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.