In uniformly hyperbolic dynamical systems, the Markov structure enables us to
learn a lot about the statistical properties: for example the existence of
equilibrium states and related thermodynamic properties, the return time
statistics, decay of correlations, and so on. In non-uniformly hyperbolic
systems - here we focus on interval maps - we can find a related Markov
extension, developed by Hofbauer and Keller, which captures all of the dynamics
of the original system and allows us to use ideas from the uniformly hyperbolic
setting. I will explain how to do this, how it is related to another kind of
Markov extension, often called Young towers, and try to give an idea of the
kind of results we are then able to prove.