Entropies of nonautonomous dynamical systems

The paper consists of two parts. In the first part, we study topological and measure-theoretical entropies of a nonautonomous dynamical system via the ideas of local entropy theory. We prove local and global variational inequalities in Theorem 3.5, which relates to the topological entropy of a nonautonomous dynamical system to its measure-theoretical entropy. Consequently, we answer affirmatively the second part of [26, Question 3.7] asked by Zhu et al. in 2012. We also show that, in general, the answer to the first part of [26, Question 3.7] is negative. In the second part, we introduce weak expansiveness for a nonautonomous system following the ideas of Bowen [4] and Misiurewicz [15]. For further understanding of the weak expansiveness of a nonautonomous system, we also introduce and study the concept of topological conditional entropy for such class of dynamical system along the lines of Misiurewicz [17]. We give a simple link relating weak expansiveness of a nonautonomous system to its topological conditional entropy. We end this paper with Question 4.8 for further understanding these concepts of a nonautonomous system.