Inducing techniques for quantitative recurrence and applications to Misiurewicz maps and doubly intermittent maps

TL;DR

This paper establishes a general method for analyzing the convergence of rare event point processes in dynamical systems, applying it to classify the limiting behavior for specific non-uniformly hyperbolic maps.

Contribution

It introduces an abstract framework linking induced systems to original systems for rare event analysis and applies it to classify limit processes in Misiurewicz and doubly intermittent maps.

Findings

Limiting processes are either homogeneous Poisson or compound Poisson with geometric multiplicity.

A dichotomy is proven for the behavior at non-periodic and periodic points.

The method allows reconstruction of multiplicity distributions for certain periodic orbits.

Abstract

We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual point $\zeta$ chosen in the phase space and then one either obtains a limiting homogeneous Poisson process at every non-periodic point $\zeta$ or a limiting compound Poisson process with geometric multiplicity distribution at every periodic point. We also highlight the reconstruction performed in order to recover the multiplicity distribution for a periodic orbit sitting outside the base of the induced map.