Lifting Measures to Inducing Schemes
Ke Zhang | Senti, Samuel | Pesin, Yakov
Inducing schemes | Markov maps | natural extensions | invariant measures | entropy
In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of [PS05, PS06]. We show that under some natural assumptions on the inducing schemes – which hold for any known examples – any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [Bru95] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multimodal maps) and for some multidimensional maps.