Proof. By Proposition 2.5(1), it holds that $\Lambda \subset X$ ; in particular, $U_X\subset U_{\Lambda }$ and $V_X\subset V_{\Lambda }$ . By hypothesis, $U_X\cup V_X\cup \{x\}$ is connected. Assume by contradiction that $x\in X\setminus \Lambda \subset C\setminus \Lambda =U_{\Lambda }\cup V_{\Lambda }$ . Without loss of generality, we can assume that $x\in U_{\Lambda }$ . Therefore,

$$ \begin{align*} U_X\cup\{x\}\cup V_X\subset U_{\Lambda}\cup V_{\Lambda}. \end{align*} $$

The set on the left-hand side is connected, while the set on the right-hand side is the disjoint union of two open sets. We then conclude that the left term is contained in one of the two open sets. Without loss of generality, we suppose that $U_X\cup \{x\}\cup V_X\subset U_{\Lambda }$ . In particular, $V_X\subset U_{\Lambda }\cap V_{\Lambda }=\emptyset $ , which is the required contradiction.

Proof. Observe that, since $X\subset \mathrm {int}({\mathbb A})$ , clearly both ${\mathbb T}\times \{1\}$ and ${\mathbb T}\times \{-1\}$ are contained in ${\mathbb A}\setminus X$ . By contradiction, assume that X does not separate the annulus, that is, ${\mathbb A}\setminus X$ is connected. Indeed, it cannot disconnect ${\mathbb A}$ into more than two connected components, by the dissipative character of f. Moreover, since $\Lambda \subset X$ , it holds that ${\mathbb A}\setminus X$ is contained in either $U_{\Lambda }$ or $V_{\Lambda }$ , the two connected components of ${\mathbb A}\setminus \Lambda $ . Without loss of generality, assume that ${\mathbb A}\setminus X\subset U_{\Lambda }$ . Thus, since ${\mathbb T}\times \{1\}\subset {\mathbb A}\setminus X$ , we have ${{\mathbb T}\times \{1\}\subset U_{\Lambda }\cap V_{\Lambda }=\emptyset} $ . This provides the required contradiction and concludes the proof.

Proof. Let $p \in \Lambda $ be a hyperbolic periodic point of saddle type, with period $q \geq 1$ . We denote by $0<\mu _1<1<\mu _2$ the eigenvalues of $Df^q(p)$ . By Hartman–Grobman theorem, the dynamics can be linearized near p. Let then U be an open neighborhood of p, and $\Psi \colon U\to \mathbb R^2$ be a homeomorphism such that $\Psi (p)=0$ and $\Psi \circ f^q|_U= A\circ \Psi |_U$ , where

$$ \begin{align*} A=\begin{bmatrix} \mu_1 & 0 \\ 0 & \mu_2 \end{bmatrix},\quad 0<\mu_1<1<\mu_2. \end{align*} $$

Let us assume that $\Lambda $ does not locally coincide with the local stable manifold of p, that is, for any neighborhood V of p, there exists a point in $\Lambda \setminus \mathcal {W}_{\mathrm {loc}}^s(p;f^q)$ . Let $\epsilon>0$ be such that the ball of radius $\epsilon $ centered at the origin $B(0,\epsilon )$ is contained in $\Psi (U)$ . Then there exists $n_0\in \mathbb N$ such that for every $n\geq n_0$ , there exists a point $q_n\in \Lambda \cap U$ such that:

• • $\Psi (q_n)=(x_n,y_n)\in B(0,\epsilon )$ ;

• • $|y_n|\in (({\epsilon}/ {2})\mu _2^{-n},\epsilon \mu _2^{-n})$ .

