S-Directional Entropy And Directional Entropy Dimension Of Z~d-Actions

Entropy is an important numerical invariant to measure the complexity of dynamicalsystems. For the positive entropy systems, the larger the entropy is, the more complexthe system is. However, for the dynamical systems of zero entropy, people often applys- entropy(0 < k < 1) and entropy dimension to describe their complexity. The thesiswill study basic properties of s- directional entropy and directional entropy dimensionof Z~d- actions. Moreover, applying them, we discuss the continuity about direction of s- directional entropy. There are two parts of this thesis.Firstly, we study s- topological directional entropy and topological directional en-tropy dimension of Z~d- actions for topological dynamical systems. On the one hand, weintroduce the concepts of s- topological entropy and s- topological directional entropyof Z~d- actions on compact metric space. More, we discuss its four basic properties byregarding ?? dimensional s- topological entropy of Z~d- actions as a function defined onthe compact subsets of R??. Also, we study the continuity of s- topological directionalentropy applying the techniques of “coding”and “shading”proposed by Boyle andLind. On the other hand, we introduce the notion of topological entropy dimension andtopological directional entropy dimension of Z~d- actions. Further, we study their basicproperties.Secondly, we study s- measure-theoretic directional entropy and measure-theoreticdirectional entropy dimension of Z~d- actions for measure-preserving dynamical systems.On one hand, we introduce the concepts of s- measure-theoretic entropy and s- measure-theoretic directional entropy and study their important basic properties. On the otherhand, we define measure-theoretic entropy dimension and measure-theoretic directionalentropy dimension and obtain their basic properties.