The Entropy Of Set-valued Dynamical Systems And Its Applications

The entropy for set-valued dynamical systems and related properties are studied. The thesis is organized as follows:In the Introduction, the background and the main contents of entropy for set-valued dynamical systems are introduced.In Chapter 1, we briefly describe some basic concepts and the known results for the topological dynamical systems and set-valued dynamical systems.In Chapter 2, several different entropies for set-valued dynamical systems and its related properties are introduced, and we discuss the relationships between the several entropies for set-valued dynamical systems.In Chapter 3, the notions of expansiveness, continuum-wise expansiveness and specification for set-valued functions are introduced. As the applications, we study the relationships among the notions and the entropy for set-valued dynamical systems. We generalize the definition of expansiveness for single-valued functions, and establish the relationships between expansiveness of set-valued functions and the shift map of its inverse limit space. We prove that an expansive set-valued function has finite set-valued entropy. We give a new proof of the result which states that a set-valued map with the specification property has positive K.T set-valued entropy.In Chapter 4, we describe the notion of pre-entropy for a continuous map, and discuss the relationship between pre-entropy of f and K.T set-valued entropy of its inverse f-1.