Observe that each $q_n$ does not belong to the local stable manifold of p and that ${f^{qk}(q_n)\in U}$ for all $0\leq k\leq n$ . Thus, we can consider the sequence $(f^{qn}(q_n))_n$ of points in $\Lambda $ . Denote $\Psi \circ f^{qn}(q_n)=:(X_n,Y_n)$ ; in particular, for every $n \in \mathbb {N}$ , it holds that ${|X_n|<\epsilon \mu _1^n}$ and $|Y_n|\in ({\epsilon}/ {2},\epsilon)$ . Therefore, up to passing to a subsequence, the sequence $(f^{qn}(q_n))_n$ converges to a point $Q\neq p$ belonging to the local unstable manifold of p. Since $\Lambda $ is closed and invariant, we conclude that

(2.3) $$ \begin{align} f^{qn}(Q)\in(\Lambda\cap \mathcal{W}^u(p;f^q))\setminus \{p\} \quad \text{for all } n \in \mathbb{Z}. \end{align} $$

Let us now prove that the branch $\mathcal {B}^u(p,Q)$ of $\mathcal {W}^u_{\mathrm {loc}}(p;f^q)\setminus \{p\}$ containing Q is contained in $\Lambda $ , and therefore, by the invariance of $\Lambda $ , that the whole branch of the unstable manifold of p also does. Assume this is not the case: since $\Lambda $ is closed and since, by equation (2.3), in any neighborhood of p, there exists a point in $\Lambda \cap \mathcal {W}^u_{\mathrm {loc}}(p;f^q)$ , for every neighborhood $V\subset U$ of p, we can find two points $q_1=\Psi ^{-1}(0,a_1),q_2=\Psi ^{-1}(0,a_2)\in (\Lambda \cap U)\setminus \{p\}$ belonging to the local unstable manifold of p such that

$$ \begin{align*} \Gamma_0:=\Psi^{-1}(\{ 0 \}\times (a_1,a_2))\subset V,\quad \text{and}\quad \Gamma_0\cap\Lambda=\emptyset. \end{align*} $$

Let $\mathcal {U}$ be a bounded neighborhood of $\Lambda $ such that $f^{-1}|_{\mathcal {U}}$ is a diffeomorphism onto its image. Up to considering V small enough, we have that

$$ \begin{align*} \bigcup_{n \in \mathbb{N}}f^{-qn}(\Lambda\cup \Gamma_0)\subset \mathcal{U}. \end{align*} $$

Let $O_0\subset \mathcal {U}$ be a bounded open set among the connected components of $C\setminus (\Lambda \cup \Gamma _0)$ , with $\Gamma _0 \subset \partial O_0$ . Then, the family $(f^{-qn}(O_0))_{n \in \mathbb {N}}$ is uniformly bounded in measure, which contradicts the dissipative character of f.

Proof. Let z be a point of transverse intersection between $\mathcal {B}$ and $\mathcal {W}^u(p;f^q)$ . The standard $\unicode{x3bb} $ -lemma (see e.g. [Reference Palis and de MeloPdM82, Ch. 2.7]) guarantees that there exists a one-dimensional disk $D\subset \mathcal {B}$ whose past iterates $(f^{-qn}(D))_{n \geq 0}$ accumulate $\mathcal {W}^s(p;f^q)$ . As $f^{-qn}(D)\subset \Lambda $ for every $n \geq 0$ and since $\Lambda $ is closed, we deduce that $\overline {\mathcal {W}^s(p;f^q)}\subset \Lambda $ . Let us denote by $\mathcal {B}'$ the branch of $\mathcal {W}^u(p;f^q)\setminus \{p\}$ containing z. By invariance of $\Lambda $ , all the points $(f^{-qn}(z))_{n \geq 0}$ belong to $\mathcal {W}^u(p;f^q)\cap \Lambda $ ; in particular, for any $\delta>0$ , $\Lambda \cap B(p,\delta )$ contains points of the branch $\mathcal {B}'\subset \mathcal {W}^u(p;f^q)\setminus \{p\}$ . Thus, arguing as in the proof of Proposition 2.8, we deduce that $\mathcal {B}'\subset \Lambda $ . Another application of the $\unicode{x3bb} $ -lemma (now for a small disk $D'\subset \mathcal {B}'$ ) gives that, in fact, $\overline {\mathcal {W}^u(p;f^q)}\subset \Lambda $ . By f-invariance of $\Lambda $ , we conclude that $\overline {\mathcal {W}^u(\mathcal {O}_f(p))\cup \mathcal {W}^s(\mathcal {O}_f(p))}\subset \Lambda $ .

