On the Lagrange and Markov Dynamical Spectra for Anosov Flows in Dimension 3

Abstract

We consider the Lagrange and the Markov dynamical spectra associated with a conservative Anosov flow on a compact manifold of dimension 3 (including geodesic flows of negative curvature and suspension flows). We show that for a large set of real functions and typical conservative Anosov flows, both the Lagrange and Markov dynamical spectra have a non-empty interior.

Appendix

1.1 Proof of Separation Lemma 3.5

The main goal of this section is to prove the Lemma 3.5, so we remember the relation (3)

$$\begin{aligned} \displaystyle \Lambda \subset \bigcup ^{l}_{i=1}\phi ^{(-\gamma ,\gamma )}(\text {int} \,{\Sigma _{i}}):=\bigcup ^{l}_{i=1}U_{\Sigma _{i}}. \end{aligned}$$

To prove Lemma 3.5 we need to understand what happens when two GCS as in the above relation intersect.

Note that if two sections $\Sigma _i$, $\Sigma _j$ has nonempty intersection, then we can consider two disjoint cases:

• 1. The intersection $\Sigma _i\cap \Sigma _j$ is transverse to the foliation ${\mathcal {F}}^{s}$, i.e. for any $x\in \text {int}\,\Sigma _i\cap \text {int}\,\Sigma _j$ there is a $C^0$-curve $\xi _x\subset \text {int} \,\Sigma _i\cap \text {int}\,\Sigma _j$ which is transverse to ${\mathcal {F}}^{s}$, then the Proposition 1 implies that $\text {int}\,\Sigma _i\cap \text {int}\,\Sigma _j$ is an open set of $\Sigma _i$ and $\Sigma _j$.

• 2. The intersection $\Sigma _i\cap \Sigma _j$ is not transverse to the foliation ${\mathcal {F}}^{s}$, i.e., there may be points in $\Sigma _i\cap \Sigma _j$ in the following two situations:

  • (i) For every point $x \in \Sigma _i\cap \Sigma _j$ there is not a curve $\xi _x\subset \text {int} \,\Sigma _i\cap \text {int}\,\Sigma _j$ transverse to ${\mathcal {F}}^{s}$, this implies $\Sigma _i\cap \Sigma _j$ does not contains open sets of $\Sigma _i$ and $\Sigma _j$.

  • (ii) The intersection $\Sigma _i\cap \Sigma _j$ contains an open set of $\Sigma _i$ and $\Sigma _j$ and also contains points as in (i).

The next task is to understand the cases (i) and (ii) of item 2. First, we shall prove the separation in Lemma 3.5 in the case that all intersections of the sections $\Sigma _{i}$ satisfy item 1. After that, we will make the separation in Lemma 3.5 when appears intersections in conditions 1 or 2.

Lemma 5.1

Let $B_i=\{j: \Sigma _i\cap \Sigma _j\ne \emptyset \ \ \text {and} \ \ \Sigma _{i}, \, \Sigma _{j} \ \ \text {satisfies the condition 1} \}$. Then there is $\delta ^{\prime }>0$ such that for every $j\in B_i$, $\phi ^{\delta }(\Sigma _{i})\cap \Sigma _{j}=\emptyset$ for all $0<\delta \le \delta ^{\prime }$.

Proof

Suppose otherwise, then for all n sufficiently large, there is $z_{i}^{n}\in \Sigma _{i}$ such that $\displaystyle \phi ^{\frac{1}{n}}(z^{n}_{i})\in \bigcup \nolimits _{j\in B_i}\Sigma _{j}$. Passing to a subsequence if necessary, we assume that $\phi ^{\frac{1}{n}}(z^{n}_{i})\in \Sigma _{j_0}$ for some $j_0\in B_i$. Since $\Sigma _{i}$ is a compact set, we can assume that $z^{n}_{i}$ converges to $z_{i}$ as $n\rightarrow \infty$, thus $\phi ^{\frac{1}{n}}(z^{n}_{i})$ converge to $z_{i}$ as $n\rightarrow \infty$. This implies that $z_{i}\in \Sigma _{i}\cap \Sigma _{j_0}$.

By Remark 5, we can assume that $z_{i}\in \text {int}\,\Sigma _{j_0}$, then by definition of $C^0$-transverse (Definition 1), there are $r>0$ small enough and $\eta >0$ such that $B_{r}(z_{i})$ (the open ball of radius r and center $z_{i}$), satisfies

$\phi ^{t}(B_{r}(z_i)\cap \Sigma _{j_0})\cap \Sigma _{j_0}=\emptyset$

for all $0<t\le \eta$.

Moreover, since $\Sigma _{i}$ and $\Sigma _{j_0}$ satisfy the condition 1, then we have $(B_{r}(z_i)\cap \Sigma _{i})\setminus \{z_i\}\subset \Sigma _{j_0}$. Taking n large enough such that $z^{n}_{i}\in B_{r}(z_i)\cap \Sigma _{i}$ and $\frac{1}{n}<\eta$. So $\phi ^{\frac{1}{n}}(z^{n}_{i})\notin \Sigma _{j_0}$ is a contradiction, thus we conclude the lemma.

Remark 18

It is worth to note that, if $\delta _{ij}:=d(\Sigma _i,\Sigma _j)>0$, then

$$\begin{aligned} \phi ^{t}(\Sigma _i)\cap \Sigma _j=\emptyset \ \ \text {for all} \ \ 0\le t< \delta _{ij}. \end{aligned}$$

The following lemma proves that the GCS as in (3) can be taken disjoint if all possible intersections of $\Sigma _{i}$ and $\Sigma _{j}$ satisfy the condition 1.

Lemma 5.2

Assuming (3) and suppose that all possible intersections of sections $\{\Sigma _{i}:i=1,\ldots ,l\}$ satisfy the condition 1. Then, there are GCS  ${\widetilde{\Sigma }}_{i}$ such that $\displaystyle \Lambda \subset \bigcup \nolimits ^{l}_{i=1}\phi ^{(-\gamma ,\gamma )}({\widetilde{\Sigma }}_{i})$ with the property ${\widetilde{\Sigma }}_{i}\cap {\widetilde{\Sigma }}_{j}=\emptyset$ for all $i,j\in \{1,\ldots ,l\}$.

Proof

Consider the set $B_1=\{j:\Sigma _{1}\cap \Sigma _{j}\ne \emptyset \}$ and $\displaystyle \delta _1=\inf \nolimits _{j\notin B_1}d(\Sigma _1,\Sigma _j)$, then by Lemma 5.1, there exist $t_1< \min \{\delta _1, \gamma \}$ such that

$$\begin{aligned} \phi ^{t_1}(\Sigma _1)\cap \Sigma _j=\emptyset \ \ \text {for all} \ \ j\ge 1. \end{aligned}$$

Put ${\widetilde{\Sigma }}_{1}:=\phi ^{t_1}(\Sigma _1)$ and $\beta _2=d({\widetilde{\Sigma }}_{1},\Sigma _2)$. Analogously, we consider the set $B_2=\{j\ge 2:\Sigma _{2}\cap \Sigma _{j}\ne \emptyset \}$ and $\displaystyle \delta _2=\min \{\inf \nolimits _{j\notin B_2}d(\Sigma _2,\Sigma _j), \beta _2\}$, then by Lemma 5.1, there exist $t_2< \min \{\delta _2, \gamma \}$ such that

$$\begin{aligned} \phi ^{t_2}(\Sigma _2)\cap \Sigma _j=\emptyset \ \ \text { for all } \ \ j\ge 2 \ \ \text { and } \ \ \phi ^{t_2}(\Sigma _2)\cap {\widetilde{\Sigma }}_{1}=\emptyset . \end{aligned}$$

We can continue with this process and obtain by induction a finite sequences of positive number $\delta _i, \beta _i$ and $t_i$ define by $\beta _i=\displaystyle \min \nolimits _{1\le j<i}d(\phi ^{t_j}(\Sigma _j),\Sigma _i)$, $\displaystyle \delta _i=\min \{\inf \nolimits _{j\notin B_i}d(\Sigma _i,\Sigma _j), \beta _i\}$, where $B_i=\{j\ge i:\Sigma _{i}\cap \Sigma _{j}\ne \emptyset \}$ and $t_i<\min \{\delta _i, \gamma \}$ with the properties

$$\begin{aligned} \phi ^{t_i}(\Sigma _i)\cap \Sigma _j=\emptyset \ \ \text {for all} \ \ j\ge i \ \ \text {and} \ \ \phi ^{t_i}(\Sigma _i)\cap \phi ^{t_j}({\Sigma }_{j})=\emptyset \ \ \text {for all} \ \ j\le i. \end{aligned}$$

Put ${\widetilde{\Sigma }}_{i}:=\phi ^{t_i}(\Sigma _i)$, then it is easy to see that the set of sections $\{{\widetilde{\Sigma }}_{1}, {\widetilde{\Sigma }}_{2},\dots , {\widetilde{\Sigma }}_{l}\}$ satisfies the conditions of lemma. $\square$

The following lemma show that if two GCS satisfy the condition 2 (ii), then a small translate in the time on one of two sections makes the resulting sections satisfy the condition 2 (i).

Lemma 5.3

Let $\Sigma _i$, $\Sigma _j$ be as in (3) satisfying the condition ${2(ii) }$, then there is $t'$ small such $\phi ^{t'}(\Sigma _i)$ and $\Sigma _j$ satisfy the condition $2\,(i)$, i.e., $\phi ^{t'}(\Sigma _i) \cap \Sigma _j$ does not contain an open set of $\phi ^{t'}(\Sigma _i)$ nor $\Sigma _j$.

