On topological entropy and topological pressure of non-autonomous iterated function systems
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In this paper we introduce the notions of topological entropy and topological pressure for non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets. NAIFSs differ from the usual (autonomous) iterated function systems, they are given \cite{LGMU} by a sequence of collections of continuous maps on a compact topological space, where maps are allowed to vary between iterations. Several basic properties of topological pressure and topological entropy of NAIFSs are provided. Especially, we generalize the classical Bowen's result to NAIFSs ensures that the topological entropy is concentrated on the set of nonwandering points. Then, we define the notion of specification property, under which, the NAIFSs have positive topological entropy and all points are entropy points. In particular, each NAIFS with the specification property is topologically chaotic. Additionally, the $\ast$-expansive property for NAIFSs is introduced. We will prove that the topological pressure of any continuous potential can be computed as a limit at a definite size scale whenever the NAIFS satisfies the $\ast$-expansive property. Finally, we study the NAIFSs induced by expanding maps. We prove that these NAIFSs having the specification and $\ast$-expansive properties.
Abstract
In this paper we introduce the notions of topological entropy and topological pressure for non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets. NAIFSs differ from the usual (autonomous) iterated function systems, they are given \cite{LGMU} by a sequence of collections of continuous maps on a compact topological space, where maps are allowed to vary between iterations. Several basic properties of topological pressure and topological entropy of NAIFSs are provided. Especially, we generalize the classical Bowen's result to NAIFSs ensures that the topological entropy is concentrated on the set of nonwandering points. Then, we define the notion of specification property, under which, the NAIFSs have positive topological entropy and all points are entropy points. In particular, each NAIFS with the specification property is topologically chaotic. Additionally, the $\ast$-expansive property for NAIFSs is introduced. We will prove that the topological pressure of any continuous potential can be computed as a limit at a definite size scale whenever the NAIFS satisfies the $\ast$-expansive property. Finally, we study the NAIFSs induced by expanding maps. We prove that these NAIFSs having the specification and $\ast$-expansive properties.
참고문헌 (46)
- R. L. Adler, "Topological Entropy", Transactions of the American Mathematical Society, 114(2), pp.309~319 (1965), JSTOR[DOI]
- L. Alseda, "Combinatorial dynamics and entropy in di-mension one" (1993), World Scientiffc Publishing Co., Inc
- Andrzej Biś, "Entropies of a semigroup of maps", Discrete and Continuous Dynamical Systems, 11(2-3), pp.639~648 (2004), American Institute of Mathematical Sciences (AIMS)[DOI]
- Andrzej Biś, "Un analogue du principe variationnel pour les actions de groupe et pseudogroupe", Annales de l’institut Fourier, 63(3), pp.839~863 (2013), Cellule MathDoc[DOI]
- A. Bis, "Some remarks on topological entropy of a semigroup of con-tinuous maps", Cubo, 8(2), pp.63~71 (2006)
- L. S. Block, "Dynamics in One Dimension" (1992), Springer-Verlag
- Thiago Bomfim, "The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows", Journal of Differential Equations, 267(1), pp.228~266 (2019), Elsevier BV[DOI]
- R. Bowen, "Topological entropy and axiom A", Global Analysis (Proc. Sympos. Pure Math), XIV, pp.23~41 (1968), Amer. Math. Soc
- Rufus Bowen, "Entropy for Group Endomorphisms and Homogeneous Spaces", Transactions of the American Mathematical Society, 153, pp.401~414 (1971), JSTOR[DOI]
- Rufus Bowen, "Topological Entropy for Noncompact Sets", Transactions of the American Mathematical Society, 184, pp.125~136 (1973), JSTOR[DOI]
- R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" (1975), Springer-Verlag
- Rufus Bowen, "The ergodic theory of AxiomA flows", Inventiones Mathematicae, 29(3), pp.181~202 (1975), Springer Science and Business Media LLC[DOI]
- D. BURAGO, "Semi-dispersing billiards of infinite topological entropy", Ergodic Theory and Dynamical Systems, 26(01), pp.45~52 (2006), Cambridge University Press (CUP)[DOI]
- E. I. Dinaburg, "On the relations among various entropy characteristics of dynamical systems", Math. USSR Izv, 5(2), pp.337~378 (1971)
- T. Downarowicz, "Positive topological entropy implies chaos DC2", Proceedings of the American Mathematical Society, 142(1), pp.137~149 (2014), American Mathematical Society (AMS)[DOI]
