S-Entropy And Entropy Dimension Of Fixed-point Free Flows

Entropy is an important numerical invariant to measure the complexity of dynamical systems. For the system with positive entropy, the larger the entropy is, the more complex the system is. Over the years, people often apply s-entropy and entropy dimension to describe the complexity of dynamical systems of zero entropy. The thesis will study basic properties of s-entropy and entropy dimension of fixed-point free flows. There are three parts of this thesis.Firstly, we study s-topological entropy and topological entropy dimension of fixed-point free flows. On the one hand, we introduce the concepts of s-topological entropy of fixed-point free flows and reparametrizations of orbits on compact metric space. More-over, we introduce the concepts of H(<p, s) and T(<p, s) by weakly spanning sets and tracing sets. Also, we study their basic properties and prove that they are equivalent. On the other hand, we introduce the notion of topological entropy dimension of fixed-point free flows. Further, we study their basic properties.Secondly, we study s-topological entropy of fixed-point free expansive flows. We get the following result:there exists ε> 0 such thatThirdly, we study s-measure-theoretic entropy and measure-theoretic entropy di-mension of fixed-point free flows. On the one hand, we introduce the concept of s-measure-theoretic entropy and study their basic properties. On the other hand, we define measure-theoretic entropy dimension and obtain their basic properties.