Semi-stable Laws for Intermittent Maps

A number of important results on statistical properties of intermittent maps (expanding everywhere except at a neutral fixed point) have been established in the last few decades.

The intermittent phenomenon might lead to the existence of an absolutely continuous invariant measure with infinite mass. In this case, it is known that Birkoff's ergodic theorem is not very informative, as the ergodic average converges to zero almost everywhere. However, the asymptotic behaviour of the (infinite measure preserving) dynamical system in a neighbourhood of the neutral fixed point can be used to establish a version of the central limit theorem for systems with infinite measure. The asymptotic behaviour of the dynamics at the neutral fixed point also allows one to establish other limit theorems (such as arc-sine laws) specific to dynamical systems with infinite measure/null recurrent Markov chains.

The main question is:
What sort of statistical properties can be obtained by allowing small perturbation of the dynamics near the neutral fixed point (the type of perturbation is to be determined)?

A standard procedure in studying dynamical systems with infinite measure is to induce to a 'good' set with finite, positive measure and exploit the so created 'good' properties of the induced map to understand the original dynamical system. As such, a major part of a suitable analysis/method required for the proposed topic can be carried out in terms of the finite measure preserving system.

The long term aim of this project is to develop a method required for the analysis of infinite measure dynamical systems lacking 'standard' properties (such as regular variation).