Proof

By contradiction, assume that for all t small enough, we have that $\text {int} \, \phi ^{t}(\Sigma _i)\cap \text {int} \, \Sigma _j$ contains an open set of $\phi ^{t}(\Sigma _i)$ and $\Sigma _j$, then there is a non-degenerate interval $I^{j}_{t}\subset W_{\epsilon }^{u}(x_j)\subset \Sigma _j$ and a non-degenerate interval $I^i_t\subset {W^{u}_{\epsilon }}(x_i)\subset \Sigma _i$ such that the set

$$\begin{aligned} \Delta _t:=\bigcup _{z\in \phi ^{t}(I^{j}_{t}))}W^s_{\epsilon }(z)\cap \bigcup _{w\in I^{i}_{t}}W^s_{\epsilon }(w) \end{aligned}$$

(12)

contains an open set of $\phi ^{t}(\Sigma _i)$ and $\Sigma _j$.

Claim: The family of intervals $I^{j}_{t}$ is pairwise disjoint.

Proof of claim

Otherwise, assume that there is $x\in I^{j}_{t}\cap I^j_{t'}\subset W^{u}_{\epsilon }(x_j)$ with $t\ne t'$, then by (12) there are $y\in I^{i}_{t}$ and $z\in I^{i}_{t'}$ such that $\phi ^{t}(x)\in W^{s}_{\epsilon }(y)$ and $\phi ^{t'}(x)\in W^{s}_{\epsilon }(z)$. Since $t, t'$ are small, then we have that $\phi ^{-t'}(z)\in W^{s}_{loc}(\phi ^{-t}(y))$. Also, since $y, \, z \in W^{u}_{\epsilon }(x_i)$, then $z\in W^{s}_{2\epsilon }(y)$, which implies that $\phi ^{-t'}(z)\in W^{u}_{loc}(\phi ^{-t'}(y))$. Since $t'-t$ is also small, then we have

$$\begin{aligned} \phi ^{-t'}(z)\in \phi ^{t'-t}(W^{u}_{loc}(\phi ^{-t'}(y)))\cap W_{loc}^{s}(\phi ^{-t'}(y)), \end{aligned}$$

which implies that $t'=t$ and $z=y$, since the stable and unstable manifold theorem, which is a contradiction. $\square$

Note that the above claim provides a contradiction because does not exist uncountable many disjoint non-empty open intervals $I_t^j$. Since each of them would contain a rational number, proving an uncountable family of distinct rational numbers. So the proof is complete.

$\square$

Remark 19

The last lemma implies that we can always assume that the sections $\Sigma _i$ and $\Sigma _j$ satisfy conditions 1) or 2(i).

The next step is to understand what happens to the cross-sections that intersect as in case 2(i).

Assume that $\Sigma _i$ and $\Sigma _j$ satisfy the condition 2(i), then $\Sigma _i \cap \Sigma _j$ is a family of curves, $\Gamma _{ij}$. By construction of GCS of Lemma 3.4, each curve $c\in \Gamma _{ij}$ is a leaf of the foliation $\{{\mathcal {F}}^s_x\cap \Sigma _i : x\in W^u_{\epsilon }(x_i)\}$, by abuse of notation we write ${\mathcal {F}}^{s}\cap \Sigma _i$. Remember that $\Sigma _i=\Sigma _{x_i}$, thus we consider the projection $\pi ^{s}_i:\Sigma _i\rightarrow W^u_{\epsilon }(x_i)$ along ${\mathcal {F}}^{s}$ (see Fig. 7).

Fig. 7

Intersection in the case 2(i)

Proposition 2

In the above conditions $\pi ^{s}_i(\Sigma _i\cap \Sigma _j)= \{W^{s}_{\epsilon }(x)\cap W^u_{\epsilon }(x_i):x\in \Sigma _i \cap \Sigma _j\}$ is a compact set.

Proof

Indeed, we need only to show that $\pi ^{s}_i(\Sigma _i\cap \Sigma _j)$ is a closed set. Let $x_n\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$, such that $x_n \rightarrow x$, then there is $y_n\in \Sigma _i\cap \Sigma _j$ such that $W^{s}_{\epsilon }(y_n)\cap W^u_{\epsilon }(x_i)=\{x_n\}$, since $\Sigma _i\cap \Sigma _j$ is compact, then we can assume that $y_{n_k} \rightarrow y\in \Sigma _i\cap \Sigma _j$. Moreover, by continuity of foliation ${\mathcal {F}}^s$, we have that $W^{s}_{\epsilon }(y_{n_k})\cap W^u_{\epsilon }(x_i)\rightarrow \pi ^{s}_{i}(y)$ and $W^{s}_{\epsilon }(y_{n_k})\cap W^u_{\epsilon }(x_i)=\{x_{n_k}\}\rightarrow x$, so $x=\pi ^{s}_{i}(y)\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$. $\square$

Remark 20

It is worth noting that the proof of the previous proposition, actually shows that $\pi ^{s}_i$ is a continuous map.

Let $x \in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$, then, by transversality of the flow with both sections, there is $\delta >0$ such that

$$\begin{aligned} \phi ^{\delta }(W^s_{\epsilon }(x)\cap \Sigma _i)\cap \Sigma _j=\emptyset , \end{aligned}$$

and by continuity we have that there is $U_x$ neighborhood of $W^s_{\epsilon }(x)\cap \Sigma _i$ on $\Sigma _i$ such that

$$\begin{aligned} \phi ^{\delta }(U_x\cap \Sigma _i)\cap \Sigma _j=\emptyset . \end{aligned}$$

(13)

The neighborhood $\displaystyle U_x$ can be taken in the form $U_x = \bigcup _{z\in I_x} W^{s}_{\epsilon }(z)\cap \Sigma _i$, where $I_{x}\subset W^{u}_{\epsilon }(x_i)$ is an interval centered in x.

Suppose that ${\mathcal {F}}^{s}_x\cap \Sigma _{i}\cap \Lambda =\emptyset$ for some $x\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$, then since $\Lambda$ is a compact set there is an open set $V_{x}$ containing ${\mathcal {F}}^{s}_x\cap \Sigma _i$ with $V_{x}\cap \Lambda =\emptyset$. Therefore, $\Sigma _{i}$ can be subdivided into two GCS, $\Sigma _{i}^{1}$ and $\Sigma _{i}^{2}$, such that $\Sigma _{i}^{r}$ and $\Sigma _j$ intersecting as the case 2(i), for $r=1,2$. In other words, if ${\mathcal {F}}^{s}_x\cap \Sigma _{i}\cap \Lambda =\emptyset$ for some $x\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$, then we return to the case 1) or 2(i) with one more section.

Remark 21

The above observation implies that, without loss of generality, we can assume that for any $x\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$ there is $p_{x}\in {\mathcal {F}}^{s}_x\cap \Sigma _i\cap \Lambda$.

Lemma 5.4

If $\Sigma _i$ and $\Sigma _j$ are two GCS as in condition 2(i). Given $\delta >0$, $0<\delta <\frac{\gamma }{2}$ (with $\gamma$ as in (3)), then for $x\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$, there is a GCS, ${\widetilde{\Sigma }}_{x}\subset U_x\cap {\Sigma _i}$ containing ${\mathcal {F}}^{s}_x\cap \Sigma _i$, such that $\Sigma _{i}$ is subdivided into three disjoint GCS, including ${\widetilde{\Sigma }}_{x}$. Denoted by $\Sigma _{i}^{\#}$ the set of complementary sections of ${\widetilde{\Sigma }}_{x}$ in the above subdivision of $\Sigma _{i}$, then

• (1) $\phi ^{\delta }({\widetilde{\Sigma }}_{x})\cap \Sigma _j=\emptyset$.

• (2) $\Lambda \cap \phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(int (\Sigma _i))\subset \Lambda \cap \left( \phi ^{(-\gamma ,\gamma )}\left( \phi ^{\delta }(int ({\widetilde{\Sigma }}_{x}))\right) \cup \displaystyle \bigcup _{\Sigma \in \Sigma _{i}^{\#}}\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})} (int (\Sigma ))\right) .$

Proof

By Remark 21, we can assume that for any $x\in \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$ there is $p_{x}\in {\mathcal {F}}^{s}_x\cap \Sigma _i\cap \Lambda$. Consider ${\mathcal {F}}^{u}_{loc}(p_{x})$, then by Remark 6 we can find open sets $V_{p_x}^{+}$ and $V_{p_x}^{-}$ in each side of $W^{u}_{loc}(p_{x})\setminus \{p_x\}$ sufficiently close to $W^{s}_{loc}(p_{x})$ with diameter sufficiently large and $V_{p_x}^{\pm }\cap \Lambda =\emptyset$. Denote by ${\widetilde{V}}_{p_x}^{\pm }$ the projection along to the flow of $V_{p_x}^{\pm }$ over $\Sigma _{i}$, respectively. Therefore, by Remark 6 we can take ${\widetilde{V}}_{p_x}^{\pm }$ such that ${\widetilde{V}}_{p_x}^{\pm }\cap \Sigma _{i}\subset U_x$ and ${\widetilde{V}}^{\pm }_{p_x}$ crosses $\Sigma _{i}$. Using ${\widetilde{V}}^{\pm }_{p_x}$ we can construct the GCS ${\widetilde{\Sigma }}_{x}$ such that ${\widetilde{\Sigma }}_{x}\subset U_x$ and by (13), ${\widetilde{\Sigma }}_{x}$ satisfies the item 1) of lemma.

