Which subsets and when orbits of non-uniformly hyperbolic systems prefer to visit: operator renewal theory approach

The paper addresses for the first time some basic questions in the theory of finite time dynamics and finite time predictions for slowly mixing non-uniformly hyperbolic dynamical systems. It is concerned with transport in phase spaces of such systems, and analyzes which subsets and when the orbits prefer to visit. An asymptotic expansion of the decay of polynomial escape rates is obtained, which also allows for finding asymptotics of the first hitting probabilities. Our approach is based on the construction of new operator renewal equations for open dynamical systems and on their spectral analysis. In order to do this, we generalize the Keller-Liverani perturbation technique. Applications to a large class of one-dimensional non-uniformly expanding systems are considered.