The Invariant Curve Of Critically Finite Rational Maps

In this paper, we study the existence of invariant curve of critically finite rational map. We prove that if f∈F where F is a family of special postcritical finite rational maps, then there exists sufficiently large n such that fn has a invariant curve.The thesis is organized as follows:In chapter1, we briefly recall the origin, developments and main objective in complex dynamics, and then we introduce the backgrounds and main results of the thesis.In chapter2, some preliminary notions and results in complex dynamics and Thurston maps, which will be used in our reviewed.In chapter3, we find two fn-isotopic invariant curves Co, C1for f∈F and suf-ficiently n. Based of the function family F, we prove that fn is equivalent to an expanding Thurston map.In chapter4, Starting from C0,C1, we construct a sequence of good isotopic in-variant curves and prove tha a fn-invariant compact set C.In chapter5, we prove that the set C is indeed a curve, which complete the proof of the main theorem.In chapter6, we study the Markov property of interval maps and give the equiva-lent conditions respectively.