To prove item 2) note simply that $\delta <\frac{\gamma }{2}$ and $\ \phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(\text {int}(\Sigma _i))\cap \Lambda =\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}( \text {int}(\Sigma _{i})\cap \Lambda )$, which is a consequence of $\Lambda$ be invariant by the flow. $\square$

As the GCS ${\widetilde{\Sigma }}_{x}$ obtained in the last lemma is contained in $U_x\cap {\Sigma _i}$, then there is an interval centered in x, ${\widetilde{I}}_x\subset I_x\subset W^{u}_{\epsilon }(x_i)$ such that

$$\begin{aligned} {\widetilde{\Sigma }}_x = \displaystyle \bigcup _{z\in {\widetilde{I}}_x}W^{s}_{\epsilon }(z)\cap \Sigma _i. \end{aligned}$$

Moreover, since $\pi ^{s}_i(\Sigma _i\cap \Sigma _j)\subset \bigcup {\widetilde{I}}_{x}$, then the compactness $\pi ^{s}_i(\Sigma _i\cap \Sigma _j)$ from Proposition 2, there is a finite set of points $\{x^1,\dots ,x^m\}\subset \pi ^{s}_i(\Sigma _i\cap \Sigma _j)$ such that

$$\begin{aligned} \pi ^{s}_i(\Sigma _i\cap \Sigma _j)\subset \bigcup _{r=1}^{m} {{\widetilde{I}}}_{x^r}. \end{aligned}$$

Thus by first part of Lemma 5.4, given $\delta >0$ small enough, holds that

$$\begin{aligned} \phi ^{\delta }({\widetilde{\Sigma }}_{x^{r}})\cap \Sigma _j=\emptyset , \ \ \ r=1,\dots ,m \ \ \text {and} \ \ \phi ^{\delta }({\widetilde{\Sigma }}_{x^{r}})\cap \phi ^{\delta } ({\widetilde{\Sigma }}_{x^{r'}})=\emptyset , \ \ r\ne r'.\qquad \end{aligned}$$

(14)

In the above conditions, we prove the following

Lemma 5.5

If $\Sigma _i$ and $\Sigma _j$ are two GCS as in the condition 2(i). Given $0<\delta <\frac{\gamma }{2}$ (with $\gamma$ as in (3)) there are GCS, ${\widetilde{\Sigma }}_{x^r}\subset U_{x^r}$ containing ${\mathcal {F}}^{s}_{x^r}\cap \Sigma _i$, $r=1,\dots , m$ and such that $\Sigma _{i}$ is subdivided into $2m+1$ disjoint GCS, including ${\widetilde{\Sigma }}_{x^r}$, $r\in \{1,\dots ,m\}$. Denote by $\Sigma _{i}^{\#}$ the complement of the set $\{{\widetilde{\Sigma }}_{x^r}\}_{r=1}^{m}$ in the above subdivision of $\Sigma _{i}$, then

• (1) $\phi ^{\delta }({\widetilde{\Sigma }}_{x^r})\cap \Sigma _j=\emptyset$, $r\in \{1,\dots ,m\}$ and $\Sigma _j\cap \Sigma =\emptyset$ for any $\Sigma \in \Sigma _{x}^{\#}$.

• (2) $\phi ^{\delta }({\widetilde{\Sigma }}_{x^r})\cap \phi ^{\delta }({\widetilde{\Sigma }}_{x^{r'}})=\emptyset$, $r\ne r'$ and $\phi ^{\delta }({\widetilde{\Sigma }}_{x^r})\cap \Sigma _{i}=\emptyset$, $r\in \{1,\dots ,m\}$.

• (3) $\Lambda \cap \phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(int (\Sigma _i))\subset \Lambda \cap \displaystyle \bigg (\bigcup \nolimits ^{m}_{r=1}\phi ^{(-\gamma ,\gamma )} \left( \phi ^{\delta }(int ({\widetilde{\Sigma }}_{x^r}))\right) \cup \bigcup \nolimits _{\Sigma \in \Sigma _{i}^{\#}}\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})} (int (\Sigma ))\bigg ).$

Proof

Given $0<\delta <\frac{\gamma }{2}$ small enough. The conditions (1) and (2) are an immediate consequence of (14). To prove item (3) note simply that $\delta <\frac{\gamma }{2}$ and $\Lambda \cap \phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(\text {int}(\Sigma _i))=\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(\Lambda \cap \text {int}(\Sigma _{i}))$, which is a consequence of $\Lambda$ be invariant by the flow.    $\square$

Remark 22

Let $\Sigma '$ be such that GCS such that $\Sigma '\cap \Sigma _i=\emptyset$. Then we can take $\delta <d(\Sigma ',\Sigma _i)$, such that $\phi ^{\delta }({\widetilde{\Sigma }}_{x^r})\cap \Sigma '=\emptyset$,   $r\in \{1,\dots ,m\}$, where ${\widetilde{\Sigma }}_{x^r}$ as in the Lemma   5.5.

To give a complete proof of Lemma 3.5, we must now treat the more general case of Lemma 5.5, where three or more sections intersect as in case 2(i)

To reinforce the idea, we recall the Eq. (3)

$$\begin{aligned} \displaystyle \Lambda \subset \bigcup ^{l}_{i=1}\phi ^{(-\gamma ,\gamma )}(\text {int}\,{\Sigma _{i}}) =\bigcup ^{l}_{i=1}U_{\Sigma _{i}}. \end{aligned}$$

Now we will prove that the GCS in (3) can be taken disjoint, even if some of the cross-sections are in condition 2(i).

Lemma 5.6

Let $\Sigma _{i}$ be a GCS as in (3). Let $B_{i}{=}\{j: \Sigma _{i} \ \ \text {intersects} \ \ \Sigma _{j} \ \ \text {as the case} \, \, \, 2(i) \}$. Then, $\Sigma _{i}$ can be subdivided in a finite number of GCS $\{\Sigma _{i}^{s}:s=1,\dots ,n\}$ such that for each s, there is $0<\delta _{s}<\frac{\gamma }{2}$ such that

• (1) $\phi ^{\delta _{s}}(\Sigma _{i}^{s})\cap \Sigma _{j}=\emptyset$, $j\in B_{i}$ and $\displaystyle \phi ^{\delta _{s}}(\Sigma _{i}^{s})\cap \phi ^{\delta _{s'}}(\Sigma _{i}^{s'})=\emptyset$, $s\ne s'$.

• (2) $\displaystyle \Lambda \cap \bigcup \nolimits _{j\in B_{i}\cup \{i\}}\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})} (int {\Sigma _{j}})\subset \Lambda \cap \bigg (\bigcup \nolimits _{j\in B_{i}}\phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})} (int \Sigma _{j})\cup \bigcup \nolimits _{s=1}^{n} \phi ^{(-\gamma ,\gamma )}\left( int (\phi ^{\delta _s}(\Sigma _{i}^{s})) \right) \bigg )$.

Proof

The proof is by induction on $\#B_{i}$. The case $\#B_{i}=1$ is true by the Lemma 5.4. Suppose the statement is true for $\#B_{i}<q$ and we prove for $\#B_{i}=q$. In fact:

Let $k\in B_{i}$, then by Lemma 5.4, given $0<\delta <\frac{\gamma }{2}$, there are a finite number of GCS $\left\{ {\widetilde{\Sigma }}^{r}_{k}\subset \Sigma _{k}: r\in \{1,\dots ,r_{k}\}\right\}$ such that

$$\begin{aligned}&\displaystyle \phi ^{\delta }({\widetilde{\Sigma }}^{r}_{k})\cap \Sigma _{k}=\emptyset , \ \text {also} \ \ \displaystyle \phi ^{\delta }({\widetilde{\Sigma }}^{r}_{k})\cap \Sigma _{i}=\emptyset \ \text {for any} \ r, \ \text {and} \ \Sigma _{i}\cap \Sigma =\emptyset , \ \ \Sigma \in \Sigma _{k}^{\#}.\\ \end{aligned}$$

(15)

$$\begin{aligned}&\Lambda \cap \phi ^{(-\frac{\gamma }{2},\frac{\gamma }{2})}(\text {int}\,\Sigma _k)\subset \Lambda \cap \left( \bigcup \nolimits ^{r_k}_{r=1}\phi ^{(-\gamma ,\gamma )} \left( \phi ^{\delta }(\text {int}\,{\widetilde{\Sigma }}_{k}^r)\right) \cup \bigcup \nolimits _{\Sigma \in \Sigma _{k}^{\#}}\phi ^{(-\gamma ,\gamma )}(\text {int}\,\Sigma )\right) ,\nonumber \\ \end{aligned}$$

(16)

where $\Sigma _{k}^{\#}$ is as in the Lemma 5.4.

Consider now the collection of GCS

$$\begin{aligned} \left\{ \Sigma _{i},\Sigma _{j}, \phi ^{\delta }({\widetilde{\Sigma }}_{k}^r), \Sigma _{k}^{\#}:j\in B_{i}{\setminus } \{k\} \ \text {and} \ \ r\in \{1,\dots ,r_{k}\}\right\} . \end{aligned}$$

For this new collection of GCS, we have $\#B_{i}<q$. Therefore, by the induction hypothesis, the lemma is true for $\left\{ \Sigma _j:j\in B_{i}\cup \{i\}\setminus \{k\}\right\}$ and by (15) and (16) we have the lemma proved. $\square$

Remark 23

Let $\Sigma '$ be a GCS as in (3) such that $\Sigma '\cap \Sigma _{i}=\emptyset$. Then by Remark 22, we can take $\delta _{s}$ less than $d(\Sigma _{i},\Sigma ')$. So $\phi ^{\delta _{s}}(\Sigma _{i}^{s})\cap \Sigma '=\emptyset$ for any $s\in \{1,\dots ,m\}$, $\Sigma _{i}^{s}$ as in the Lemma 5.6.

