
exists $r>0$ such that if $z\in\partial V\setminus\\{\tau\\}$ is $r$-close to
$\tau$, then $C^{s}_{\frac{\varepsilon}{2}}(z)$ intersects
$u(a^{\prime}_{1},a^{\prime}_{2};y)$. Since $\operatorname{Spin}(f)$ is at
most countable (see Corollary 3.8), there exists

$z\in B_{r}(\tau)\cap\partial V\setminus\\{\tau\\}$

such that $C^{s}_{\varepsilon}(z)$ does not contain spines. Since
$\operatorname{diam}(W)<\delta$, Lemma 2.13 ensures that
$C^{s}_{\varepsilon}(z)$ separates $W$. Since $C^{s}_{\varepsilon}(z)\subset
U$, it follows that

$\\#(C^{s}_{\frac{\varepsilon}{2}}(z)\cap
u(a^{\prime}_{1},a^{\prime}_{2};y))\geq 2.$

∎

Figure 19. Long bi-asymptotic sectors close to $C^{s}_{\varepsilon}(x)$.

##  4\. $Cw_{F}$-hyperbolicity and $cw_{2}$-hyperbolicity

In this section we prove Theorem 1.1. We first prove that a cwF-hyperbolic
surface homeomorphism that is not cw2-expansive must contain infinitely many
spines, and then we prove the finiteness of the set of spines for a
$cw_{3}$-hyperbolic homeomorphism using the long bi-asymptotic sectors we
constructed at the end of Section 3. These two results ensure that
$cw_{3}$-hyperbolicity implies $cw_{2}$-hyperbolicity on surfaces. If a cwF-
hyperbolic surface homeomorphism is not cw1-hyperbolic, then there exists
arbitrarily small bi-asymptotic sectors, but this does not necessarily imply
the existence of an infinite number of spines, since all these sectors can
converge to the same single spine, as in the pseudo-Anosov diffeomorphism of
$\mathbb{S}^{2}$. If the homeomorphism is not $cw_{2}$-expansive, then for
each $\varepsilon>0$ there exists $x\in S$ such that
$\\#(C^{s}_{\varepsilon}(x)\cap C^{u}_{\varepsilon}(x))\geq 3$. We note below
that either there exists two bi-asymptotic sectors with disjoint interiors and
connected by their boundaries (see Figure 20) or there exists a non-regular
bi-asymptotic sector. In the first case, Lemma 3.6 ensures that each of these
sectors must contain a spine, and in the second case we will prove that inside
any non-regular sector there exist at least two distinct spines. This ensures
the existence of an infinite number of spines, since these sectors cannot
accumulate in a pair of two spines when $\varepsilon$ converges to zero.

Figure 20. Examples of bi-asymptotic sectors and their spines.

######  Proposition 4.1.

If $a^{s}$ and $a^{u}$ bound a non-regular bi-asymptotic sector $D$ with
$\operatorname{diam}(D)\leq\delta$, then there exist at least two distinct
spines in $\operatorname{Int}D$.

###### Proof.

Let $a_{1}$ and $a_{2}$ be the end points of $a^{s}$ and $a^{u}$ and assume
that

$C^{s}_{D}(a_{1})\cap\operatorname{Int}D\neq\emptyset\,\,\,\,\,\,\text{and}\,\,\,\,\,\,C^{u}_{D}(a_{1})\cap\operatorname{Int}D\neq\emptyset.$

Lemma 2.13 ensures that both $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ and
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ either separate $D$ or contain a spine
in $\operatorname{Int}D$. If both separate $D$, then there exist two bi-
asymptotic sectors with disjoint interiors and connected by their boundaries,
and Lemma 3.6 ensures the existence of two distinct spines in
$\operatorname{Int}D$ (see Figure 21).

Figure 21.

