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Entropy continuity for interval maps with holes

Published online by Cambridge University Press:  24 January 2017

OSCAR F. BANDTLOW  and
HANS HENRIK RUGH
OSCAR F. BANDTLOW
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E3 4NS, UK email o.bandtlow@qmul.ac.uk
HANS HENRIK RUGH
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France email hans-henrik.rugh@math.u-psud.fr
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Abstract

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole at a uniform rate. We show that under suitable conditions the topological entropy varies locally Hölder continuously with the local Hölder exponent depending itself on the value of the topological entropy.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2036 - 2061
Copyright
© Cambridge University Press, 2017 

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References

Baladi, V. and Keller, G.. Zeta functions and transfer operators for piecewise monotone transformations. Comm. Math. Phys. 127 (1990), 459–477.Google Scholar
Bahsoun, W., Bose, C. and Froyland, G. (Eds). Ergodic Theory, Open Dynamics, and Coherent Structures. Springer, New York, 2014.Google Scholar
Boyarsky, A. and Gora, P.. Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser, Boston, 1997.Google Scholar
Bruin, H., Demers, M. and Melbourne, I.. Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. & Dynam. Sys. 30 (2010), 687–728.Google Scholar
Bunimovich, L. and Yurchenko, A.. Where to place a hole to achieve a maximal escape rate. Israel J. Math. 182 (2008), 229–252.Google Scholar
Carminati, C. and Tiozzo, G.. The local Hölder exponent for the dimension of invariant subsets of the circle. Ergod. Th. & Dynam. Sys.; doi:10.1017/etds.2015.135, published online 8 March 2016.Google Scholar
Chatelin, F.. Spectral Approximation of Linear Operators. Academic Press, New York, 1983.Google Scholar
Chernov, N. and Markarian, R.. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Brasil. Mat. 28 (1997), 271–314.Google 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.Google 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.Google Scholar
Cristadoro, G., Knight, G. and Degli Esposti, M.. Follow the fugitive: an application of the method of images to open systems. J. Phys. A 46 (2013), 272001, 8pp.Google Scholar
Demers, M. and Wright, P.. Behaviour of the escape rate function in hyperbolic dynamical systems. Nonlinearity 25 (2012), 2133–2150.Google Scholar
Demers, M., Wright, P. and Young, L-S.. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys. 294 (2010), 353–388.Google Scholar
Dettmann, C.. Open circle maps: small hole asymptotics. Nonlinearity 26 (2013), 307–317.Google Scholar
Dunford, N. and Schwartz, J. T.. Linear Operators. Part I. General Theory. Interscience, New York, 1958.Google Scholar
Ferguson, A. and Pollicott, M.. Escape rates for Gibbs measures. Ergod. Th. & Dynam. Sys. 32 (2012), 961–988.Google Scholar
Giusti, E.. Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston, 1984.Google Scholar
Guckenheimer, J.. The growth of topological entropy for one-dimensional maps. Global Theory of Dynamical Systems (Proc. Int. Conf., Northwestern University, Evanston, IL, 1979) (Lecture Notes in Mathematics, 819) . Eds. Nitecki, Z. and Robinson, C.. Springer, Berlin, 1980, pp. 216–223.Google Scholar
Hennion, H.. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118 (1993), 627–634.Google Scholar
Hofbauer, F.. Periodic points for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 5 (1985), 237–256.Google Scholar
Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.Google Scholar
Ionescu Tulcea, C. T. and Marinescu, G.. Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 (1950), 140–147.Google Scholar
Isola, S. and Politi, A.. Universal encoding for unimodal maps. J. Statist. Phys. 61 (1990), 263–291.Google Scholar
Keller, G.. On the rate of convergence to equilibrium in one-dimensional systems. Comm. Math. Phys. 96 (1984), 181–193.Google Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999), 141–152.Google Scholar
Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135 (2009), 519–534.Google 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.Google Scholar
Misiurewicz, M. and Szlenk, W.. Entropy of piecewise monotone mappings. Studia Math. 67 (1980), 45–63.Google Scholar
Pianigiani, G. and Yorke, J. A.. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351–366.Google Scholar
Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.Google Scholar
Schaefer, H. H.. Topological Vector Spaces, 2nd edn. Ed. Wolff, M. P.. Springer, New York, 1999.Google Scholar
Urbański, M.. On Hausdorff dimension of invariant sets for expanding maps of a circle. Ergod. Th. & Dynam. Sys. 6 (1986), 295–309.Google Scholar
van den Bedem, H. and Chernov, N.. Expanding maps of an interval with holes. Ergod. Th. & Dynam. Sys. 22 (2002), 637–654.Google Scholar