Large intersection classes on fractals

TL;DR

This paper introduces classes of subsets within fractal limit sets that are closed under countable intersections and possess large Hausdorff dimension, with applications across ergodic theory, Diophantine approximation, and complex dynamics.

Contribution

It defines new classes of subsets on fractals with closure properties and large dimension, expanding understanding of their structure and applications.

Findings

Classes are closed under countable intersections.

All sets in the classes have large Hausdorff dimension.

Applications demonstrated in ergodic theory, Diophantine approximation, and complex dynamics.

Abstract

We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large Hausdorff dimension. We show some applications in ergodic theory, Diophantine approximation and complex dynamics.