Nonautonomous Dynamical Systems II: Variational Principles

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. In this paper, we study measure-theoretic entropies and pressures, Bowen and packing topological entropies and pressures on $(\boldsymbol{X},\boldsymbol{T})$, and we prove that they are invariant under equiconjugacies of nonautonomous dynamical systems. By establishing Billingsley type theorems for Bowen and packing topological pressures, we obtain their variational principles, that is, given a non-empty compact subset $K \subset X_{0}$ and an equicontinuous sequence $\boldsymbol{f}= \{f_k\}_{k=0}^\infty$ of functions $f_k : X_k\to \mathbb{R}$, we have that $$P^{\mathrm{B}}(\boldsymbol{T},\boldsymbol{f},K)=\sup\{\underline{P}_{\mu}(\boldsymbol{T},\boldsymbol{f}): \mu \in M(X_{0}), \mu(K)=1\},$$ and for $\|\boldsymbol{f}\|<+\infty$ and $P^{\mathrm{P}}(\boldsymbol{T},\boldsymbol{f},K)>\|\boldsymbol{f}\|$, $$P^{\mathrm{P}}(\boldsymbol{T},\boldsymbol{f},K)=\sup\{\overline{P}_{\mu}(\boldsymbol{T},\boldsymbol{f}): \mu \in M(X_{0}), \mu(K)=1\},$$ where $\underline{P}_{\mu}$ and $\overline{P}_{\mu}$, $P^{\mathrm{B}}$ and $P^{\mathrm{P}}$ denote measure-theoretic lower and upper pressures, Bowen and packing topological pressure, respectively. The Billingsley type theorems and variational principles for Bowen and packing topological entropies are direct consequences of the ones for Bowen and packing topological pressures.