Proof. The fact that $\mathcal {W}^u(p;f^q)\setminus \{p\}$ and $\mathcal {W}^s(p;f^q)\setminus \{p\}$ intersect transversally guarantees the existence of a horseshoe $K(p)$ which is contained in the homoclinic class $H(p):=\overline {\mathcal {W}^s(\mathcal {O}_f(p))\pitchfork \mathcal {W}^u(\mathcal {O}_f(p))}$ of p (see [Reference SmaleSma65, Reference SmaleSma67] and [Reference NewhouseNew72, §2]). If $\Lambda $ contains one branch $\mathcal {B}^s\subset \mathcal {W}^s(p;f^q)\setminus \{p\}$ , then by Lemma 2.9, $\overline {\mathcal {W}^s(\mathcal {O}_f(p))\cup \mathcal {W}^u(\mathcal {O}_f(p))}\subset \Lambda $ ; otherwise, by Proposition 2.8, it holds that $\overline {\mathcal {W}^u(p;f^q)}\subset \Lambda $ . In either case, $K(p)\subset H(p)\subset \overline { \mathcal {W}^u(\mathcal {O}_f(p))}\subset \Lambda $ .

Proof. We notice that, since for any $M \in ( 0,1 ]$ ,

$$ \begin{align*} f_{\unicode{x3bb}}(\mathbb{T} \times [-M,M]) \subset \mathbb{T} \times [-\max_{\mathbb A}\unicode{x3bb} \, M,\max_{\mathbb A}\unicode{x3bb}\, M] \subset \mathbb{T} \times (-M,M), \end{align*} $$

the attractor (see equation (2.1)) corresponds to $\Lambda _{\unicode{x3bb} }^0= \mathbb {T} \times \{0\}$ . Since $\mathbb {T} \times \{0\}$ is the minimal element, with respect to the inclusion, in $\mathcal {X}(f_{\unicode{x3bb} })$ , this concludes the proof.

Proof. The statement immediately follows from Lemma 3.7. Indeed, as in §3.3, for a 2-periodic orbit $\{p,f_{\unicode{x3bb} }(p)\}=\{p,f_1(p)\}$ , denote by $\tau $ and $\mathcal {K}$ respectively the distance between the two bounces and the common curvature at these points, and as in equation (3.6), let

$$ \begin{align*} k_{1,2}(p):=(\tau \mathcal{K} + 1)^2\geq 0. \end{align*} $$

On the one hand, when $p\in \{E_1,E_2\}$ , $\tau \mathcal {K} = 2 a_2 (-{a_2}/{a_1^2}) = -2 ({a_2}/{a_1})^2 \in (-2,0)$ . Thus, $k_{1,2}(p)\in [0,1)$ and then, the $2$ -periodic orbit $\{E_1,E_2=f_{\unicode{x3bb} }(E_1)\}$ is a sink. On the other hand, when $p\in \{H_1,H_2\}$ , $\tau \mathcal {K} = 2 a_1( -{a_1}/{a_2^2}) = -2 ({a_1}/{a_2})^2 <-2$ . Thus, $k_{1,2}(p)>1$ and then, the $2$ -periodic orbit $\{H_1,H_2=f_{\unicode{x3bb} }(H_1)\}$ is a saddle.

