Random Products of Standard Maps


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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  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.


  1. [ABV00]

    Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–298 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  2. [Anz51]

    Anzai, H.: Ergodic skew product transformations on the torus. Osaka Math. J. 3(1), 83–99 (1951)

    MathSciNet  MATH  Google Scholar 

  3. [AV10]

    Avila, A., Viana, M.: Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181(1), 115–178 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  4. [BC91]

    Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991)

    MathSciNet  Article  Google Scholar 

  5. [BC14]

    Berger, P., Carrasco, P.D.: Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Commun. Math. Phys. 329(1), 239–262 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  6. [BDP02]

    Burns, K., Dolgopyat, D., Pesin, Ya.: Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108(5), 927–942 (2002)

    MathSciNet  Article  Google Scholar 

  7. [BDPP08]

    Burns, K., Dolgopyat, D., Pesin, Y., Pollicott, M.: Stable ergodicity for partially hyperbolic attractors with negative central exponents. J. Mod. Dyn. 2(1), 63–81 (2008)

    MathSciNet  Article  Google Scholar 

  8. [BDV05]

    Bonatti, C., Díaz, L., Viana, M.: Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Physics. Springer, Berlin (2005)

    Google Scholar 

  9. [Ber19]

    Berger, P.: Abundance of non-uniformly hyperbolic Hénon-like endomorphisms. Astérisque 410, 53–176 (2019)

    Google Scholar 

  10. [Bow08]

    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470. Springer (2008)

  11. [BR75]

    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  12. [BV00]

    Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Isr. J. Math. 115(1), 157–193 (2000)

    MathSciNet  Article  Google Scholar 

  13. [BV05]

    Bochi, J., Viana, M.: The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. Math. 161(3), 1423–1485 (2005)

    MathSciNet  Article  Google Scholar 

  14. [BW99]

    Burns, K., Wilkinson, A.: Stable ergodicity of skew products. Ann. Sci. de le Ecole Norm. Sup. 32(6), 859–889 (1999)

    MathSciNet  Article  Google Scholar 

  15. [BXY17]

    Blumenthal, A., Xue, J., Young, L.S.: Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. Math. 185, 285–310 (2017)

    MathSciNet  Article  Google Scholar 

  16. [BY93]

    Benedicks, M., Young, L.S.: Sinai–Bowen–Ruelle measures for certain Henon maps. Invent. Math. 112(1), 541–576 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  17. [CHHU17]

    Carrasco, P.D., Hertz, F.R., Hertz, J.R., Ures, R.: Partial hyperbolicity in dimension three. Ergod. Theory Dyn. Syst. 8, 2801–2837 (2017). https://doi.org/10.1017/etds.2016.142

    Article  MATH  Google Scholar 

  18. [Chi79]

    Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52(5), 263–379 (1979)

    ADS  MathSciNet  Article  Google Scholar 

  19. [CK15]

    Castejón, O., Kaloshin, V.: Random iteration of maps on a cylinder and diffusive behavior. Preprint at Arxiv (2015)

  20. [CP15]

    Crovisier, S., Potrie, R.: Introduction to partially hyperbolic dynamics. In: Lecture Notes for the School on Dynamical Systems, ICTP (2015)

  21. [CS08]

    Chirikov, B., Shepelyansky, D.: Chirikov standard map. Scholarpedia 3(3), 3550 (2008)

    ADS  Article  Google Scholar 

  22. [Dol00]

    Dolgopyat, D.: On dynamics of mostly contracting diffeomorphisms. Commun. Math. Phys. 213(1), 181–201 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  23. [Dol05]

    Dolgopyat, D.: Averaging and invariant measures. Mosc. Math. J. 5(3), 537–576 (2005)

    MathSciNet  Article  Google Scholar 

  24. [dSL18]

    de Simoi, J., Liverani, C.: Limit theorems for fast-slow partially hyperbolic systems. Invent. Math. 213, 811–1016 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  25. [Dua94]