We finish this section making the proof of the Lemma 3.5.

Proof of Lemma 3.5

If all the possible intersections satisfy condition 1, the result follows from Lemma 5.2. Then, we can suppose that there is i, such that the set $B_{i}=\{j: \Sigma _{i} \ \ \text {intersects} \ \ \Sigma _{j} \ \ \text {as the case}\, \, \text {2(i)}\}$ is non-empty. Without loss of generality, assume that $B_{1}\ne \emptyset$. Let us will conclude the proof by induction on l at (3).

Note that Lemma 5.4 implies the result in the case $l=2$. Therefore, suppose it is true for $k<l$ and we will prove for $k=l$. Indeed, fix $\Sigma _{1}$ and consider the set

$$\begin{aligned} T_{1}=\{j: \Sigma _{j} \ \ \text {intersects} \ \ \Sigma _{1} \ \ \text {as the case} \, \, \, \text {1}\}. \end{aligned}$$

Then by Lemma 5.1, there is $0<\delta <\frac{\gamma }{2}$ small enough, such that $\phi ^{\delta }(\Sigma _{1})\cap \Sigma _{j}=\emptyset$ for any $j\in T_{1}$.

Abusing the notation, let’s still call $B_1=\{j:\phi ^{\delta }(\Sigma _{1}) \ \ \text {intersects} \ \ \Sigma _{j} \ \ \text {as the case} \, \text {2(i)}\}$. Then by Lemma 5.5, $\phi ^{\delta }(\Sigma _{1})$ can be subdivided in a finite number of GCS $\{\Sigma _{1}^{s}:s=1,\dots ,m_0\}$ and for each s there is $0<\delta _{s}<\frac{\gamma }{2}$ such that holds 1) and 2) of Lemma 5.5. Also by Remark 23, we can assume that $\phi ^{\delta _s}(\Sigma _{1}^{s})\cap \Sigma _{j}=\emptyset$ for any $s\in \{1,\dots ,m_0\}$ and for any $j\in T_{1}\setminus \{1\}$.

Since the cardinal $\#\left( T_{1}\setminus \{1\}\cup B_{1}\right) <l$, then the set $\displaystyle \left\{ \Sigma _{j}:j\in T_{1}\setminus \{1\}\right\} \cup \left\{ \Sigma _{k}:k\in B_{1}\right\}$ satisfies the induction hypothesis, therefore there are $n(l)-1$ GCS, ${\widetilde{\Sigma }}_{i}$ with ${\widetilde{\Sigma }}_{i}\cap {\widetilde{\Sigma }}_{j}=\emptyset$ for $i\ne j$, such that

$$\begin{aligned} \Lambda \cap \bigcup _{i\in T_1\cup B_1\setminus \{1\}}\left( \phi ^{-(\gamma ,\gamma )}(\text {int}\, \Sigma _i)\right) \subset \Lambda \cap \bigcup ^{n(l)}_{i=2}\phi ^{(-2\gamma ,2\gamma )}(\text {int} \,{\widetilde{\Sigma }}_i). \end{aligned}$$

(17)

Since $\phi ^{\delta _s}(\Sigma _{1}^{s})\cap \Sigma _{j} =\emptyset$ for any $j\in T_1\cup B_1\setminus \{1\}$ and any $s\in \{1,\dots ,m\}$, then the ${\widetilde{\Sigma }}_j$ may be taken such that $\phi ^{\delta _s}(\Sigma _{1}^{s})\cap {\widetilde{\Sigma }}_i=\emptyset$ for any $s\in \{1,\dots ,m\}$ and any $i\in \{2,\dots ,n(l)\}$.

So, by the condition 2) of Lemma 5.5 and (17) we have that

$$\begin{aligned} \displaystyle \Lambda= & {} \Lambda \cap \bigcup ^{l}_{j=1}\phi ^{(-\gamma ,\gamma )} (\text {int}\,\Sigma _j)\subset \Lambda \cap \left( \bigcup ^{l}_{j=2} \phi ^{(-\gamma ,\gamma }(\text {int}\,\Sigma _{j})\cup \phi ^{(-\gamma , \gamma )}\left( \text {int}\,\phi ^{\delta }(\Sigma _{1})\right) \right) \\= & {} \displaystyle \Lambda \cap \left( \bigcup _{j\in B_{1}}\phi ^{(-\gamma ,\gamma )} (\text {int}\,\Sigma _{j})\cup \bigcup _{j\in T_{1}\setminus \{1\}} \phi ^{(-\gamma ,\gamma )}(\text {int}\,\Sigma _{j})\cup \phi ^{(-\gamma ,\gamma )}\left( \text {int}\,\phi ^{\delta } (\Sigma _{1})\right) \right) \\\subset & {} \Lambda \cap \left( \bigcup ^{n(l)}_{i=2}\phi ^{(-2\gamma ,2\gamma )}(\text {int}\, {\widetilde{\Sigma }}_i)\cup \bigcup _{s=1}^{m_0} \phi ^{(-2\gamma ,2\gamma )}\left( \text {int}\,\phi ^{\delta _s} (\Sigma _{i}^{s})\right) \right) . \end{aligned}$$

Therefore, the last inclusion concludes our proof, since $m=n(l)-1+m_0$.    $\square$

1.2 Proof of Hyperbolicity of Poincaré Map

Our main goal of this section is to prove Lemma 3.6, which has [4] as its main reference. We recall some information. Let $\Xi =\bigcup _{i=1}^{m} \Sigma _{i}$ be a finite union of cross-sections to the flow $\phi ^{t}$ given by Remark 7, which are pairwise disjoint. Sometimes, abusing of notation, we consider $\Xi =\{\Sigma _1,\ldots , \Sigma _m\}$. Let ${{\mathcal {R}}}:\Xi \rightarrow \Xi$ be a Poincaré map, that is, the map of first return to $\Xi$, ${{\mathcal {R}}}(y)=\phi ^{t_{+}(y)}(y)$, where $t_{+}(y)$ corresponds to the first time that the positive orbits of $y\in \Xi$ encounter $\Xi$.

The splitting $E^{s}\oplus \Phi \oplus E^{u}$ over a neighborhood $U_{0}$ of $\Lambda$ defines a continuous splitting $E^{s}_{\Sigma }\oplus E^{u}_{\Sigma }$ of the tangent bundle $T\Sigma$ with $\Sigma \in \Xi$ given by

$$\begin{aligned} E^{s}_{\Sigma }(y)=E^{cs}_{y}\cap T_{y}\Sigma \ \text {and} \ E^{u}_{\Sigma }(y)=E^{cu}_{y}\cap T_{y}\Sigma , \end{aligned}$$

(18)

where $E_{y}^{cs}=E^{s}_y\oplus \left\langle \Phi (y)\right\rangle$ and $E_{y}^{cu}=E^{u}_y\oplus \left\langle {\Phi }(y)\right\rangle$.

We will show that for a sufficiently large iterated of ${{\mathcal {R}}}$, ${{\mathcal {R}}}^{n}$, the splitting (18) defines a hyperbolic splitting for transformation ${{\mathcal {R}}}^{n}$ on the cross-sections, at least restricted to $\Lambda \cap \Xi$ (cf. [4, chap. 6]). To achieve this goal, we will take into consideration the following:

Remark 24

• (1) In what follows, we use $K\ge 1$ as a generic notation for large constants depending only on a lower bound for the angles between the cross-sections and the flow direction. Also depending on upper and lower bounds for the norm of the vector field on the cross-sections.

• (2) Let us consider unit vectors, $e^{s}_{x}\in E^{s}_{x}$ and $\hat{e}^{s}_{x}\in E^{s}_{\Sigma }(x)$, and write

  $$\begin{aligned} e^{s}_{x}=a_{x}\hat{e}^{s}_{x}+b_{x}\frac{\Phi (x)}{\left\| \Phi (x)\right\| }. \end{aligned}$$

  (19)

Since the angle between $E^{s}_{x}$ and $\Phi (x)$, $\angle (E^{s}_{x},\Phi (x))$, is greater than or equal to the angle between $E^{s}_{x}$ and $E^{cu}_{x}$, $\angle (E^{s}_{x},E^{cu}_{x})$, due to the fact $\Phi (x)\in E^{cu}_{x}$. The latter is uniformly bounded from zero, we have $\left| a_{x}\right| \ge \kappa$ for some $\kappa >0$ which depends only on the flow.

Let $0<\lambda <1$ be, then there is $t_1>0$ such that $\displaystyle {\lambda ^{t_{1}}<\frac{\kappa }{K}\lambda \ \ \text {and} \ \ \lambda ^{t_{1}}<\frac{\lambda }{K^{3}}}$, take n, such that $t_{n}(x):=\sum ^{n}_{i=1}{t_{i}(x)}>t_{1}$ for all $x \in \Lambda \cap \Xi$, where $t_{i}(x)=t_{+}({{\mathcal {R}}}^{i-1}(x))$.

So, we have the following proposition:

Proposition 3

Let ${{\mathcal {R}}}:\Xi \rightarrow \Xi$ be a Poincaré map and n as before. Then $D{{\mathcal {R}}}^{n}_{x}(E^{s}_{\Sigma }(x))=E^{s}_{\Sigma ^{\prime }}({{\mathcal {R}}}^{n}(x))$ at every $x \in \Sigma \in \{\Sigma _i\}_{i}$ and $D{{\mathcal {R}}}^{n}_{x}(E^{u}_{\Sigma }(x))=E^{u}_{\Sigma ^{\prime }}({{\mathcal {R}}}^{n}(x))$ at every $x\in \Lambda \cap \Sigma$ where ${{\mathcal {R}}}^{n}(x)\in \Sigma ^{\prime }\in \{\Sigma _{i}\}_i$.

