Variational Principle for Topological Pressure on Subsets of Non-autonomous Dynamical Systems

Abstract

This paper discusses a variational principle on subsets for topological pressure of non-autonomous dynamical systems. Let \((X, f_{1,\infty })\) be a non-autonomous dynamical system and \(\psi \) be a continuous potential on X, where (X, d) is a compact metric space and \(f_{1,\infty }=(f_n)_{n=1}^\infty \) is a sequence of continuous maps \(f_n: X\rightarrow X\). We define the Pesin–Pitskel topological pressure \(P_{f_{1,\infty }}^{B}(Z,\psi )\) and weighted topological pressure \(P_{f_{1,\infty }}^{\mathcal {W}}(Z,\psi )\) for any subset Z of X. Also, we define the measure-theoretic pressure \(P_{\mu ,f_{1,\infty }}(X,\psi )\) for any \(\mu \in \mathcal {M}(X)\), where \(\mathcal {M}(X)\) denotes the set of all Borel probability measures on X. Then, for any nonempty compact subset Z of X, we show the following variational principle for topological pressure

$$\begin{aligned} P_{f_{1,\infty }}^{B}(Z,\psi )=P_{f_{1,\infty }}^{\mathcal {W}}(Z,\psi )=\sup \{P_{\mu ,f_{1,\infty }}(X,\psi ):\mu \in \mathcal {M}(X), \mu (Z)=1\}. \end{aligned}$$

Moreover, we show that the Pesin–Pitskel topological pressure and weighted topological pressure can be determined by the measure-theoretic pressure of Borel probability measures. In particular, we have the same results for topological entropy.