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Fast–Slow Partially Hyperbolic Systems Versus Freidlin–Wentzell Random Systems

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Abstract

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Freidlin–Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a “sink” with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.

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Notes

  1. In fact in such papers it was assumed only \(F_\varepsilon \in {\mathcal C}^4({\mathbb T}^2,{\mathbb T}^2)\), here we need a bit more regularity.

  2. Admittedly a rather degenerate Markov process as the transition kernel is singular.

  3. \(F_*\) stands for the pushforward, namely \(F_*\mu (\varphi )=\mu (\varphi \circ F)\).

  4. Recall that \(\mu \) is a physical measure if, for all continuous g, \(\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}g\circ F_\varepsilon ^k(x)=\mu (g)\) for x belonging to a set of positive Lebesgue measure.

  5. Note that [13, Theorem 6.3], does not provide the explicit T dependence in the lower bound of the measure of \(Q_h\) stated here, however the latter is not needed to prove that the measure is strictly positive. On the other end, once there exists one trajectory in \(supp \,\ell \) that ends up in \(B(\theta _1,\frac{1}{2}{C_\#}\varepsilon ^{ 5/12})\), then trajectories that start in an \(\exp {-{c_\#}\varepsilon ^{-1} T}\) neighborhood will depart from such a trajectory less than \(\frac{1}{2}{C_\#}\varepsilon ^{ 5/12}\) in time T, hence the current claim.

  6. Related results are present in [30], where they are investigated from the point of view of viscosity solutions.

  7. We refrain from proving it but it is not very hard to check it numerically.

  8. As defined after Theorem 5

References

  1. Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alves, J.F., Luzzatto, S., Pinheiro, V.: Markov Structures and Decay of Correlations for Non-uniformly Expanding Dynamical Systems. Ann. Inst. H. Poincar Anal. Non Linéaire 22(6), 817–839 (2005)

  3. Avila, A., Gouëzel, S., Tsujii, M.: Smoothness of solenoid attractors. Discret. Contin. Dyn. Syst. 15(1), 21–35 (2006)

    Article  MATH  Google Scholar 

  4. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge (2000)

    Book  MATH  Google Scholar 

  5. Barrera, G., Jara, M.: Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems. J. Stat. Phys. 163, 113–138 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Baxendale, P.: Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Theory Relat. Fields 81, 521–554 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonatti, C., Díaz, L.J., Viana, M.: Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopedia of Mathematical Sciences. Springer, Berlin (2004)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Brin, M.: On dynamical coherence. Ergod. Theory Dyn. Syst. 23(2), 395–401 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Armando, A., Júnior, C.: Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Isr. J. Math. 130, 29–75 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. De Simoi, J., Liverani, C.: The Martingale approach after Varadhan and Dolpogpyat. In: Dolgopyat, D., Pesin, Y., Pollicott, M., Stoyanov, L (ed.) Hyperbolic Dynamics, Fluctuations and Large Deviations, Proceedings of Symposia in Pure Mathematics, AMS, vol. 89, pp 311–339 (2015)

  12. De Simoi, J., Liverani, C.: Fast-slow partially hyperbolic systems. Limit Theorems. Preprint arXiv:1408.5453

  13. De Simoi, J., Liverani, C.: Fast-slow partially hyperbolic systems. Statistical properties. Inventiones. 1–81 (2016). doi:10.1007/s00222-016-0651-y

  14. Deuschel, J.-D., Stroock, D.W.: Large deviations, Pure and Applied Mathematics, vol. 137. Academic Press Inc, Boston (1989)

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Dolgopyat, D.: Lectures on u-Gibbs states. http://www2.math.umd.edu/~dolgop/ugibbs.pdf

  17. Dolgopyat, D.: On mixing properties of compact group extensions of hyperbolic systems. Isr. J. Math. 130, 157–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 260, third edition. Springer, Heidelberg (2012). Translated from the 1979 Russian original by Joseph Szücs

  19. Giacomin, G., Poquet, C., Shapira, A.: Small noise and long time phase diffusion in stochastic limit cycle oscillators. arXiv:1512.04436

  20. Hairer, M., Mattingly, J.: Yet another look at Harris’ ergodic theorem for Markov chains, seminar on stochastic analysis, random fields and applications VI. Progr. Probab. 63, 109–117 (2011)

    MATH  Google Scholar 

  21. Hasselblatt, B., Pesin, Y.: Partially Hyperbolic Dynamical Systems. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, pp. 1–55. Elsevier B. V, Amsterdam (2006)