Moreover, we have that

$\left\| D{{\mathcal {R}}}^{n}|_{E^{s}_{\Sigma }(x)}\right\| < \lambda$ and $\left\| D{{\mathcal {R}}}^{n}|_{E^{u}_{\Sigma }(x)}\right\| >\frac{1}{\lambda }$

at every $x\in \Sigma \in \Xi$.

Proof

The differential of the map ${{\mathcal {R}}}^{n}$ at any point $x\in \Sigma$ is given by

$$\begin{aligned} D{{\mathcal {R}}}^{n}(x)=P_{{{\mathcal {R}}}^{n}(x)}\circ D\phi ^{t_{n}(x)}|_{T_{x}\Sigma }, \end{aligned}$$

where $P_{{{\mathcal {R}}}^{n}(x)}$ is the projection onto $T_{{{\mathcal {R}}}^{n}(x)}\Sigma '$ along the direction of $\Phi ({{\mathcal {R}}}^{n}(x))$.

Note that $E^{s}_{\Sigma }$ is tangent to $\Sigma \cap W^{cs}$. Since the center stable manifold $W^{cs}(x)$ is invariant, we have that the stable bundle is invariant:

$D{{\mathcal {R}}}^{n}(x)(E^{s}_{\Sigma }(x))=E^{s}_{\Sigma ^{\prime }}({{\mathcal {R}}}^{n}(x))$.

Moreover, for all $x\in \Sigma$ we have

$D\phi ^{t_{n}(x)}(E^{u}_{\Sigma }(x))\subset D\phi ^{t_{n}(x)}(E^{cu}_{x})=E^{cu}_{{{\mathcal {R}}}^{n}(x)}$,

since $P_{{{\mathcal {R}}}^{n}(x)}$ is the projection along the vector field, it sends $E^{cu}_{{{\mathcal {R}}}^{n}(x)}$ to $E^{u}_{\Sigma ^{\prime }}({{\mathcal {R}}}^{n}(x))$.

This proves that the unstable bundle is invariant restricted to $\Lambda$, that is, $D{{\mathcal {R}}}^{n}(x)(E^{u}_{\Sigma }(x))=E^{u}_{\Sigma ^{\prime }}({{\mathcal {R}}}^{n}(x))$, because they have the same dimension 1.

Next, we prove the expansion and contraction statements. We start by noting that $\left\| P_{{{\mathcal {R}}}^{n}(x)}\right\| \le K$, with $K\ge 1$. Then we consider the basis $\left\{ \frac{\Phi (x)}{\left\| \Phi (x)\right\| },e^{u}_{x}\right\}$ of $E^{cu}_{x}$, where $e^{u}_{x}$ is a unit vector in the direction of $E^{u}_{\Sigma }(x)$ and $\Phi (x)$ is the direction of flow. Since the direction of the flow is invariant by $D\phi ^t$, then the matrix of $D\phi ^{t}|_{E^{cu}_{x}}$ relative to this basis is upper triangular:

$D\phi ^{t_{n}(x)}|_{E^{cu}_{x}}=\left[ \begin{array}{cc} \frac{\left\| \Phi ({{\mathcal {R}}}^{n}(x))\right\| }{\left\| \Phi (x)\right\| } &{} * \\ &{} \\ 0 &{} a \end{array}\right] \quad$

since $D\phi ^{t_{n}(x)}(\phi (x))=\Phi (\phi ^{t_{n}(x)}(x))=\Phi ({{\mathcal {R}}}^{n}(x))$.

Then,

$$\begin{aligned} \left\| D{{\mathcal {R}}}^{n}(x)e^{u}_{x}\right\|= & {} \left\| P_{{{\mathcal {R}}}^{n}(x)} (D\phi ^{t_{n}(x)}(x))e^{u}_{x}\right\| = \left\| ae^{u}_{{{\mathcal {R}}}^{n}(x)}\right\| = \left| a\right| \\\ge & {} \frac{1}{K}\frac{\left\| \Phi (x)\right\| }{\left\| \Phi ({{\mathcal {R}}}^{n}(x)) \right\| }\left| det(D\phi ^{t_{n}(x)}|_{E^{cu}_{x}}\right| \ge \frac{1}{K^{3}}\lambda ^{-t_{n}(x)}\ge K^{-3}\lambda ^{-t_{1}}>\frac{1}{\lambda }. \end{aligned}$$

To prove that $\left\| D{{\mathcal {R}}}^{n}|_{E^{s}_{\Sigma }(x)}\right\| <\lambda$, let us consider unit vectors, $e^{s}_{x}\in E^{s}_{x}$ and $\hat{e}^{s}_{x}\in E^{s}_{\Sigma }(x)$, and write as in (19)

$$\begin{aligned} e^{s}_{x}=a_{x}\hat{e}^{s}_{x}+b_{x}\frac{\Phi (x)}{\left\| \Phi (x)\right\| }, \end{aligned}$$

with $\left| a_{x}\right| \ge \kappa$ for some $\kappa >0$ which depends only on the flow.

Then, since $\displaystyle P_{{{\mathcal {R}}}^{n}(x)}\left( \frac{\Phi ({{\mathcal {R}}}^{n}(x))}{\left\| \Phi (x)\right\| } \right) =0$ we have that

$$\begin{aligned} \displaystyle \left\| D{{\mathcal {R}}}^{n}(x)\hat{e}^{s}_{x}\right\|= & {} \left\| P_{{{\mathcal {R}}}^{n}(x)} (D\phi ^{t_{n}(x)}(x))\hat{e}^{s}_{x}\right\| \nonumber \\= & {} \left\| P_{{{\mathcal {R}}}^{n}(x)}(D\phi ^{t_{n}(x)}(x))\left[ \frac{1}{a_{x}} \left[ {e^{ss}_{x}-b_{x}\frac{\Phi (x)}{\left\| \Phi (x)\right\| }}\right] \right] \right\| \nonumber \\= & {} \frac{1}{\left| a_{x}\right| }\left\| P_{{{\mathcal {R}}}^{n}(x)}(D\phi ^{t_{n}(x)}(x)) \left[ e^{s}_{x}-b_{x}\frac{\Phi (x)}{\left\| \Phi (x)\right\| }\right] \right\| \nonumber \\= & {} \frac{1}{\left| a_{x}\right| }\left\| P_{{{\mathcal {R}}}^{n}(x)} (D\phi ^{t_{n}(x)}(x))(e^{s}_{x})-b_{x}P_{{{\mathcal {R}}}^{n}(x)} \left( \frac{\Phi (R^{n}(x))}{\left\| \Phi (x)\right\| }\right) \right\| \nonumber \\\le & {} \frac{K}{\kappa }\left\| D\phi ^{t_{n}(x)}(x)(e^{ss}_{x})\right\| \le \frac{K}{\kappa }\lambda ^{t_{n}(x)}\le \frac{K}{\kappa }\lambda ^{t_1}<\lambda \ . \end{aligned}$$

(20)

$\square$

The next step is to prove that there exists n such that ${\mathcal {R}}^n$ is defined for every point of $\Lambda \cap \Xi$ and consequently, by Proposition 3, it is a hyperbolic set for ${\mathcal {R}}^n$. Moreover, it should be a hyperbolic set for ${\mathcal {R}}$, since $\Lambda \cap \Xi$ is invariant by ${\mathcal {R}}$.

For every $x\in \Sigma \ \in \Xi$, we define $W^s(x,\Sigma )$ to be the connected component of $W^{cs}(x)\cap \Sigma$ that contains x. Given $\Sigma \ ,\Sigma ^{\prime }\in \Xi$ we set $\Sigma (\Sigma ^{\prime })_{n}=\left\{ x\in \Sigma :{{\mathcal {R}}}^{n}(x)\in \Sigma ^{\prime }\right\}$ the domain of the map ${{\mathcal {R}}}^{n}$ from $\Sigma$ to $\Sigma ^{\prime }$. Remembering relation (20), the tangent direction to each $W^{s}(x,\Sigma )$ is contracted at an exponential rate $\left\| D{{\mathcal {R}}}^{n}(x)\hat{e}^{s}_{x}\right\| \le Ce^{-\beta t_{n}(x)}$, with $C=\frac{K}{\kappa }$ and $\beta =-\log \lambda >0$. Since the cross-section of $\Xi$ are GCS and satisfies (1) for some $\delta >0$, then we can take n such that $t_{n}(x)>t_{1}$ as in Proposition 3 with $t_{1}$ satisfying

$$\begin{aligned} Ce^{-\beta t_{1}} \sup \left\{ l(W^{s}(x,\Sigma )):x\in \Sigma \right\}<\delta \ \text {and}\ \ Ce^{-\beta t_{1}}<\frac{1}{2}, \end{aligned}$$

(21)

where $l(W^{s}(x,\Sigma ))$ is the length of $W^{s}(x,\Sigma )$. Under these conditions, we have

Lemma 5.7

Let n be satisfying conditions from Proposition 3. If ${{\mathcal {R}}}^{n}:\Sigma (\Sigma ^{\prime })_{n} \rightarrow \Sigma ^{\prime }$ defined by ${{\mathcal {R}}}^{n}(x)=\phi ^{t_{n}(x)}(x)$. Then,

• (1) ${{\mathcal {R}}}^{n}(W^{s}(x,\Sigma ))\subset W^{s}({{\mathcal {R}}}^{n}(x),\Sigma ^{\prime })$ for every $x\in \Sigma (\Sigma ^{\prime })_{n}$,

• (2) $d({{\mathcal {R}}}^{n}(y),{{\mathcal {R}}}^{n}(z))\le \frac{1}{2}d(y,z)$ for every $y,z\in W^{s}(x,\Sigma )$ and $x\in \Sigma (\Sigma ^{\prime })_{n}$.

