Lifting measures to inducing schemes

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y.Pesin and S.Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J. 5(3) (2005), 669678; Y.Pesin and S.Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemeswhich hold for many known examplesany 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 [J.Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H.Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys. 168(3) (1995), 571580] 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 multi-modal maps) and for some multi-dimensional maps.