Statistical properties of intermittent maps with unbounded derivative

TL;DR

This paper investigates the ergodic and statistical behaviors of Lorenz-type maps with indifferent fixed points and unbounded derivatives, establishing polynomial decay of correlations, limit theorems, and exponential distribution of return times.

Contribution

It provides rigorous proofs of statistical properties for these maps, connecting physics-inspired models with mathematical dynamical systems theory.

Findings

Correlations decay polynomially over time.

Limit theorems such as Stable Laws and CLT hold for observables.

Return and hitting times follow exponential distributions in the limit.

Abstract

We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable Limit Theorems (convergence to Stable Laws or Central Limit Theorem) hold for H\"older continuous observables. We moreover show that the return and hitting times are in the limit exponentially distributed.