Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T15:00:25.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical properties for compositions of standard maps with increasing coefficient

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior Limit theorems Ergodic theory

Published online by Cambridge University Press:  07 February 2020

ALEX BLUMENTHAL Open the ORCID record for ALEX BLUMENTHAL [Opens in a new window]
ALEX BLUMENTHAL*
Affiliation:
University of Maryland, College Park, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Chirikov standard map is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Rigorous analysis is notoriously difficult and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing coefficient. When the coefficients increase to infinity at a sufficiently fast polynomial rate, we obtain a strong law, a central limit theorem, and quantitative mixing estimates for Holder observables. The methods used are not specific to the standard map and apply to a class of compositions of ‘prototypical’ two-dimensional maps with hyperbolicity on ‘most’ of phase space.

Keywords

non-autonomous dynamicsnon-uniform hyperbolicitystatistical properties of deterministic dynamics

MSC classification

Primary: 37C60: Nonautonomous smooth dynamical systems 37A25: Ergodicity, mixing, rates of mixing 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 981 - 1024
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aimino, R., Hu, H., Nicol, M., Török, A. and Vaienti, S.. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. A 35(3) (2015), 793–806.10.3934/dcds.2015.35.793CrossRefGoogle Scholar
Arnold, L.. Random Dynamical Systems. Springer Science & Business Media, Berlin, 2013.Google Scholar
Arnoux, P. and Fisher, A. M.. Anosov families, renormalization and non-stationary subshifts. Ergod. Th. & Dynam. Sys. 25(3) (2005), 661–709.CrossRefGoogle Scholar
Bakhtin, V. I.. Random processes generated by a hyperbolic sequence of mappings, I. Russ. Acad. Sci. Izv. Math. 44(2) (1995), 247–279.Google Scholar
Bakhtin, V. I.. Random processes generated by a hyperbolic sequence of mappings, II. Russ. Acad. Sci. Izv. Math. 44(3) (1995), 617–627.Google Scholar
Barreira, L. and Pesin, Y.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents (Encyclopedia of Mathematics and its Applications, 115) . Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. of Math. (2) 133(1) (1991), 73–169.CrossRefGoogle Scholar
Berger, P. and Turaev, D.. On Herman’s positive entropy conjecture. Adv. Math. 349 (2019), 1234–1288.CrossRefGoogle Scholar
Blumenthal, A., Xue, J. and Young, L.-S.. Lyapunov exponents and correlation decay for random perturbations of some prototypical 2D maps. Comm. Math. Phys. 359(1) (2018), 347–373.CrossRefGoogle Scholar
Blumenthal, A., Xue, J. and Young, L.-S.. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2) 185(1) (2017), 285–310.CrossRefGoogle Scholar
Chernov, N. and Dolgopyat, D.. Brownian Brownian Motion-I. American Mathematical Society, Providence, RI, 2009.CrossRefGoogle Scholar
Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127) . American Mathematical Society, Providence, RI, 2006.10.1090/surv/127CrossRefGoogle Scholar
Chirikov, B. V.. A universal instability of many-dimensional oscillator systems. Phys. Rep. 52(5) (1979), 263–379.CrossRefGoogle Scholar
Conze, J.-P. and Raugi, A.. Limit theorems for sequential expanding dynamical systems. Ergodic Theory and Related Fields: 2004–2006 Chapel Hill Workshops on Probability and Ergodic Theory, University of North Carolina, Chapel Hill, North Carolina (Contemporary Mathematics, 430) . American Mathematical Society, Providence, RI, 2007, pp. 89–121.CrossRefGoogle Scholar
De Simoi, J.. Stability and instability results in a model of Fermi acceleration. Discrete Contin. Dyn. Syst. A 25(3) (2009), 719–750.CrossRefGoogle Scholar
De Simoi, J.. Fermi acceleration in anti-integrable limits of the standard map. Comm. Math. Phys. 321(3) (2013), 703–745.CrossRefGoogle Scholar
Dolgopyat, D. I.. Averaging and invariant measures. Mosc. Math. J. 5(3) (2005), 537–576.CrossRefGoogle Scholar
Dolgopyat, D.. Bouncing balls in non-linear potentials. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 165–182.10.3934/dcds.2008.22.165CrossRefGoogle Scholar
Dolgopyat, D., Kaloshin, V. and Koralov, L.. Sample path properties of the stochastic flows. Ann. Probab. 32 (2004), 1–27.Google Scholar
Duarte, P.. Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 359–409.CrossRefGoogle Scholar
Gorodetski, A.. On stochastic sea of the standard map. Comm. Math. Phys. 309(1) (2012), 155–192.CrossRefGoogle Scholar
Kifer, Y.. Ergodic Theory of Random Transformations (Progress in Probability and Statistics, 10) . Springer Science & Business Media, Heidelberg, 2012.Google Scholar
Liverani, C.. Decay of correlations. Ann. of Math. (2) 142(2) (1995), 239–301.CrossRefGoogle Scholar
Liverani, C.. Central limit theorem for deterministic systems. Int. Conf. Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362) . Longman, London, 1996, pp. 56–75.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19(3) (1999), 671–685.CrossRefGoogle Scholar
Nicol, M., Török, A. and Vaienti, S.. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys. 38(3) (2018), 1127–1153.CrossRefGoogle Scholar
Pesin, Ya. B. and Sinai, Ya. G.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 417–438.CrossRefGoogle Scholar
Rokhlin, V. A.. On the fundamental ideas of measure theory. Mat. Sb. 67(1) (1949), 107–150.Google Scholar
Ruelle, D.. A measure associated with Axiom-A attractors. Amer. J. Math. 98 (1976), 619–654.CrossRefGoogle Scholar
Stenlund, M.. Non-stationary compositions of Anosov diffeomorphisms. Nonlinearity 24(10) (2011), 2991–3018.CrossRefGoogle Scholar
Stenlund, M.. A vector-valued almost sure invariance principle for Sinai billiards with random scatterers. Comm. Math. Phys. 325(3) (2014), 879–916.CrossRefGoogle Scholar
Stenlund, M., Young, L.-S. and Zhang, H.. Dispersing billiards with moving scatterers. Comm. Math. Phys. 322(3) (2013), 909–955.CrossRefGoogle Scholar
Young, L.-S.. Ergodic theory of differentiable dynamical systems. Real and Complex Dynamical Systems. Springer, Heidelberg, 1995, pp. 293–336.CrossRefGoogle Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.CrossRefGoogle Scholar