Some Property Of Substitution Dynamical Systems

Substitution dynamical system is an important component of nonlinear science. There is profound relationship between substitution dynamical system and many other subjects, such as number theory, fractal geometry, hamonic analysis, complex analysis, as well as combinatorial analysis and physics. Its importance has aroused the interest of many mathematicians. At present, in the usual case when alphabet is finite, the theory is well-establishde, but the research of theory on infinite alphabet has just started.In the present paper we try to introduce some properties and results of sustitution dynamical system on infinite alphabet, with a simple example named drunken man substitution. Then compare these properties and results with the corresponding classic theory, mainly involved the complexity, minimality, invariant measure, ergodic measure and entropy.The thesis is organized as follows: After an introductory Chaper I, we present some basic concepts, the marks, and some of the most basic conclusions in Chapter II. Then give some new properties and results of substitution dynamical system on infinite alphabet, and analysis the differences.