Studies On Topics For The Random Dynamical System

Earlier at the beginning of last century, Fatou and Julia Started the study of the iteration theory of a rational function as a complex analytic dynamical system. Because it is closely related to the chaos theory and fractal geometry, there are increasing interest not only in the mathematics but also in the other relating fields. As a kind of special rational functions, polynomials have a lot of special and attracting dynamical properties. Since a function behaves on the Julia set in a chaotic way which can explain many natural phenomena, the study of Julia set is very important and difficult.Now, for a bounded sequence (cn) of complex numbers, R.Briick have studied the properties of Julia sets for the random dynamical system formed by polynomials z2 + cn and obtained some important results. However there are many problems unsolved. In this thesis, we mainly study connectedness of Julia sets for the random dynamical system generated by z2 + cn and answer a question posed by R.Briick. Besides, we discuss the dynamical properties of the random 'dynamical system generated by a set of polynomials of degree n of the form zn + ak, 1zn-1 + ...... + dk, n- There are four sections in this thesis.First, we elaborate the background of this paper and introduce the relating notations and notions.In chapter 1, we discuss the condition for Julia sets for the random dynamical system generated by the quadratic polynomials z2 + cn to be totally disconnected, and construct a sequence (cn) for which the Julia set J(cn) is totally disconnected.In chapter 2, we study topological properties of Julia sets for the random dynamical system generated by z2 + cn and answer a question posed by R.Bruck.In chapter 3, we discuss the dynamical properties of the random dynamical system generated by polynomials zn + ak, 1zn-1 + ...... + ak, n of degree n.