Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems

Abstract: A nonautonomous dynamical system $(\boldsymbol{X},\boldsymbol{T})={(X_{k},T_{k})}{k=0}^{\infty}$ is a sequence of continuous mappings $T{k}:X_{k} \to X_{k+1}$ along a sequence of compact metric spaces $X_{k}$. In this paper, we study the nonautonomous symbolic systems $(\boldsymbol{\Sigma}(\boldsymbol{m}),\boldsymbol{\sigma})$ and nonautonomous expansive dynamical systems. We prove homogeneous properties and provide the formulae for topological pressures $\underline{P},\overline{P},P^{\mathrm{B}},P^{\mathrm{P}}$ on symbolic systems for potentials $\boldsymbol{f}={f_{k} \in C(\Sigma_{k}^{\infty}(\boldsymbol{m}),\mathbb{R})}_{k=0}^{\infty}$ with strongly bounded variation. We also give the formulae for the measure-theoretic pressures of nonautonomous Bernoulli measures and obtain equilibrium states in nonautonomous symbolic systems for certain classes of potentials. Finally, we prove the existence of generators for pressure in strongly uniformly expansive (sue) systems. We show that all nonautonomous sue systems have symbolic extensions, and that a class of nonautonomous sue systems on 0-topological-dimensional spaces $X_{k}$ may be embedded in autonomous systems.