The Study Of Non-uniform Hyperbolicity Conditions In Interval Maps

In one-dimensional dynamics,there are a lot of widely used non-uniform hyperbolicity conditions,such as Collet-Eckmann condition,summability condition,backward contraction and large derivative condition.Up to now,some equivalence conclusions have been obtained by studying the relationship between these different conditions.These conclusions also play an important role in the study of other problems.For instance,Juan Rivera-Letelier and Shen Weixiao proved that for the unicritical polynomial having a priori bounds,the unique conformal measure of minimal exponent has no atom at the critical point.During dealing with this problem,they made use of the equivalence between polynomial backward contraction and summability condition.In this article,we will study some non-uniform hyperbolicity conditions in 3interval maps with non-flat critical points and with all periodic points hyperbolic repelling.Specific arrangements are as follows:In chapter one,we introduce some basic definitions and common conclusions,such as the real Schwarz lemma and Koebe distortion principle,and introduce the main conclusions.In chapter two,we prove that the polynomial backward contraction is equivalent to the summabitity condition under certain conditions.This conclusion deals with the first step that the first major conclusion need.In chapter three,we primarily prove the relationship between Poincaréseries and polynomial backward contraction.Then we prove that it can be obtained that a function satisfies the summability condition with some certain constant exponent from the convergent Poincaréseries by use of the result of chapter two.In chapter four,we prove the second main conclusion: if the interval map f satisfies the backward contraction,f satisfies the large derivative condition.