Functional norms for Young towers

We introduce functional norms for hyperbolic Young towers which allow us to directly study the transfer operator on the full tower. By eliminating the need for secondary expanding towers commonly employed in this context, this approach simplifies and expands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an example, we prove large-deviation estimates with a uniform rate function for a large class of non-invariant measures and show how to translate these to the underlying system.

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), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many 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 [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], 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), 571–580] 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.

Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.