We let $\left\{ U_{\Sigma _{i}}:i=1,\dots ,m\right\}$ be a finite cover of $\Lambda$, as in the Lemma 3.5 where the $\Sigma _{i}$ is a GCS for each i, and we set $T_{3}$ to be an upper bound for the time it takes any point $z\in U_{\Sigma _{i}}$ to leave this tubular neighborhood under the flow, for any $i=1,\dots ,l$. We assume, without loss of generality, that $t_{1}$ in Proposition 3 and (21) is bigger than $T_{3}$ and we consider n of Lemma 5.7. If the point z never returns to one of the cross-sections, then the map ${{\mathcal {R}}}$ is not defined at z. Moreover, by the Lemma 5.7, if ${{\mathcal {R}}}^{n}$ is defined for $x\in \Sigma$ for some $\Sigma \in \Xi$, then ${{\mathcal {R}}}^{n}$ is defined for every point in $W^{s}(x,\Sigma )$. Hence, the domain of ${{\mathcal {R}}}^{n}|\Sigma$ consists of strips of $\Sigma$. The smoothness of $(t,x)\longrightarrow \phi ^{t}(x)$ ensure that the strips

$\Sigma (\Sigma ^{\prime })_{n}=\left\{ x\in \Sigma :{{\mathcal {R}}}^{n}(x)\in \Sigma ^{\prime }\right\}$

have non-empty interior in $\Sigma$ for every $\Sigma , \Sigma ^{\prime } \in \Xi$. Note that by the Tubular Flow Theorem and the smoothness of the flow, the map ${{\mathcal {R}}}$ is locally smooth for all points $x\in \text {int}\, \Sigma$ such that ${{\mathcal {R}}}(x)\in \text {int}\, \Xi$, where $\displaystyle \text {int}\,\Xi =\{\text {int}\,\Sigma _{i}\}^{m}_{i=1}$. We will denote $\displaystyle \partial ^{j}\Xi = \{\partial ^{j}\Sigma _{i}\}^{l}_{i=1}$ for $j=s,u$.

Lemma 5.8

The set of discontinuities of ${{\mathcal {R}}}$ in $\Xi \setminus (\partial ^{s}\Xi \cup \partial ^{u}\Xi )$ is contained in the set of point $x\in \Xi \setminus (\partial ^{s}\Xi \cup \partial ^{u}\Xi )$ such that, ${{\mathcal {R}}}(x)$ is defined and belongs to $(\partial ^{s}\Xi \cup \partial ^{u}\Xi )$.

Proof

Let x be a point in $\Sigma \setminus (\partial ^{s}\Sigma \cup \partial ^{u}\Sigma )$ for some $\Sigma \in \Xi$, not satisfying the condition. Then ${{\mathcal {R}}}(x)$ is defined and ${{\mathcal {R}}}(x)$ belongs to the interior of some cross-section $\Sigma ^{\prime }$. By the smoothness of the flow, we have that ${{\mathcal {R}}}$ is smooth in a neighborhood of x in $\Sigma$. Hence, any discontinuity point for ${{\mathcal {R}}}$ must be in the condition of the Lemma. $\square$

Let $D_{j}\subset \Sigma _{j}$ be the set of points sent by ${{\mathcal {R}}}^{n}$ into stable boundary points of some GCS of $\Xi$, if we define the set

$$\begin{aligned} L_{j}=\left\{ W^{s}(x,\Sigma _{j}):x\in D_{j}\right\} , \end{aligned}$$

then the Lemma 5.7 implies that $L_{j}=D_{j}$. Let $B_{j}\subset \Sigma _{j}$ be the set of points sent by ${{\mathcal {R}}}^{n}$ into unstable boundary points of some GCS of $\Xi$. Denote

$$\begin{aligned} \Gamma _{j}=\bigcup _{x\in D_{j}}W^{s}(x,\Sigma _{j})\cup B_{j} \ \ \text {and}\ \ \Gamma =\bigcup \Gamma _{j}\cup (\partial ^{s}\Xi \cup \partial ^{u}\Xi ). \end{aligned}$$

Then, ${{\mathcal {R}}}^{n}$ is smooth in the complement $\Xi \setminus \Gamma$ of $\Gamma$. Observe that if $x\in D_{j}$ for some $j\in \left\{ 1,\dots ,l\right\}$, then

$$\begin{aligned} {{\mathcal {R}}}^{n}(W^{s}(x,\Sigma _{j}))\subset \partial ^{s}\Sigma ^{\prime } \ \ \text {for some} \ \ \Sigma ^{\prime }\in \Xi . \end{aligned}$$

We know that $\partial ^{s}\Xi \cap \Lambda =\emptyset$, then ${{\mathcal {R}}}^{n}(W^{s}(x,\Sigma _{j}))\cap \Lambda =\emptyset$ for all $x \in D_j$, which implies that $W^{s}(x,\Sigma _{j})\cap \Lambda =\emptyset$ for all $x\in D_{j}$. However, if $x\in B_{j}$, then ${{\mathcal {R}}}^{n}(x)\in \partial ^{u}\Sigma ^{\prime }$ for some $\Sigma ^{\prime }\in \Xi$ and again we know that $\partial ^{u}\Xi \cap \Lambda =\emptyset$, this implies that $B_{j}\cap \Lambda =\emptyset$. Therefore, $\Gamma _{j}\cap \Lambda =\emptyset$ for all $j\in \left\{ 1,\dots ,l\right\}$, so $\Gamma \cap \Lambda =\emptyset$. The latter arguments proved the following:

Lemma 5.9

If $x\in \Lambda \cap \Xi$, then ${{\mathcal {R}}}^{n}(x)$ is defined and ${{\mathcal {R}}}^{n}(x)\in int \,\Xi$.

Proof of Lemma 3.6

Note simply that by Lemma 5.9 the set $\Lambda \cap \Xi$ is an invariant set for ${{\mathcal {R}}}^{n}$ and by Proposition 3, $\Lambda \cap \Xi$ is hyperbolic set for ${{\mathcal {R}}}^{n}$ and since $\Lambda \cap \Xi$ is invariant for ${{\mathcal {R}}}$, then $\Lambda \cap \Xi$ is hyperbolic for ${{\mathcal {R}}}$. $\square$

1.2.1 Conservative Poincaré Map

Recall that $\phi =(\phi ^t)_{t\in {\mathbb {R}}}$ preserves a volume form $\omega$ on M. We define the 2-form on $\Xi$, $\omega _{_{\Xi }}:T\Xi \times T\Xi \rightarrow {\mathbb {R}}$, defined by $\omega _{_{\Xi }})_{_{\theta }}(u,v)= \omega (u,v,\Phi (\theta ))$, where $u,v\in T_{\theta }\Xi$ and $T\Xi =\bigcup _{\theta } T_{\theta }\Xi$ is the tangent bundle of $T\Xi$.

Lemma 5.10

The 2-form $\omega _{_{\Xi }}$ is a volume form on $\Xi$ which is invariant by ${\mathcal {R}}$.

Proof

Note simply that

$$\begin{aligned} (\omega _{_{\Xi }})_{{\mathcal {R}}(\theta )}(D{\mathcal {R}}(u), D{\mathcal {R}}(v))= & {} \omega _{\phi ^{t_{+}(\theta )}(\theta )}(D{\mathcal {R}}(u), D{\mathcal {R}}(v), \Phi (\phi ^{t_{+(\theta )}}(\theta ))) \\= & {} \omega (P_{{{\mathcal {R}}}(\theta )}(D\phi ^{t_{+}(\theta )}(u)), P_{{{\mathcal {R}}}(\theta )}(D\phi ^{t_{+}(\theta )}(v)), \Phi (\phi ^{t_{+}}(\theta ))), \end{aligned}$$

where $P_{{{\mathcal {R}}}(\theta )}$ is the projection onto $T_{{{\mathcal {R}}}(\theta )}\Xi$ along the direction of $\Phi ({{\mathcal {R}}}^{n}(x))$ (see proof of Propostion 3). Note that $D\phi ^{t_{+}(\theta )}(u)= P_{{{\mathcal {R}}}(\theta )}(D\phi ^{t_{+}(\theta )}(u)+ A\,\Phi (\phi ^{t_{+}(\theta )}(\theta ))$ and $D\phi ^{t_{+}(\theta )}(v)= P_{{{\mathcal {R}}}(\theta )}(D\phi ^{t_{+}(\theta )}(v))+ B\,\Phi (\phi ^{t_{+}(\theta )}(\theta ))$, for some constants A and B. Since $\omega$ is a volume form, then

$$\begin{aligned}&\omega (P_{{{\mathcal {R}}}(\theta )}(D\phi ^{t_{+}(\theta )}(u)), P_{{{\mathcal {R}}} (\theta )}(D\phi ^{t_{+}(\theta )}(v)), \Phi (\phi ^{t_{+}}(\theta )))\\&\quad = \omega (D\phi ^{t_{+}(\theta )}(u),D\phi ^{t_{+}(\theta )}(v), \Phi (\phi ^{t_{+}}(\theta )))\\&\quad =\omega (u,v,\Phi (\theta ))= (\omega _{_{\Xi }})_{\theta }(u,v). \end{aligned}$$

Therefore, the 2-form $\omega _{_{\Xi }}$ is an invariant form for ${\mathcal {R}}$. As $\Xi$ is transverse to the flow and $\omega$ is a volume form, then it is easy to see that $\omega _{_{\Xi }}$ is also a volume form for $\Xi$. $\square$

1.3 Regular Cantor Sets

Let ${\mathbb {A}}$ be a finite alphabet, ${\mathbb {B}}$ a subset of ${\mathbb {A}}^{2}$, and $\Sigma _{{\mathbb {B}}}$ the subshift of finite type of ${\mathbb {A}}^{{\mathbb {Z}}}$ with allowed transitions ${\mathbb {B}}$. We will always assume that $\Sigma _{{\mathbb {B}}}$ is topologically mixing and that every letter in ${\mathbb {A}}$ occurs in $\Sigma _{{\mathbb {B}}}$.

