Random Products of Standard Maps

Abstract

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics is given by (a continuum of parameters in) the Froeschlé family. These types of coupled systems appear as some induced maps in models for the study of Arnold diffusion. Consequently, we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics. The methods are also suitable for studying cocycles over shift spaces, and do not demand any low dimensionality condition on the fiber.

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Notes

  1. 1.

    Here [r] denotes the integer part of r.

  2. 2.

    If \(\Lambda \subset M\) is f-invariant, we say that a df-invariant sum \(E\oplus F\subset T_{\Lambda }M\) is a dominated splitting if there exists \(0<K<1\) such that for \(n\ge 0\), \(p\in \Lambda \) and unit vectors \(v\in E(p),w\in F(p)\) it holds \(\left\| d_pf^n(v)\right\| \le K\cdot \left\| d_pf^n(w)\right\| \).

  3. 3.

    Meaning that they are induced by linear maps \({\mathbb {R}}^2\rightarrow {\mathbb {R}}^{2e}\).

  4. 4.

    Here \(*\) denotes concatenation of paths.

  5. 5.

    cf. Lemma 8 and Corollary 6 in [BC14]: observe that by Corollary 4.3 the length of the admissible curves is bounded.

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Acknowledgements

The results here presented are based on previous joint work with Pierre Berger, and are deeply influenced by several discussion that the author had with him during these times. I would like to thank Pierre for sharing his ideas with me. Initial stages of this paper were prepared while I was visiting PUCV-Valparaiso; I would like to thank Carlos Vazquez for his encouragement and generosity in these moments. Also, I would like to thank Enrique Pujals and Jiagang Yang for encouragement and the interest deposited in this project. Finally, I would like to express my sincere thanks to the referees who not only gave me many suggestions to improve the presentation and caught inaccuracies-plain errors, but also gave me ideas on how to improve the results appearing on previous versions.

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Communicated by C. Liverani

Appendix

Appendix

Given a coupled family \(\{f_r=A_r\times _{\varphi _r} S_r\}_{r}\) over an hyperbolic base, it is desirable to have some conditions that will imply non-uniform hyperbolicity of the family instead of only positive exponents along the fiber direction. We discuss a possible approach here.

We assume that both \(\{S_r\}_{r},\{S_r^{-1}\}_{r}\) satisfy S-1,S-2, \(A_r\in SL(2,{\mathbb {Z}})\) hyperbolic, and define the following conditions.

  • A-3

    • \(\displaystyle {\frac{\left\| dS^{-1}_r\right\| ^3\cdot \left\| dS_r\right\| }{\lambda _r\left\| \varphi _r|E^s_{A_r}\right\| }\xrightarrow [r\longrightarrow +\infty ]{}0,\frac{\left\| dS^{-1}_r\right\| \cdot \left\| dS_r\right\| \cdot \left\| \varphi _r|E_{A_r}^u\right\| }{\lambda _r}\xrightarrow [r\longrightarrow +\infty ]{}0}\).

    • There exists \(q\in {\mathbb {N}}\) such that

      $$\begin{aligned} \frac{\left\| dS_r\right\| ^{3q}\left\| d^2S_r^{-1}\right\| ^{{3q}}}{\lambda _r}\xrightarrow [r\longrightarrow +\infty ]{}0. \end{aligned}$$
  • A-4

    • \(\displaystyle {\min _{1\le i\le e}\left\| P_i\circ dS_r^{-1}\circ \varphi _r|E^s_{A_r}\right\| >0}\).

    • \(\displaystyle {\max _{1\le \i \le e}\frac{\left\| P_i\circ dS_r^{-1}\circ \varphi _r|E^u_{A_r}\right\| +\left\| P_i\circ dS_r^{-1}\right\| }{\lambda ^2\left\| P_i\circ dS^{-1}\circ \varphi _r|E^s_{A_r}\right\| }\xrightarrow [r\longrightarrow +\infty ]{}0}\).

    • \(\displaystyle {\frac{\min _{1\le i\le e}\left\| P_i\circ dS^{-1}\circ \varphi _r|E^s_{A_r}\right\| }{\max _{1\le i\le e} \left\| P_i\circ dS^{-1}\circ \varphi _r|E^s_{A_r}\right\| }\xrightarrow [r\longrightarrow +\infty ]{} 1.}\)

Definition 5.3

We say the the coupling in a family of skew-products \(\{f_r=A_r\times _{\varphi _r} S_r\}_{r}\) is bi-adapted if it is adapted and moreover the family satisfies A-3, A-4 above. In this case we also say that \(\{f_r\}\) is a bi-adapted family.

The lack is symmetry between these and A-1, A-2 comes from the different form of df and \(df^{-1}\); compare (23) with (26).

It is direct (although somewhat tedious) to check that if \(f_r\) verifies A-1 to A-4 then \(df_r^{-1}\) verifies A-1, A-2. We can thus apply our Main Theorem to both \(\{f_r\}_{r}, \{f_r^{-1}\}_{r}\) and deduce the following.

Corollary B

Assume that \(\{f_r=A_r\times _{\varphi _r} S_r:M={\mathbb {T}}^l\times {\mathbb {T}}^{2e}\rightarrow M\}_{r}\) is a bi-adapted family with \(A_r\in SL(2,{\mathbb {Z}})\) hyperbolic and the families \(\{S_r\}_r,\{S_r^{-1}\}_r\) satisfy conditions S-1, S-2. Then there exists \(r_0\) such that for every \(r\ge r_0\) there exists \(Q(r)>0\) satisfying for Lebesgue almost every \(m\in M\)

$$\begin{aligned} v\in T_mM{\setminus }\{0\}\quad \Rightarrow \lim _{n\longrightarrow \infty }\left| \frac{\log \left\| d_mf^n(v)\right\| }{n}\right| >Q(r). \end{aligned}$$

In particular \(f_r\) is NUH and has a physical measure. The same holds for any \({\widetilde{f}}\) in a \({\mathcal {C}}^2\) neighborhood \({\mathcal {U}}_r\) of \(f_r\).

The previous Corollary is given for completeness. In practice however, checking A-4 could be difficult since it depends on the relation between \(dS_r^{-1}\) and \(d\varphi _r|E^u_A\), and this control may not be achievable. This is why in the examples given in Section 3 we appeal to other arguments to deal with the inverse map.

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Carrasco, P.D. Random Products of Standard Maps. Commun. Math. Phys. 377, 773–810 (2020). https://doi.org/10.1007/s00220-020-03765-6

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