    Google Scholar 

  22. Hirayama, M., Pesin, Y.: Non-absolutely continuous foliations. Isr. J. Math. 160(1), 173–187 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds (Lecture Notes in Mathematics 583). Springer, New York (1977)

    Book  Google Scholar 

  24. Kleptsyn, V., Volk, D.: Physical measures for nonlinear random walks on interval. Mosc. Math. J. 14(2), 339–365 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Kleptsyn, V.A., Nalskii, M.B.: Contraction of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38(4), 267–282 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kifer, J.I.: On the asymptotic behavior of transition densities of processes with small diffusion, Akademija Nauk SSSR. Teorija Verojatnosteĭ i ee Primenenija 21(3), 527–536 (1976)

    Google Scholar 

  27. Kifer, Y.: The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Isr. J. Math. 40, 74–96 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kifer, Y.: Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem. Amer. Math. Soc. 201(944), viii+129 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Sérgio, R.: Fenley. Quasi-isometric foliations. Topology 31(3), 667–676 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wendell, H., Panagiotis, E.: PDE-viscosity solution approach to some problems of large deviations. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 13, 171–192 (1986)

    MathSciNet  MATH  Google Scholar 

  31. Gouëzel, S.: Decay of correlations for nun uniformly expanding systems. Bull. Soc. Math. Fr. 134(1), 1–31 (2006)

    Article  MATH  Google Scholar 

  32. Hertz, F.R., Hertz, J.R., Ures, R.: A non-dynamically coherent example on \({\mathbb{T}}^3\). Ann. Inst. H. Poincar Anal. Non Linéaire 33(4), 10231032 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. Fr. 134(1), 1–31 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergod. Theory Dyn. Syst. 26(1), 189–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liverani, C.: Central Limit Theorem for Deterministic Systems. International Conference on Dynamical Systems (Montevideo, 1995), 5675, Pitman Res. Notes Math. Ser., 362. Longman, Harlow (1996)

  36. John, M.: Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intell. 19(2), 3032 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Ruelle, D., Wilkinson, A.: Absolutely singular dynamical foliations. Commun. Math. Phys. 219(3), 481–487 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Shub, M., Sullivan, D.: Expanding endomorphisms of the circle revisited. Ergod. Theory Dyn. Syst. 5(6), 285–289 (1985)

    MathSciNet  MATH  Google Scholar 

  39. Shub, M., Wilkinson, A.: Pathological foliations and removable zero exponents. Invent. Math. 139(3), 495–508 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Masato, T.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1), 37–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Villani, C.: Hypocoercivity. Mem. Amer. Math. Soc. 202(950), iv+141 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Baraviera, A.T., Bonatti, C.: Removing zero lyapunov exponents. Ergod. Theory Dyn. Syst. 23(12), 1655–1670 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ponce, G., Tahzibi, A.: Central lyapunov exponent of partially hyperbolic diffeomorphisms of \({{\mathbb{T}}^3}\). Proc. Amer. Math. Soc. 142(9), 3193–3205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ponce, G., Tahzibi, A., Varao, R.: Minimal yet measurable foliations. J. Mod. Dyn. 8, 93–107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Saghin, R., Xia, Z.: Geometric expansion, lyapunov exponents and foliations. Annales de l’Institut Henri Poincare (C) non linear. Analysis 26(2), 689–704 (2009)

    MATH  Google Scholar 

  46. Tsujii, M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1), 37–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Varao, R.: Center foliation: absolute continuity, disintegration and rigidity. Ergod. Theory Dyn. Syst. 36, 256–275 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work has been supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953). D. V. has been partially funded by the Russian Academic Excellence Project ‘5–100’.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St George St, Toronto, ON, M5S 2E4, Canada

    Jacopo de Simoi

  2. Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133, Roma, Italy

    Carlangelo Liverani

  3. Institut Camille Jordan UMR 5208, Université de Lyon Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne, France

    Christophe Poquet

  4. Centre for Cognition and Decision making, National Research University Higher School of Economics, Moscow, Russian Federation

    Denis Volk

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  1. Jacopo de Simoi
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  2. Carlangelo Liverani
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Correspondence to Carlangelo Liverani.

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de Simoi, J., Liverani, C., Poquet, C. et al. Fast–Slow Partially Hyperbolic Systems Versus Freidlin–Wentzell Random Systems. J Stat Phys 166, 650–679 (2017). https://doi.org/10.1007/s10955-016-1628-3

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