标题： 非自治动力系统 I：拓扑压力和熵 显示英文标题

标题： Nonautonomous Dynamical Systems I: Topological Pressures and Entropies

摘要： 设 $\boldsymbol{X}=\{X_{k}\}_{k=0}^{\infty}$ 是一列紧致度量空间 $X_{k}$ 和 $\boldsymbol{T}=\{T_{k}\}_{k=0}^{\infty}$ 是一列连续映射 $T_{k}:X_{k} \to X_{k+1}$。 对偶 $(\boldsymbol{X},\boldsymbol{T})$ 被称为非自治动力系统。 我们的主要目的是研究非自治动力系统上拓扑压和熵的变分原理。 在本文中，我们引入了一种多种拓扑压力（$\underline{Q}$，$\overline{Q}$，$\underline{P}$，$\overline{P}$，$P^{\mathrm{B}}$和$P^{\mathrm{P}}$）对于势能$\boldsymbol{f}=\{f_{k} \in C(X_{k},\mathbb{R})\}_{k=0}^{\infty}$在子集$Z \subset X_{0}$上，类似于分形维数，并我们提供了各种关键性质，这对于研究非自治动力系统上的变分原理至关重要。 特别是，我们得到了这些压力的幂规则和乘积规则，并且我们还证明了它们在非自治动力系统的等共轭下以及在$\boldsymbol{f}$上的等连续性下是不变量。 从分形维数的角度来看，这些压力是描述非自治动力系统的“维数”，我们获得了类似于分形维数的压力的各种性质。 显示英文摘要

摘要： 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.