An expansive map of type $\Sigma _{{\mathbb {B}}}$ is a map g with the following properties:

• (i) the domain of g is a disjoint union $\displaystyle \bigcup \nolimits _{{\mathbb {B}}}I(a,b)$. Where for each (a, b),   I(a, b) is a compact subinterval of $I(a) := [0,1]\times \{a\}$;

• (ii) for each $(a,b) \in {\mathbb {B}}$, the restriction of g to I(a, b) is a smooth diffeomorphism onto I(b) satisfying $|Dg(t)| > 1$ for all t.

The regular Cantor set associated to g is the maximal invariant set

$$\begin{aligned} K = \bigcap _{n\ge 0} g^{-n}\bigg (\bigcup _{{\mathbb {B}}} I(a,b)\bigg ). \end{aligned}$$

Let $\Sigma ^+_{{\mathbb {B}}}$ be the unilateral subshift associated to $\Sigma _{{\mathbb {B}}}$. There exists a unique homeomorphism $h:\Sigma ^{+}_{{\mathbb {B}}} \rightarrow K$ such that

$$\begin{aligned} h({\underline{a}}) \in I(a_0), \text { for } {\underline{a}} = (a_0,a_1,\dots ) \in \Sigma ^+_{{\mathbb {B}}} \ \ and \ \ h\circ \sigma =g \circ h, \end{aligned}$$

where $\sigma ^{+}:\Sigma _{{\mathbb {B}}}^{+} \rightarrow \Sigma _{{\mathbb {B}}}^{+}$, is defined as follows $\sigma ^{+}((a_{n})_{n\ge 0})=(a_{n+1})_{n\ge 0}$.

1.4 Expanding Maps Associated to a Horseshoe

Let $\Lambda$ be a horseshoe associated to $C^{2}$-diffeomorphism $\varphi$ on a surface M and consider a finite collection $(R_{a})_{a\in {\mathbb {A}}}$ of disjoint rectangles of M, which are a Markov partition of $\Lambda$. Define the sets

$$\begin{aligned} W^{s}(\Lambda ,R)= & {} \bigcap _{n\ge 0}\varphi ^{-n}\left( \bigcup _{a\in {\mathbb {A}}}R_{a}\right) , \\ W^{u}(\Lambda ,R)= & {} \bigcap _{n\le 0}\varphi ^{-n}\left( \bigcup _{a\in {\mathbb {A}}}R_{a}\right) . \end{aligned}$$

There is a $r>1$ and a collection of $C^{r}$-submersions $(\pi _{a}:R_{a}\rightarrow I(a))_{a\in {\mathbb {A}}}$, satisfying the following property:

If $z,z^{\prime }\in R_{a_{0}}\cap \varphi ^{-1}(R_{a_{1}})$ and $\pi _{a_{0}}(z)=\pi _{a_{0}}(z^{\prime })$, then we have

$$\begin{aligned} \pi _{a_{1}}(\varphi (z))=\pi _{a_{1}}(\varphi (z^{\prime })). \end{aligned}$$

In particular, the connected components of $W^{s}(\Lambda ,R)\cap R_{a}$ are the level lines of $\pi _{a}$. Then we define a mapping $g^{u}$ of class $C^{r}$ (expansive of type $\Sigma _{{\mathbb {B}}}$) by the formula

$$\begin{aligned} g^{u}(\pi _{a_{0}}(z))=\pi _{a_{1}}(\varphi (z)) \end{aligned}$$

for $(a_{0},a_{1})\in {\mathbb {B}}$, $z\in R_{a_{0}}\cap \varphi ^{-1}(R_{a_{1}})$. The regular Cantor set $K^{u}$ defined by $g^{u}$, describes the geometry transverse of the stable foliation $W^{s}(\Lambda ,R)$. Analogously, we can describe the geometry transverse of the unstable foliation $W^{u}(\Lambda ,R)$ using a regular Cantor set $K^{s}$ defined by a mapping $g^{s}$ of class $C^{r}$ (expansive of type $\Sigma _{{\mathbb {B}}}$).

Also, the horseshoe $\Lambda$ is locally the product of two regular Cantor sets $K^{s}$ and $K^{u}$. So, the Hausdorff dimension of $\Lambda$, $HD(\Lambda )$ is equal to $HD(K^{s}\times K^{u})$, but for regular Cantor sets, we have that $HD(K^{s}\times K^{u})=HD(K^{s})+HD(K^{u})$. Thus $HD(\Lambda )=HD(K^{s})+HD(K^{u})$ (cf. [24, chap 4]).

1.5 Intersections of Regular Cantor Sets and Property V

Let r be a real number $> 1$, or $r=+\infty$. The space of $C^r$ expansive maps of type $\Sigma$ (cf. Subsection 3), endowed with the $C^r$ topology, will be denoted by $\Omega _\Sigma ^r$ . The union $\Omega _\Sigma = \displaystyle \bigcup \nolimits _{r>1} \Omega _\Sigma ^r$ is endowed with the inductive limit topology.

Let $\Sigma ^- = \{(\theta _n)_{n\le 0}\,, (\theta _i,\theta _{i+1}) \in {\mathbb {B}}\text { for } i < 0\}$. We equip $\Sigma ^-$ with the following ultrametric distance: for $\underline{\theta } \ne \underline{{{\widetilde{\theta }}}} \in \Sigma ^-$, set

$$\begin{aligned} d(\underline{\theta },\underline{{{\widetilde{\theta }}}}) = \left\{ \begin{array}{lll} \ \ \ 1 &{}\quad \text{ if } \ \theta _0 \ne {\widetilde{\theta }}_0;\\ |I(\underline{\theta } \wedge \underline{{{\widetilde{\theta }}}})|&{}\quad \text{ otherwise }\end{array} \right. , \end{aligned}$$

where $\underline{\theta } \wedge \underline{{{\widetilde{\theta }}}} = (\theta _{-n},\dots ,\theta _0)$ if ${{\widetilde{\theta }}}_{-j} = \theta _{-j}$ for $0 \le j \le n$ and ${{\widetilde{\theta }}}_{-n-1} \ne \theta _{-n-1}$ .

Now, let $\underline{\theta } \in \Sigma ^-$; for $n > 0$, let $\underline{\theta }^n = (\theta _{-n},\dots ,\theta _0)$, and let $B(\underline{\theta }^n)$ be the affine map from $I(\underline{\theta }^n)$ onto $I(\theta _0)$ such that the diffeomorphism $k_n^{\underline{\theta }} = B(\underline{\theta }^n) \circ f_{\underline{\theta }^n}$ is orientation preserving.

We have the following well-known result (cf. [25]):

Proposition

Let $r \in (1,+\infty )$, $g \in \Omega _\Sigma ^r$.

• 1. For any $\underline{\theta } \in \Sigma ^-$, there is a diffeomorphism $k^{\underline{\theta }} \in \text {Diff}_+^{\ r}(I(\theta _0))$ such that $k_n^{\underline{\theta }}$ converges to $k^{\underline{\theta }}$ in $\text {Diff}_{+}^{\ r'}(I(\theta _0))$, for any $r'< r$, uniformly in $\underline{\theta }$. The convergence is also uniform in a neighborhood of g in $\Omega _\Sigma ^r$ .

• 2. If r is an integer or $r = +\infty$,   $k_n^{\underline{\theta }}$ converge to $k^{\underline{\theta }}$ in $\text {Diff}_+^r(I(\theta _0))$. More precisely, for every $0 \le j \le r-1$, there is a constant $C_j$ (independent on $\underline{\theta }$) such that

  $$\begin{aligned} \left| D^j \, \log \, D \left[ k_n^{\underline{\theta }} \circ (k^{\underline{\theta }})^{-1}\right] (x)\right| \le C_j|I(\underline{\theta }^n)|. \end{aligned}$$

  It follows that $\underline{\theta } \rightarrow k^{\underline{\theta }}$ is Lipschitz in the following sense: for $\theta _0 = {{\widetilde{\theta }}}_0$, we have

  $$\begin{aligned} \left| D^j \, \log \, D\big [k^{\underline{{{\widetilde{\theta }}}}} \circ (k^{\underline{\theta }})^{-1}\big ](x)\right| \le C_j\,d(\underline{\theta }, \underline{{{\widetilde{\theta }}}}). \end{aligned}$$

Let $r \in (1,+\infty ]$. For $a \in {\mathbb {A}}$, we denote by ${{{\mathcal {P}}}}^{r}(a)$ the space of $C^r$-embeddings of I(a) into ${\mathbb {R}}$, endowed with the $C^r$ topology. The affine group $Aff({\mathbb {R}})$ acts by composition on the left on ${{{\mathcal {P}}}}^r(a)$, the quotient space being denoted by $\overline{{{\mathcal {P}}}}^r(a)$. We also consider ${{{\mathcal {P}}}}(a) = \displaystyle \bigcup \nolimits _{r>1} {{{\mathcal {P}}}}^r(a)$ and $\overline{{{\mathcal {P}}}}(a) = \displaystyle \bigcup \nolimits _{r>1} \overline{{{\mathcal {P}}}}^r(a)$, endowed with the inductive limit topologies.

