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

This paper discusses a variational principle on subsets for topological pressure of non-autonomous dynamical systems. Let \((X, f_{1,\infty })\) $(X,f_{1,\infty })$ be a non-autonomous dynamical system and \(\psi \) $\psi$ be a continuous potential on X, where (X, d) is a compact metric space and \(f_{1,\infty }=(f_n)_{n=1}^\infty \) $f_{1,\infty }={(f_n)}_{n=1}^{\infty }$ is a sequence of continuous maps \(f_n: X\rightarrow X\) $f_n:X\to X$ . We define the Pesin–Pitskel topological pressure \(P_{f_{1,\infty }}^{B}(Z,\psi )\) $P_{f_{1,\infty }}^B(Z,\psi )$ and weighted topological pressure \(P_{f_{1,\infty }}^{\mathcal {W}}(Z,\psi )\) $P_{f_{1,\infty }}^{\mathcal{W}}(Z,\psi )$ for any subset Z of X. Also, we define the measure-theoretic pressure \(P_{\mu ,f_{1,\infty }}(X,\psi )\) $P_{\mu ,f_{1,\infty }}(X,\psi )$ for any \(\mu \in \mathcal {M}(X)\) $\mu \in \mathcal{M}(X)$ , where \(\mathcal {M}(X)\) $\mathcal{M}(X)$ 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 \(\begin{aligned} P_{f_{1,\infty }}^{B}(Z,\psi )=P_{f_{1,\infty }}^{\mathcal {W}}(Z,\psi )=\sup \{P_{\mu ,f_{1,\infty }}(X,\psi ):\mu \in \mathcal {M}(X), \mu (Z)=1\}. \end{aligned}\) $$\begin{array}{c}\hfill P_{f_{1,\infty }}^B(Z,\psi )=P_{f_{1,\infty }}^{\mathcal{W}}(Z,\psi )=sup\{P_{\mu ,f_{1,\infty }}(X,\psi ):\mu \in \mathcal{M}(X),\mu (Z)=1\}.\end{array}$$ 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.