Shrinking target problem for random iterated function systems

TL;DR

This paper investigates the Hausdorff dimension of sets in random iterated function systems related to shrinking target problems, extending ubiquity theorems to more general targets and recurrence properties.

Contribution

It introduces a novel approach extending ubiquity theorems to analyze the Hausdorff dimension in random IFS with general targets and recurrence conditions.

Findings

Derived Hausdorff dimension formulas for shrinking target sets

Extended ubiquity theorem to broader target classes

Applicable to sets with Poincaré recurrence properties

Abstract

We describe the shrinking target problem for random iterated function systems which semi-conjugate to a random subshifts of finite type. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to an more general targets and the sets based on the quantitative Poincar\'e recurrence properties.