Notation 4.5. For $i = 1,2$ , we denote by $\mathcal {W}^s(H_i;f_{\unicode{x3bb} }^2)$ (respectively $\mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)$ ) the (one-dimensional) stable (respectively unstable) manifold of $H_i$ for $f_{\unicode{x3bb} }^2$ . To lighten the notation, for $*=s,u$ , $i=1,2$ , we also denote by $\mathcal {W}^*(\mathcal {O}_{\unicode{x3bb} }(H_i))$ the union $\mathcal {W}^*(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^*(H_2;f_{\unicode{x3bb} }^2)$ . Similarly, let $\mathcal {W}^s(E_i;f_{\unicode{x3bb} }^2)$ be the (two-dimensional) stable manifold of $E_i$ for $f_{\unicode{x3bb} }^2$ . Again, to lighten the notation, for $i=1,2$ , we denote by $\mathcal {W}^s(\mathcal {O}_{\unicode{x3bb} }(E_i)) $ the union $\mathcal {W}^s(E_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^s(E_2;f_{\unicode{x3bb} }^2)$ .

Proof. For $(x,v)\in \mathbb {A}$ , let $(x',v'):=f_{\unicode{x3bb} } (x,v)$ . We first observe that

$$ \begin{align*} B(x'-x)\cdot (x+x') = Bx' \cdot x + Bx' \cdot x' - Bx \cdot x -Bx \cdot x' = 0,\end{align*} $$

since $x,x' \in \mathcal {E}$ and the matrix B is symmetric. As $x'-x$ is collinear to the vector v, the previous relation yields $Bv \cdot (x+x')=0$ ; by the symmetry of B, we thus obtain

(4.3) $$ \begin{align} -Bx' \cdot v=Bx \cdot v. \end{align} $$

Moreover, due to the reflection law, $Bx' \cdot (v+v') < 0$ , except when $v'$ is collinear with the normal at $x'$ , in which case, $Bx' \cdot (v+v')=0$ . By equation (4.3), we conclude that

$$ \begin{align*} L(x',v')=Bx' \cdot v' \le Bx\cdot v=L(x,v), \end{align*} $$

with equality exactly when the bounce at $x'$ is perpendicular to $\mathcal {E}$ . This means that $|L(x',v')-L(x,v)|\ll 1$ if and only if $d((x,v),f_{\unicode{x3bb} }^{-1}({\mathbb A}_{\perp }))\ll 1$ .

Proof. Fix $(s,r)\in \mathbb {A}$ . By Remark 4.9(a), the function $\mathcal {L}_{\unicode{x3bb} }$ is a Lyapunov function for $f_{\unicode{x3bb} }$ whose neutral set is $\mathrm {II}$ ; consequently, the omega-limit set $\omega _{f_{\unicode{x3bb} }}(s,r)$ satisfies $\omega _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\mathcal {L}_{\unicode{x3bb} })=\mathrm {II}$ . For each $n\geq 0$ , we set $u_{n}:=\mathcal {L}_{\unicode{x3bb} }(f_{\unicode{x3bb} }^{n}(s,r)). $ The sequence $(u_{n})_{\geq 0}$ is decreasing and bounded from below by $\min _{\mathbb {A}} \mathcal {L}_{\unicode{x3bb} }>-\infty $ , and hence is convergent. In particular, $\lim _{n \to +\infty } (u_n-u_{n+1})= 0$ ; by equation (4.4), we deduce that

$$ \begin{align*}\lim_{n \to +\infty}\,d(f_{\unicode{x3bb}}^{n}(s,r),\mathrm{II})=0,\end{align*} $$

where d is the usual distance inherited from ${\mathbb A}$ . Recall that $\mathrm {II}$ is formed of only four different points and let $\epsilon :=\tfrac 13\min _{p\neq q \in \mathrm {II}}d(p,q)>0$ . From the previous limit, there exists $n_0 \in \mathbb {N}$ such that for any $n \geq n_0$ , we have $d(f_{\unicode{x3bb} }^{n}(s,r),\mathrm {II})<\epsilon $ . Actually, for $n \geq n_0$ , there exists a unique point $p(s,r,n)\in \mathrm {II}$ such that $d(f_{\unicode{x3bb} }^{n}(s,r),p(s,r,n))<\epsilon $ . In particular, since the bounce at $f_{\unicode{x3bb} }^{n+1}(s,r)$ gets closer and closer to being perpendicular as $n\to +\infty $ , we have that

$$ \begin{align*}\lim_{n\to+\infty}\,d(f_{\unicode{x3bb}}^{n+2}(s,r),f_{\unicode{x3bb}}^{n}(s,r))=0.\end{align*} $$

