Maximal invariant sets for restrictions of tent and unimodal maps

Abstract

The aim of this paper is to study the restriction of a tent mapf: I → I to a subintervalJ ∋ I, and to characterize when the maximal invariant off⋎J is a countable set. We shall extend our results for a certain class of unimodal maps, via a semi-conjugacy. In the case of tent maps there is a critical case where the maximal invariant set is a Cantor set with Hausdorff dimension zero. The caracterization of this critical case leads to interesting arithmetical phenomena, particularly in the case of the complete tent map.