    Duarte, P.: Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. Henri Poincare 11(4), 359–409 (1994)

    MathSciNet  Article  Google Scholar 

  26. [Fro72]

    Froeschlé, C.: Numerical study of a four-dimensional mapping. Astron. Astrophys. 16, 172 (1972)

    ADS  MathSciNet  MATH  Google Scholar 

  27. [Gol01]

    Gole, C.: Symplectic Twist Maps: Global Variational Techniques (Advanced Series in Nonlinear Dynamics). World Scientific Pub Co Inc, Singapore (2001)

    Google Scholar 

  28. [Gor12]

    Gorodetski, A.: On stochastic sea of the standard map. Commun. Math. Phys. 309(1), 155–192 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  29. [HPS77]

    Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer (1977)

  30. [Jak81]

    Jakobson, M.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  31. [KKM17]

    Korepanov, A., Kosloff, Z., Melbourne, I.: Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems. Studia Math. 238(1), 59–89 (2017)

    MathSciNet  Article  Google Scholar 

  32. [Kni96]

    Knill, O.: Topological entropy of standard type monotone twist maps. Trans. AMS 348(8), 2999–3013 (1996)

    MathSciNet  Article  Google Scholar 

  33. [Led84]

    Ledrappier, F.: Quelques proprietes des exposants caracteristiques. In: Hennequin, P.L. (ed.) École d’Été de Probabilités de Saint-Flour XII-1982, pp. 305–396. Springer, Berlin (1984)

    Google Scholar 

  34. [Mar16]

    Marin, K.: \({\cal{C}}^r\)-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms. Comment. Math. Helv. 91(2), 357–396 (2016)

    MathSciNet  Article  Google Scholar 

  35. [Oba20]

    Obata, D.: On the stable ergodicity of Berger–Carrasco’s example. Ergod. Theory Dyn. Syst. 40(4), 1008–1056 (2020)

    MathSciNet  Article  Google Scholar 

  36. [Ose68]

    Oseledets, V.I.: A multiplicative ergodic theorem Trans. Lyapunov characteristic numbers for dynamical systems. Mosc. Math. Soc. 19, 197–231 (1968)

    MATH  Google Scholar 

  37. [PC10]

    Pesin, Y., Climenhaga, V.: Open problems in the theory of non-uniform hyperbolicity. Discrete Contin. Dyn. Syst. 27(2), 589–607 (2010)

    MathSciNet  Article  Google Scholar 

  38. [Pes77]

    Pesin, Y.: Characteristic Lyapunov exponents, and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)

    Article  Google Scholar 

  39. [Pes04]

    Pesin, Y.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics. European Mathematical Society (2004)

  40. [Rob98]

    Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (Studies in Advanced Mathematics), 2nd edn. CRC Press, Boca Raton (1998)

    Google Scholar 

  41. [RQ92]

    Roberts, J.A.G., Quispel, G.R.W.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216(2–3), 63–177 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  42. [Rue76]

    Ruelle, D.: A measure associated with axiom-a attractors. Am. J. Math. 98(3), 619 (1976)

    MathSciNet  Article  Google Scholar 

  43. [Shu86]

    Shub, M.: Global Stability of Dynamical Systems. Springer, Berlin (1986)

    Google Scholar 

  44. [Via97]

    Viana, M.: Multidimensial non-hyperbolic attractors. Publ. Math IHES 85(1), 63–96 (1997)

    Article  Google Scholar 

  45. [WLL90]

    Wood, B.P., Lichtenberg, A.J., Lieberman, M.A.: Arnold diffusion in weakly coupled standard maps. Phys. Rev. A 42(10), 5885–5893 (1990)

    ADS  MathSciNet  Article  Google Scholar 

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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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Correspondence to Pablo D. Carrasco.

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Communicated by C. Liverani



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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