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Asymptotic escape rates and limiting distributions for multimodal maps

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  09 March 2020

MARK F. DEMERS Open the ORCID record for MARK F. DEMERS [Opens in a new window]  and
MIKE TODD Open the ORCID record for MIKE TODD [Opens in a new window]
MARK F. DEMERS
Affiliation:
Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA email mdemers@fairfield.edu
MIKE TODD
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK email m.todd@st-andrews.ac.uk
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Abstract

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.

Keywords

multimodal mapsHofbauer extensionopen systemsescape ratetransfer operator

MSC classification

Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37D35: Thermodynamic formalism, variational principles, equilibrium states 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1656 - 1705
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Afraimovich, V. S. and Bunimovich, L. A.. Which hole is leaking the most: a topological approach to study open systems. Nonlinearity 23(3) (2010), 643–656.CrossRefGoogle Scholar
Altmann, E. G., Portela, J. S. E. and Tél, T.. Leaking chaotic systems. Rev. Mod. Phys. 85 (2013), 869–918.CrossRefGoogle Scholar
Bruin, H., Demers, M. F. and Melbourne, I.. Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. & Dynam. Sys. 30 (2010), 687–728.CrossRefGoogle Scholar
Bruin, H., Demers, M. F. and Todd, M.. Hitting and escaping statistics: mixing, targets and holes. Adv. Math. 328 (2018), 1263–1298.CrossRefGoogle Scholar
Bruin, H., Luzzatto, S. and van Strien, S.. Decay of correlations in one-dimensional dynamics. Ann. Sci. Éc. Norm. Supér. 36 (2003), 621–646.CrossRefGoogle Scholar
Bruin, H., Rivera-Letelier, J., Shen, W. and van Strien, S.. Large derivatives, backward contraction and invariant densities for interval maps. Invent. Math. 172 (2008), 509–533.CrossRefGoogle Scholar
Bahsoun, W. and Vaienti, S.. Metastability of certain intermittent maps. Nonlinearity 25(1) (2012), 107–124.CrossRefGoogle Scholar
Bahsoun, W. and Vaienti, S.. Escape rates formulae and metastability for randomly perturbed maps. Nonlinearity 26(5) (2013), 1415–1438.CrossRefGoogle Scholar
Bunimovich, L. A. and Yurchenko, A.. Where to place a hole to achieve a maximal escape rate. Israel J. Math. 182(1) (2011), 229–252.CrossRefGoogle Scholar
Chernov, N. and Markarian, R.. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997), 271–314.CrossRefGoogle Scholar
Collet, P., Martínez, S. and Schmitt, B.. The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7 (1994), 1437–1443.CrossRefGoogle Scholar
Chernov, N., Markarian, R. and Troubetzkoy, S.. Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049–1073.CrossRefGoogle Scholar
Demers, M. F.. Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Th. & Dynam. Sys. 25 (2005), 1139–1171.CrossRefGoogle Scholar
Demers, M. F.. Dispersing billiards with small holes. Ergodic Theory, Open Dynamics and Coherent Structures (Springer Proceedings in Mathematics, 70) . Eds. Bahsoun, W., Bose, C. and Froyland, G.. Springer, New York, 2014, pp. 137–170.CrossRefGoogle Scholar
Demers, M. F. and Fernandez, B.. Escape rates and singular limiting distributions for intermittent maps with holes. Trans. Amer. Math. Soc. 368 (2016), 4907–4932.CrossRefGoogle Scholar
Dettmann, C. P. and Georgiou, O.. Survival probability for the stadium billiard. Physica D 238 (2009), 2395–2403.CrossRefGoogle Scholar
Dettmann, C. P., Georgiou, O., Knight, G. and Klages, R.. Dependence of chaotic diffusion on the size and position of holes. Chaos 22 (2012), 023132.Google Scholar
Demers, M. F., Ianzano, C., Mayer, P., Morfe, P. and Yoo, E.. Limiting distributions for countable state topological Markov chains with holes. Discrete Contin. Dyn. Sys. 37(1) (2017), 105–130.CrossRefGoogle Scholar