This figure illustrates the case where $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$
does not intersect $C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$. In the case they
intersect, in only a finite number of points by cwF-expansive- ness, then the
existence of an intersection $a_{n}$ such that $s(a_{2},a_{n};a_{1})$ and
$u(a_{2},a_{n};a_{1})$ bound a regular sector, ensures the existence of two
sectors with disjoint interiors and connected by their boundaries. Indeed, if
$C^{s}_{D}(a_{1})\setminus\\{a^{s}\cup s(a_{2},a_{n};a_{1})\\}$ does not
intersect $u(a_{2},a_{n};a_{1})$, then

$C^{s}_{D}(a_{1})\setminus\\{a^{s}\cup
s(a_{2},a_{n};a_{1})\\}\,\,\,\,\,\,\text{and}\,\,\,\,\,\,C^{u}_{D}(a_{1})\setminus\\{a^{u}\cup
u(a_{2},a_{n};a_{1})\\}$

bound a sector with interior disjoint from the first one (see Figure 22).

Figure 22.

If $C^{s}_{D}(a_{1})\setminus\\{a^{s}\cup s(a_{2},a_{n};a_{1})\\}$ intersects
$u(a_{2},a_{n};a_{1})$ at $a_{m}$, then it also forms a sector with disjoint
interior from the first one since the regular intersection in $a_{n}$ ensures
that $s(a_{n},a_{m};a_{1})$ is outside the interior of the first sector. In
this case, the sector bounded by $s(a_{2},a_{m};a_{1})$ and
$u(a_{2},a_{m};a_{1})$ is not regular at $a_{m}$ (see Figure 23). Since both
$C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ and
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ separate $D$, there exists an
intersection $a_{j}$ such that $s(a_{2},a_{j};a_{1})$ and
$u(a_{2},a_{j};a_{1})$ bound a regular sector. Indeed, if $a_{m}$ is a non-
regular intersection as above, then $C^{s}_{D}(a_{1})\setminus\\{a^{s}\cup
s(a_{2},a_{m};a_{1})\\}$ intersects $u(a_{2},a_{m};a_{1})$ at $a_{j}$, and the
sector formed by $s(a_{2},a_{j};a_{1})$ and $u(a_{2},a_{j};a_{1})$ is regular
since the intersection at $a_{j}$ comes from inside the non-regular sector
bounded by $s(a_{2},a_{m};a_{1})$ and $u(a_{2},a_{m};a_{1})$ (see Figure 23).

Figure 23.

If both $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ and
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ do not separate $D$, then both end in
spines. If these spines are actually the same spine and
$C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ and
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ do not intersect before the spine, then
they bound a bi-asymptotic sector with this spine as one of the end points of
the sector, so Lemma 3.6 ensures the existence of a spine in the interior of
this sector that is, hence, distinct from the first spine (see Figure 24).

Figure 24.

If $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ and
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ intersect before the spine in the end,
then either there is an intersection bounding a regular sector and we argument
as in the case above to create two sectors with disjoint interiors, or there
are only non-regular intersections and we create a sector between the last one
before the spine and the spine. In both cases a second spine appears. Now
assume that $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ separates $D$ but
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ does not. If
$C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ does not intersect
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ in $\operatorname{Int}D$, then
$C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ forms a bi-asymptotic sector with a
sub-arc of $a^{u}$ that does not contain the spine in
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$. Then Lemma 3.6 ensures the existence
of a spine in this sector that is, hence, distinct from the first spine (see
Figure 14). If $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ intersects
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ at $a_{i}\in\operatorname{Int}D$, and
$s(a_{2},a_{i},a_{1})$ and $u(a_{2},a_{i},a_{1})$ bound a regular sector, then
there is a spine at the interior of this sector that is distinct from the
spine in $C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ (see Figure 10(a)). If
$s(a_{2},a_{i},a_{1})$ and $u(a_{2},a_{i},a_{1})$ bound a non-regular sector,
then the spine in $C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ is inside this
sector, but since $C^{s}_{D}(a_{1})\setminus\\{a^{s}\\}$ separates $D$, it
must intersect $C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ an additional time
creating a sector that does not contain the spine in
$C^{u}_{D}(a_{1})\setminus\\{a^{u}\\}$ in its interior (see Figure 25).

Figure 25.

∎

We are ready to prove our main theorem.

