Structure and dimension of invariant subsets of expanding Markov maps and joint invariance

Abstract

A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyze the structure of the invariant subsets of a single map. For a mapping of the circle of class , , we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, , as well as the continuous dependence of the dimension on the endpoints of the subinterval .