Dettmann, C. P. and Rahman, M. R.. Survival probability for open spherical billiards. Chaos 24 (2014), 043130.CrossRefGoogle ScholarPubMed
Dobbs, N. and Todd, M.. Free energy jumps up. Preprint, arXiv:1512.09245.Google Scholar
Dolgopyat, D. and Wright, P.. The diffusion coefficient for piecewise expanding maps of the interval with metastable states. Stochastics Dyn. 12 (2012), paper 1150005.CrossRefGoogle Scholar
Demers, M. F. and Todd, M.. Equilibrium states, pressure and escape for multimodal maps with holes. Israel J. Math. 221(1) (2017), 367–424.CrossRefGoogle Scholar
Demers, M. F. and Todd, M.. Slow and fast escape for open intermittent maps. Comm. Math. Phys. 351(2) (2017), 775–835.CrossRefGoogle Scholar
Demers, M. F. and Wright, P.. Behavior of the escape rate function in hyperbolic dynamical systems. Nonlinearity 25 (2012), 2133–2150.CrossRefGoogle Scholar
Demers, M. F., Wright, P. and Young, L.-S.. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys. 294 (2010), 353–388.CrossRefGoogle Scholar
Freitas, A. C. M., Freitas, J. M. and Todd, M.. The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics. Comm. Math. Phys. 321 (2013), 483–527.CrossRefGoogle Scholar
Freitas, A. C. M., Freitas, J. M. and Todd, M.. Speed of convergence for laws of rare events and escape rates. Stochastic Process. Appl. 125 (2015), 1653–1687.CrossRefGoogle Scholar
Ferrari, P. A., Kesten, H., Martínez, S. and Picco, P.. Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 (1995), 501–521.CrossRefGoogle Scholar
Froyland, G., Murray, R. and Stancevic, O.. Spectral degeneracy and escape dynamics for intermittent maps with a hole. Nonlinearity 24 (2011), 2435–2463.CrossRefGoogle Scholar
Ferguson, A. and Pollicott, M.. Escape rates for Gibbs measures. Ergod. Th. & Dynam. Sys. 32 (2012), 961–988.CrossRefGoogle Scholar
Gonzalez-Tokman, C., Hunt, B. and Wright, P.. Approximating invariant densities for metastable systems. Ergod. Th. & Dynam. Sys. 34 (2014), 1230–1272.Google Scholar
Hofbauer, F.. Piecewise invertible dynamical systems. Probab. Theory Relat. Fields 72 (1986), 359–386.CrossRefGoogle Scholar
Hofbauer, F. and Raith, P.. Topologically transitive subsets of piecewise monotonic maps, which contain no periodic points. Monatsh. Math. 107 (1989), 217–239.CrossRefGoogle Scholar
Iommi, G. and Todd, M.. Natural equilibrium states for multimodal maps. Comm. Math. Phys. 300 (2010), 65–94.CrossRefGoogle Scholar
Iommi, G. and Todd, M.. Thermodynamic formalism for interval maps: inducing schemes. Dyn. Syst. 28 (2013), 354–380.CrossRefGoogle Scholar
Keller, G.. Lifting measures to Markov extensions. Monatsh. Math. 108 (1989), 183–200.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) XXVIII (1999), 141–152.Google Scholar
Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3) (2009), 519–534.CrossRefGoogle Scholar
Liverani, C. and Maume-Deschamps, V.. Lasota–Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Inst. Henri Poincaré Probab. Stat. 39 (2003), 385–412.CrossRefGoogle Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials. Comm. Math. Phys. 328 (2014), 397–419.CrossRefGoogle Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
Przytycki, F. and Rivera-Letelier, J.. Geometric pressure for multimodal maps of the interval. Mem. Amer. Math. Soc. 259(1246).Google Scholar
Pollicott, M. and Urbanski, M.. Open Conformal Systems and Perturbations of Transfer Operators (Lecture Notes in Mathematics, 2206) . Springer, Berlin, 2018.Google Scholar
Pianigiani, G. and Yorke, J.. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351–366.Google Scholar
Rivera-Letelier, J. and Shen, W.. Statistical properties of one-dimensional maps under weak hyperbolicity assumptions. Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 1027–1083.CrossRefGoogle Scholar
Vere-Jones, D.. Geometric ergodicity in denumerable Markov chains. Quart. J. Math 13 (1962), 7–28.CrossRefGoogle Scholar
Young, L. S.. Some large deviation results for dynamical systems. Trans. Amer. Math. Soc. 318 (1990), 525–543.Google Scholar