Extreme Value Laws in Dynamical Systems for Non-smooth Observations

TL;DR

This paper establishes the equivalence between hitting time statistics and Extreme Value Laws in non-smooth measure dynamical systems, extending previous results to a broader class of measures with good mixing properties.

Contribution

It proves the equivalence of hitting time statistics and Extreme Value Laws for non-absolutely continuous measures and applies this to systems with equilibrium states.

Findings

Extreme Value Laws hold for systems with non-smooth measures.

Equivalence between hitting time statistics and Extreme Value Laws is established.

Results apply to systems with good mixing properties.

Abstract

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.