Study Of Non-Linearizability And Small Cycles Property Of Some Rational Maps

The linearization problem is a very important problem in complex dynam-ics.The famous Douady's conjecture is about this problem.At present,the study of the linearization problem are mainly focused on polynomials,and the study of that on rational maps is scarcely undertaken.In this thesis,we study the linearization problem on rational maps from the perspective of non-linearizability.Moreover,we study its related problem about small cycles property.In the first part of this thesis,we focus on rational maps with additional conditions for the critical orbits(here after called saturated rational maps).We use the perturbation technique,the quasiconformal surgery and the Yoccoz's geometric construction method to prove that a class of saturated rational maps are non-linearizable at any irrationally indifferent fixed point with a non-Brjuno rotation number,and have infinitely many cycles in any neighborhood of this fixed point.As a corollary,the Douady's conjecture is true for this class of rational maps.We also give a family of quadratic rational maps and a family of cubic rational maps belonging to this category.The results of this part extend Geyer's result about the linearization problem on saturated polynomials to a class of saturated rational maps.Furthermore,we can obtain that the Cremer point of these saturated rational maps must have the small cycles property.In the second part of this thesis,we focus on rational maps with additional conditions on rotation numbers.We construct a special set of irrational rotation numbers.We prove that any rational map is not linearizable near the fixed point with these rotation numbers,and have infinitely many cycles in any neighborhood of this fixed point.In the third part of this thesis,we focus on polynomials.Firstly,whenis a non-Brjuno rotation number,with the help of equipotentials and external rays of polynomials and by adding dynamic conditions to the cubic polynomials,we obtain a class of cubic polynomials that are not linearizable near the fixed point with multiplier=0)²⁴)and has infinitely many cycles in any neighborhood of each these fixed points.Secondly,with the help of Yoccoz's method,we discuss high-degree polynomials.Let P_{,(9)be a holomorphic family of polynomials of degree(9 whose elements has 0 as a fixed point with multiplier=0)²⁴}.We prove that whenis a non-Brjuno rotation number,P₍,(9)contains a holomorphic subfamily of complex dimension(9-1 whose elements are non-linearizable at 0 and has infinitely many periodic orbits in any neighborhood of 0.The conclusions of this part enrich the results about the study of the linearization problem on polynomials.The method of this part can be extended to the study of non-linearizability and small cycles property on other high-degree polynomials.