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 analyse 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 .