Nonautonomous Dynamical Systems I: Topological Pressures and Entropies

Let $\boldsymbol{X}=\{X_{k}\}_{k=0}^{\infty}$ be a sequence of compact metric spaces $X_{k}$ and $\boldsymbol{T}=\{T_{k}\}_{k=0}^{\infty}$ a sequence of continuous mappings $T_{k}:X_{k} \to X_{k+1}$. The pair $(\boldsymbol{X},\boldsymbol{T})$ is called a nonautonomous dynamical system. Our main object is to study the variational principles of topological pressures and entropies on nonautonomous dynamical systems. In this paper, we introduce a variety of topological pressures ($\underline{Q}$,$\overline{Q}$,$\underline{P}$,$\overline{P}$,$P^{\mathrm{B}}$ and $P^{\mathrm{P}}$) for potentials $\boldsymbol{f}=\{f_{k} \in C(X_{k},\mathbb{R})\}_{k=0}^{\infty}$ on subsets $Z \subset X_{0}$ analogous to fractal dimensions, and we provide various key properties which are crucial for the study of the variational principles on nonautonomous dynamical systems. Especially, we obtain the power rules and product rules of these pressures, and we also show they are invariants under equiconjugacies of nonautonomous dynamical systems and equicontinuity on $\boldsymbol{f}$. From a fractal dimension point of view, these pressures are kinds of 'dimensions' describing the nonautonomous dynamical systems, and we obtain various properties of pressures analogous to fractal dimensions.