Let us then fix $n_1\geq n_0$ such that for any $n\geq n_1$ , it holds that

$$ \begin{align*} d(f_{\unicode{x3bb}}^{n+2}(s,r),f_{\unicode{x3bb}}^{n}(s,r))<\epsilon. \end{align*} $$

Then, for any $n\geq n_1$ , we have

$$ \begin{align*} &d(p(s,r,n),p(s,r,n+2))\\ &\leq d(p(s,r,2n),f_{\unicode{x3bb}}^{n}(s,r))+d(f_{\unicode{x3bb}}^{n}(s,r),f_{\unicode{x3bb}}^{n+2}(s,r))+d(f_{\unicode{x3bb}}^{n+2}(s,r),\\&\quad\ p(s,r,n+2))<3 \epsilon. \end{align*} $$

By the choice of $\epsilon> 0$ , it follows that $p(s,r,n+2)=p(s,r,n)$ for any $n \geq n_1$ . Let us then set $p_+=p_+(s,r):=p(s,r,2n)$ for any $2n\geq n_1$ (it is well defined by the previous discussion). Then we conclude that

$$ \begin{align*} \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n}(s,r)=p_{+},\quad \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n+1}(s,r)=f_{\unicode{x3bb}}(p_{+}). \end{align*} $$

In particular, we deduce also that $p_+(f_{\unicode{x3bb} }(s,r))=f_{\unicode{x3bb} }(p_+(s,r))$ and that $\omega _{f_{\unicode{x3bb} }}(s,r)=\{p_+,f_{\unicode{x3bb} }(p_+)\}$ .

The sets $\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$ are $f_{\unicode{x3bb} }$ -invariant; moreover, $f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$ , respectively $f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0}$ , is invertible, and $\tilde {\mathcal {L}}_{\unicode{x3bb} }:=-\mathcal {L}_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$ , respectively $\tilde {\mathcal {L}}_{\unicode{x3bb} }^0:=-\mathcal {L}_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0}$ , is a Lyapunov function for $(f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }})^{-1}$ , respectively $(f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0})^{-1}$ , see Remark 4.9(c). Since $\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$ are compact, for any ${(s,r)\in \Lambda _{\unicode{x3bb} }}$ , respectively $(s,r)\in \Lambda _{\unicode{x3bb} }^0$ , we have that the alpha-limit set $\alpha _{f_{\unicode{x3bb} }}(s,r)$ satisfies $\emptyset \neq \alpha _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\tilde {\mathcal {L}}_{\unicode{x3bb} })\cap \Lambda _{\unicode{x3bb} }\subset \mathrm {II}$ , respectively $\emptyset \neq \alpha _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\tilde {\mathcal {L}}_{\unicode{x3bb} }^0)\cap \Lambda _{\unicode{x3bb} }^0\subset \mathrm {II}$ ; in particular, we deduce that $\emptyset \neq \Lambda _{\unicode{x3bb} } \cap \mathrm {II} \subset \Lambda _{\unicode{x3bb} }^0 \cap \mathrm {II}$ . Fix $(s,r)\in \Lambda _{\unicode{x3bb} }$ , respectively ${(s,r)\in \Lambda _{\unicode{x3bb} }^0}$ . Arguing as above, we see that there exists $p_-=p_-(s,r)\in \mathrm {II}\cap \Lambda _{\unicode{x3bb} }$ , respectively $p_-=p_-(s,r)\in \mathrm {II}\cap \Lambda _{\unicode{x3bb} }^0$ , such that

$$ \begin{align*} \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n}(s,r)=p_{-}, \quad \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n-1}(s,r)=f_{\unicode{x3bb}}(p_{-}).\end{align*} $$

Then, it holds that $\alpha _{f_{\unicode{x3bb} }}(s,r)=\{p_-,f_{\unicode{x3bb} }(p_-)\}$ . There are two cases for the point ${(s,r)\in \Lambda _{\unicode{x3bb} }}$ , respectively $\Lambda ^0_{\unicode{x3bb} }$ :