Remark 25

In [20] was considered ${{{\mathcal {P}}}}^{r}(a)$ for $r \in (1,+\infty ]$, but all the definitions and results involving ${{{\mathcal {P}}}}^{r}(a)$ can be obtained considering $r\in [1,+\infty ]$.

Let ${\mathcal {A}} =(\underline{\theta }, A)$, where $\underline{\theta } \in \Sigma ^-$ and A is now an affine embedding of $I(\theta _0)$ into ${\mathbb {R}}$. We have a canonical map

$$\begin{aligned} {{\mathcal {A}}}\rightarrow & {} {{{\mathcal {P}}}}^r = \bigcup _{{\mathbb {A}}} {{{\mathcal {P}}}}^r(a)\\ (\underline{\theta },A)\mapsto & {} A\circ k^{\underline{\theta }} \ \ (\in {{{\mathcal {P}}}}^r(\theta _0)). \end{aligned}$$

Now assume we are given two sets of data $({\mathbb {A}},{\mathbb {B}},\Sigma ,g)$, $({{\mathbb {A}}}',{{\mathbb {B}}}',\Sigma ',g')$ defining regular Cantor sets K, $K'$.

We define as in the previous the spaces ${\mathcal {P}} = \displaystyle \bigcup \nolimits _{{\mathbb {A}}}{{\mathcal {P}}}(a)$ and ${{{\mathcal {P}}}}'= \displaystyle \bigcup \nolimits _{{{\mathbb {A}}}'} {{{\mathcal {P}}}}(a')$.

A pair $(h,h')$, $(h \in {{{\mathcal {P}}}}(a), h'\in {{{\mathcal {P}}}} '(a'))$ is called a smooth configuration for $K(a)=K\cap I(a)$, $K'(a')=K'\cap I(a')$. Actually, rather than working in the product ${{\mathcal {P}}} \times {{{\mathcal {P}}}}'$, it is better to go to the quotient Q by the diagonal action of the affine group $Aff({\mathbb {R}})$. Elements of Q are called smooth relative configurations for K(a), $K'(a')$.

We say that a smooth configuration $(h,h') \in {{{\mathcal {P}}}}(a)\times {{{\mathcal {P}}}}(a')$ is

• linked if $h(I(a)) \cap h'(I(a')) \ne \emptyset$;

• intersecting if $h(K(\underline{a})) \cap h'(K(\underline{a}')) \ne \emptyset$, where $K(\underline{a})=K\cap I(\underline{a})$ and $K(\underline{a}')=K\cap I(\underline{a}')$;

• stably intersecting if it is still intersecting when we perturb it in ${{\mathcal {P}}}\times {{\mathcal {P}}}'$, and we perturb $(g,g')$ in $\Omega _\Sigma \times \Omega _{\Sigma '}$ .

All these definitions are invariant under the action of the affine group and, therefore, make sense for smooth relative configurations.

As in previous, we can introduce the spaces ${{\mathcal {A}}}$, ${{{\mathcal {A}}}}'$ associated to the limit geometries of g,  $g'$, respectively. We denote by ${{\mathcal {C}}}$ the quotient of ${{\mathcal {A}}}\times {{{\mathcal {A}}}}'$ by the diagonal action on the left of the affine group. An element of ${{\mathcal {C}}}$, represented by $(\underline{\theta },A) \in {{\mathcal {A}}}$,   $(\underline{\theta }', A') \in {{{\mathcal {A}}}}'$, is called a relative configuration of the limit geometries determined by $\underline{\theta }$, $\underline{\theta }'$. We have canonical maps

$$\begin{aligned} {{\mathcal {A}}}\times {{{\mathcal {A}}}}'\rightarrow & {} {{\mathcal {P}}}\times {{{\mathcal {P}}}}'\\ {{\mathcal {C}}}\rightarrow & {} Q \end{aligned}$$

allowing to define linked, intersecting, and stably intersecting configurations at the level of ${{\mathcal {A}}}\times {{{\mathcal {A}}}}'$ or ${{\mathcal {C}}}$.

Remark

For a configuration $((\underline{\theta }, A), (\underline{\theta }',A'))$ of limit geometries, one could also consider the weaker notion of stable intersection obtained by considering perturbations of g, $g'$ in $\Omega _\Sigma \times \Omega _{\Sigma '}$ and perturbations of $(\underline{\theta },A)$, $(\underline{\theta }',A')$ in ${{\mathcal {A}}}\times {{{\mathcal {A}}}}'$. We do not know of any example of expansive maps g, $g'$, and configurations $(\underline{\theta },A)$, $(\underline{\theta }',A')$ which are stably intersecting in the weaker sense, but not in the stronger sense.

We consider the following subset V of $\Omega _\Sigma \times \Omega _{\Sigma '}$ . A pair $(g, g')$ belongs to V if for any $[(\underline{\theta },A), (\underline{\theta }',A')] \in {{\mathcal {A}}} \times {{{\mathcal {A}}}}'$ there is a translation $R_t$ (in ${\mathbb {R}}$) such that $(R_t\circ A \circ k^{\underline{\theta }}, A'\circ k^{\prime \underline{\theta }'})$ is a stably intersecting configuration.

Definition 10

We say that a pair $(\psi , \Lambda )$, where $\Lambda$ is a horseshoe for $\psi$, has the property V if the stable and unstable cantor sets have the property V in the above sense.

The more important result in this setting is

Theorem 5.1

(Moreira-Yoccoz [21]) Let $\varphi$ be a $C^{\infty }$ diffeomorphism with a horseshoe $\Lambda$. Let $K^s$, $K^u$ are the stable and unstable Cantor sets respectively. Suppose that $HD(K^s)+HD(K^u)>1$. If  ${\mathcal {U}}$ is sufficiently small neighborhood $\varphi$ in $Diff^{\infty }(M)$, there is an open and dense set ${{\mathcal {U}}}^{*}\subset {\mathcal {U}}$ such that, for every $\psi \in {{\mathcal {U}}}^{*}$ the pair $(\psi ,\Lambda _{\psi })$ has the property V.

1.6 The Birkhoff Invariant

Let $f:({\mathbb {R}}^2,0)\rightarrow ({\mathbb {R}}^2,0)$ be a germ of diffeomorphism area-preserving (in dimension two is symplectic) and 0 a hyperbolic fixed point with eigenvalues $\lambda$ and $\lambda ^{-1}$, then the Birkhoff normal form (cf. [18]) says that there is an area-preserving change of coordinates $\eta$ such that $\eta ^{-1}\circ f \circ \eta =N$, where $N(x,y)=(U(xy)x,U^{-1}(xy)y)$ and U(xy) is a power series $\lambda +U_2xy+\cdots$ convergent in a neighborhood of $x=y=0$. In other words, in this coordinates f can be written by

$$\begin{aligned} f(x,y)=(\lambda x(1+axy+{\mathcal {O}}(\Vert (x,y)\Vert ^{4})),\lambda ^{-1} y(1-axy+{\mathcal {O}}(\Vert (x,y)\Vert ^{4})))\qquad \end{aligned}$$

(22)

and the number a is called the Birkhoff Invariant of f.

Lemma 5.11

The Birkhoff invariant for area-preserving diffeomorphisms in $({\mathbb {R}}^2,0)$ only depends on  3-jets in 0, $J^{3}(0)$. Moreover, the set of diffeomorphism area-preserving in $({\mathbb {R}}^2,0)$ such that the Birkhoff invariant is non-zero is open, dense, and invariant in $J^{3}(0)$.

Proof

For the proof of [18, Theorem 1 and 2], we have the first part and opening. For density, suppose that for some $f:({\mathbb {R}}^2,0)\rightarrow ({\mathbb {R}}^2,0)$, the Birkhoff invariant is zero, then for $\epsilon >0$ we consider the function $N_\epsilon (x,y):=(\lambda x(1+{\mathcal {O}}(\Vert (x,y)\Vert ^{4})),\lambda ^{-1} y(1+{\mathcal {O}}(\Vert (x,y)\Vert ^{4})))+\epsilon (x^2y,-xy^2)$, then the function $f_{\epsilon }=\eta \circ N_\epsilon \circ \eta ^{-1}$ is area-preserving diffeomorphism close to f with the Birkhoff invariant $\epsilon$.

Let f, g be as above and suppose that the Birkhoff invariant for f is non-zero, then $g^{-1}\circ f\circ g$ has the Birkhoff invariant non zero. Indeed, by the Birkhoff Normal Form [18, Theorem 1], there is an area-preserving change of coordinates $\eta$ such that $\eta ^{-1}\circ g^{-1}\circ f\circ g \circ \eta$ has the form (22), then $(g\circ \eta )^{-1}\circ f \circ (g\circ \eta )$ has the form (22). In other words, there is another area-preserving change of coordinates $g\circ \eta$ such that f has the form (22), but by the unicity of the Birkhoff normal form (see [18, page 674]), we have that the Birkhoff invariant of $g^{-1}\circ f\circ g$ is equal to the Birkhoff invariant of f, therefore non-zero. $\square$