- E. Eberlein, "On topological entropy of semigroups of commuting transformations", International Conference on Dynamical Systems in Mathematical Physics, Asterisque,, 40, pp.17~62 (1976), Soc. Math. France
- L. Wayne Goodwyn, "Topological Entropy Bounds Measure-Theoretic Entropy", Proceedings of the American Mathematical Society, 23(3), pp.679~688 (1969), JSTOR[DOI]
- James Propp, "Topological entropy for nonuniformly continuous maps", Discrete and Continuous Dynamical Systems, 22(1, pp.2, September) (2008), 201~213[DOI]
- "Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics"[URL]
- Wen Huang, "A local variational principle of pressure and its applications to equilibrium states", Israel Journal of Mathematics, 161(1), pp.29~74 (2007), Springer Science and Business Media LLC[DOI]
- X. Huang, "Topological pressure of nonautonomous dynamical systems", Nonlinear Dyn. Syst. Theory, 8(1), pp.43~48 (2008)
- P. E. Kloeden, "Synchronization of nonautonomous dynamical systems", Electron. J. Differential Equations, 2003(39), pp.10~ (2003)
- P. E. Kloeden, "Nonautonomous attractors of switching systems", Dynamical Systems, 21(2), pp.209~230 (2006), Informa UK Limited[DOI]
- S. Kolyada, "Topological entropy of nonautonomous dynamical systems", Random Comput. Dynam, 4(2-3), pp.205~233 (1996)
- Dongkui Ma, "Topological pressure and topological entropy of a semigroup of maps", Discrete & Continuous Dynamical Systems - A, 31(2), pp.545~556 (2011), American Institute of Mathematical Sciences (AIMS)[DOI]
- "Speciffcation and thermodynamic properties of non-autonomous dynamical systems"[URL]
- ERIC J. OLSON, "Generalised Cantor sets and the dimension of products", Mathematical Proceedings of the Cambridge Philosophical Society, 160(1), pp.51~75 (2016), Cambridge University Press (CUP)[DOI]
- Ya B Pesin, "Dimension type characteristics for invariant sets of dynamical systems", Russian Mathematical Surveys, 43(4), pp.111~151 (1988), IOP Publishing[DOI]
- Y. B. Pesin, "Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics" (1997), University of Chicago Press
- Y. B. Pesin, "Topological pressure and the variational principle for noncompact sets", Funktsional. Anal. i Prilozhen, 18(4), pp.50~63 (1984)[DOI]
- Martin Rasmussen, "Transit times and mean ages for nonautonomous and autonomous compartmental systems", Journal of Mathematical Biology, 73(6-7), pp.1379~1398 (2016), Springer Science and Business Media LLC[DOI]
- L. Rempe-Gillen, "Non-autonomous conformal iterated function sys-tems and Moran-set constructions", Trans. Amer. Math. Soc, 368(3), pp.1979~2017 (2016)[DOI]
- J. C. Robinson, "Strict inequality in the box-counting dimension product formulas", Real Anal. Exchange, 38(1), pp.95~119 (2012)
- Fagner B. Rodrigues, "Specification and thermodynamical properties of semigroup actions", Journal of Mathematical Physics, 57(5), pp.052704~ (2016), AIP Publishing[DOI]
- David Ruelle, "Statistical Mechanics on a Compact Set with Z p Action Satisfying Expansiveness and Specification", Transactions of the American Mathematical Society, 185, pp.237~251 (1973), JSTOR[DOI]
- David Ruelle, "A Measure Associated with Axiom-A Attractors", American Journal of Mathematics, 98(3), pp.619~654 (1976), JSTOR[DOI]
- Hua Shao, "Estimations of topological entropy for non-autonomous discrete systems", Journal of Difference Equations and Applications, 22(3), pp.474~484 (2015), Informa UK Limited[DOI]
- Ja. G. Sinai, "Gibbs measures in ergodic theory", Uspehi Mat. Nauk, 27(4), pp.21~64 (1972)
- N. Sumi, "Partial hyperbolicity and speciffcation", Proc. Amer. Math. Soc, 144(3), pp.1161~1170 (2016)[DOI]
- Jingru Tang, "Some properties on topological entropy of free semigroup action", Dynamical Systems, 33(1), pp.54~71 (2017), Informa UK Limited[DOI]
- DANIEL J. THOMPSON, "A thermodynamic definition of topological pressure for non-compact sets", Ergodic Theory and Dynamical Systems, 31(2), pp.527~547 (2011), Cambridge University Press (CUP)[DOI]
- Daniel J. Thompson, "Irregular sets, the 𝛽-transformation and the almost specification property", Transactions of the American Mathematical Society, 364(10), pp.5395~5414 (2012), American Mathematical Society (AMS)[DOI]
- M. Viana, "Foundations of Ergodic Theory" (2016), Cambridge University Press
- P. Walters, "An Introduction to Ergodic Theory" (1982), Springer-Verlag
- Kenichiro Yamamoto, "On the weaker forms of the specification property and their applications", Proceedings of the American Mathematical Society, 137(11), pp.3807~3807 (2009), American Mathematical Society (AMS)[DOI]
- FANPING ZENG, "Pre-image pressure and invariant measures", Ergodic Theory and Dynamical Systems, 27(03), pp.1037~1052 (2007), Cambridge University Press (CUP)[DOI]