J. Korean Math. Soc. 2019; 56(6): 1561-1597
Online first article July 17, 2019 Printed November 1, 2019
https://doi.org/10.4134/JKMS.j180788
Copyright © The Korean Mathematical Society.
Fatemeh H. Ghane, Javad Nazarian Sarkooh
Ferdowsi University of Mashhad; Ferdowsi University of Mashhad
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.
Keywords: non-autonomous iterated function system, topological entropy, topological pressure, entropy point, specification property, nonwandering point
MSC numbers: 37B55, 37B40, 37D35
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