###### Proof of Theorem 1.1.

Assume that $f$ is a cwF-hyperbolic homeomorphism with a finite number of
spines. If $f$ is not $cw_{2}$-expansive, then for each $\alpha\in(0,\delta)$
there exists $x\in S$ such that $\\#(C^{s}_{\alpha}(x)\cap
C^{u}_{\alpha}(x))\geq 3$. Using an order $<$ in $C^{s}_{\alpha}(x)$ we can
choose three consecutive points $a_{1},a_{2},a_{3}\in C^{s}_{\alpha}(x)\cap
C^{u}_{\alpha}(x)$, that is, $a_{1}<a_{2}<a_{3}$ and there are no points of
$C^{s}_{\alpha}(x)\cap C^{u}_{\alpha}(x)$ in $(a_{1},a_{2})$ and
$(a_{2},a_{3})$. This ensures that the stable/unstable arcs connecting $a_{1}$
to $a_{2}$, and also $a_{2}$ to $a_{3}$, form bi-asymptotic sectors (this is
not true in the case there are intersections in $(a_{1},a_{2})$ or
$(a_{2},a_{3})$ as in Figure 23). If the by-asymptotic sector formed by the
stable/unstable arcs from $a_{1}$ to $a_{2}$ is regular in $a_{2}$, then the
stable and unstable arcs from $a_{2}$ to $a_{3}$ form a bi-asymptotic sector
with interior disjoint from the interior of the sector from $a_{1}$ to
$a_{2}$. This ensures the existence of two distinct spines $\alpha$-close. If
the intersection in $a_{2}$ is not regular, then Proposition 4.1 ensures the
existence of two distinct spines inside the non-regular sector. This proves
that for each $\alpha\in(0,\delta)$ there exist two distinct spines
$\alpha$-close, and, hence, we obtain an infinite number of distinct spines
for $f$, contradicting the assumption. This proves that $f$ is cw2-hyperbolic.

Now we prove that cw3-hyperbolicity implies finiteness of the number of
spines. This is the only step of the proof that we do not know how to prove
assuming cwF-hyperbolicity. Let $f$ be a $cw_{3}$-hyperbolic homeomorphism and
assume the existence of an infinite number of distinct spines. For each
$\alpha\in(0,\varepsilon)$ choose $\delta_{\alpha}\in(0,\alpha)$ satisfying
the Lemmas 2.2, 2.4, and 2.5. Consider $x_{1}$ and $x_{2}$ spines such that
there exists $y\in C^{s}_{\alpha}(x_{1})\cap
C^{u}_{\alpha}(x_{2})\neq\emptyset$ with

$\operatorname{diam}s(x_{1},y;x_{1})<\frac{\delta_{\alpha}}{4}\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\operatorname{diam}u(x_{2},y;x_{2})<\frac{\delta_{\alpha}}{4}.$

Note that $y$ is not a spine, since it is contained in the local stable
continua of a spine. Lemma 3.9 ensures the existence of long bi-asymptotic
sectors close to $C^{s}_{\varepsilon}(x_{1})$ and $C^{u}_{\varepsilon}(x_{2})$
intersecting in four distinct points (see Figure 26). Since this can be done
for any $\alpha>0$, it follows that $f$ is not cw3-hyperbolic.

Figure 26.

∎

## Acknowledgements

Bernardo Carvalho was supported by Progetto di Eccellenza MatMod@TOV grant
number PRIN 2017S35EHN, and by CNPq grant number 405916/2018-3. Rodrigo Arruda
and Alberto Sarmiento were also supported by Fapemig grant number
APQ-00036-22.

## References

R. Arruda

Departamento de Matemática,

Universidade Federal de Minas Gerais - UFMG

Av. Antônio Carlos, 6627 - Campus Pampulha

Belo Horizonte - MG, Brazil.

B. Carvalho

Dipartimento di Matematica,

Università degli Studi di Roma Tor Vergata

Via Cracovia n.50 - 00133

Roma - RM, Italy

A. Sarmiento

Departamento de Matemática,

Universidade Federal de Minas Gerais - UFMG

Av. Antônio Carlos, 6627 - Campus Pampulha

Belo Horizonte - MG, Brazil.