Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems

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 with a sequence of compact metric spaces $X_{k}$. In this paper, we study the nonautonomous symbolic dynamical systems and nonautonomous expansive dynamical systems. We first study the homogeneous properties of pressures in nonautonomous symbolic systems $(\boldsymbol{\Sigma}(\boldsymbol{m}),\boldsymbol{\sigma})$, and we simplify the formulae of Bowen, packing, lower and upper topological pressures for potentials $\boldsymbol{f}=\{f_{k} \in C(\Sigma_{k}^{\infty}(\boldsymbol{m}),\mathbb{R})\}_{k=0}^{\infty}$ with strongly bounded variation. Then we apply a law of large numbers to obtain the formulae for the lower and upper measure-theoretic pressures with respect to nonautonomous Bernoulli measures and obtain Bowen equilibrium states and packing equilibrium states for potentials in nonautonomous symbolic systems. Finally, we study the generators in nonautonomous expansive systems $(\boldsymbol{X},\boldsymbol{T})$, and we obtain that $(\boldsymbol{X},\boldsymbol{T})$ is expansive if and only if it has a generator. Moreover, strongly uniformly expansive $(\boldsymbol{X},\boldsymbol{T})$ is equisemiconjugate to a subsystem of the nonautonomous symbolic dynamical system.