Research On Wild Attractor

In this thesis, we talk about unimodal interval maps relative to the wild attractor. First, we study unimodal interval maps which have wild attractors. We prove that above maps satisfies Tsujii’s slow recurrence condition. Then we simplify Tsujii’theorem, and prove a class of unimodal maps which have wild attractors are stochastic stability. Then, we consider the existence of wild attrac-tors, we study a new class of maps, which is different from Bruin’s Fibonacci-like. We prove part of them have no acip when the critical order is large enough.The thesis is organized as follows:In chapter1, we briefly recall the origin developments and the main research contents, then we introduce the backgrounds, the status and main results. We also give the main results of this thesis.In chapter2, we give some notions and results of interval dynamical systems, ergodic theory and real analysis.In chapter3, we consider the interval unimodal maps which have wild attrac-tors, then we will use principal nest, child and enhanced nest to study the status of orbit of the critical point, and prove the critical point satisfies the Tsujii’s slow recurrence condition.In chapter4, we introduce the basic theory and relative contents of stochastic stability. Then we give the proof of stochastic stability of infinitely renormalizable maps as a application of basic theory.In chapter5, we give a sufficient condition of judging stochastic stability of maps of adding types, and prove the stochastic stability of the unimodal interval maps which have wild attractors under some measure conditions.In chapter6, we study a new class of maps which is different from Bruin’s. These maps satisfy sn+1=sn+msn-1, where sn denote the return time of c to In-1. If m is even, we prove these maps have no acip when the critical order is large enough.