Distortion Results and Invariant Cantor Sets of Unimodal Maps

Topological Conditions for the Existence of Absorbing Cantor Sets

This paper deals with strange attractors of S-unimodal maps f . It generalizes results from [BKNS] in the sense that very general topological conditions are given that either i) guarantee the existence of an absorbing Cantor set provided the critical point of f is sufficiently degenerate, or ii) prohibit the existence of an absorbing Cantor set altogether. As a byproduct we obtain very weak top...

Exactness and maximal automorphic factors of unimodal interval maps

We study exactness and maximal automorphic factors of C3 unimodal maps of the interval. We show that for a large class of infinitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic non-singular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact...

(Non)invertibility of Fibonacci-like unimodal maps restricted to their critical omega-limit sets

A Fibonacci(-like) unimodal map is defined by special combinatorial properties which guarantee that the critical omega-limit set ω(c) is a minimal Cantor set. In this paper we give conditions to ensure that f |ω(c) is invertible, except at a subset of the backward critical orbit. Furthermore, any finite subtree of the binary tree can appear for some f as the tree connecting all points at which ...

Minimal Cantor Systems and Unimodal Maps

Invariant Measures of Minimal Post-critical Sets of Logistic Maps

We construct logistic maps whose restriction to the ω-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor s...