The Dynamical Properties Of A Class Of Rational Functions With Degree Two

The aim of this dissertation is to study the dynamical properties of a function in a class of rational functions,and the family has one parameter.With the parameter changed the properties of the function also changed.Fistly we obtain a class of rational functions,whose Fatou sets has infinite components but only contain one forward invariant component. Sullivan conjected that for any rational function of degree d(> 2),the numbers of distinct cycles of different type have the upper bound 2(d-l).And Shishikura proved this conjecture in 1980's.Obviously,the numbers of distinct cycles of different type have the natural bound O.While this bound is rather imprecise.For polynomials,whose Fatou sets have infinite components ,there are at least 2 distinct cycles.And a nature problem is that whether the bound is 2 for any rational function if its Fatou set has infinite components? We will find a negative answer in this article. And the example in this article shows that for common rational functions the precise bound is 1.Secondly,we get two rational functions whose Julia sets are the entire complex sphere. They are the new rational functions and they don't conjugate to the former functions whose Julia sets are the Riemann sphere. And moreover,we have a class of rational functions whose degrees are 2 and they share the same Julia set.However, not only two functions in this family are permutable.Finally,we obtain a geometrically finite rational function of simple form and it has the buried points.