• • either $\alpha _{f_{\unicode{x3bb} }}(s,r) \cap \omega _{f_{\unicode{x3bb} }}(s,r)\neq \emptyset $ , and then the whole orbit $(f_{\unicode{x3bb} }^k(s,r))_{k \in \mathbb {Z}}$ is in the neutral set $\mathcal {N}(\mathcal {L}_{\unicode{x3bb} })$ , that is, $(s,r)=p_-(s,r)=p_+(s,r)$ is $2$ -periodic;

• • otherwise, $(s,r)\notin \mathrm {II}$ (this case clearly occurs, as $\Lambda _{\unicode{x3bb} }$ separates $\mathbb {A}$ , while $\mathrm {II}$ is a finite set) and the $2$ -periodic orbits $\alpha _{f_{\unicode{x3bb} }}(s,r)=\{p_-,f_{\unicode{x3bb} }(p_-)\}$ and $\omega _{f_{\unicode{x3bb} }}(s,r)=\{p_+,f_{\unicode{x3bb} }(p_+)\}$ are distinct. By Remark 4.9(b), and since the orbits of $p_+$ and $p_-$ are different, we have that $\mathcal {L}_{\unicode{x3bb} }(p_-)>\mathcal {L}_{\unicode{x3bb} }(p_+)$ and actually, $\alpha _{f_{\unicode{x3bb} }}(s,r)=\{H_1,H_2\}$ and $\omega _{f_{\unicode{x3bb} }}(s,r)=\{E_1,E_2\}$ . As $\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$ are closed, we also deduce that $\mathrm {II}\subset \Lambda _{\unicode{x3bb} }\subset \Lambda _{\unicode{x3bb} }^0$ .

Notation 4.12. Given some small $\delta>0$ , we denote by $\mathcal {W}_{\delta }^{u}(H_i;f_{\unicode{x3bb} }^2)$ the $\delta $ -local unstable manifold of $H_i$ with respect to $f_{\unicode{x3bb} }^2$ , that is, the set

$$ \begin{align*}\mathcal{W}_{\delta}^{u}(H_i;f_{\unicode{x3bb}}^2) := \{(s,r)\in{\mathbb A} :d(f_{\unicode{x3bb}}^{2n}(s,r),H_i)\leq \delta \text{ for all } n\leq 0\}.\end{align*} $$

Similarly, the $\delta $ -local stable manifold of $H_i$ with respect to $f_{\unicode{x3bb} }^2$ is

$$ \begin{align*}\mathcal{W}_{\delta}^{s}(H_i;f_{\unicode{x3bb}}^2):=\{(s,r)\in{\mathbb A}:d(f_{\unicode{x3bb}}^{2n}(s;r),H_i)\leq \delta \text{ for all } n\geq 0\}.\end{align*} $$

For $*=s,u$ , $i=1,2$ , the notation $\mathcal {W}_{\delta }^{*}(\mathcal {O}_{\unicode{x3bb} }(H_i))$ refers to $\mathcal {W}_{\delta }^{*}(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}_{\delta }^{*}(H_2;f_{\unicode{x3bb} }^2)$ . We denote by

$$ \begin{align*}\mathcal{W}_{\delta}^{s}(E_i;f_{\unicode{x3bb}}^2):=\{(s,r)\in{\mathbb A}:d(f_{\unicode{x3bb}}^{2n}(s,r),E_i)\leq\delta\text{ for all } n\geq 0\},\end{align*} $$

the $\delta $ -local stable manifold of $E_i$ with respect to $f_{\unicode{x3bb} }^2$ . Similarly, for $i=1,2$ , the notation $\mathcal {W}_{\delta }^{s}(\mathcal {O}_{\unicode{x3bb} }(E_i))$ refers to $\mathcal {W}_{\delta }^{s}(E_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}_{\delta }^{s}(E_2;f_{\unicode{x3bb} }^2)$ .