Dynamically defined subsets of generic self-affine sets

TL;DR

This paper establishes sharp bounds on the Hausdorff dimension of shrinking target and recurrence sets within generic self-affine sets, extending previous results through mild conditions on affine mappings.

Contribution

Introduces mild conditions on affine maps that enable precise Hausdorff dimension bounds for dynamical subsets of self-affine sets, broadening existing theories.

Findings

Bounds are sharp for generic self-affine sets.

Mild assumptions extend previous recurrence results.

Results significantly expand the understanding of dynamical subsets.

Abstract

In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to bound the Hausdorff dimension of cylindrical shrinking target and recurrence sets. For generic self-affine sets in the sense of Falconer, that is by randomising the translation part of the affine maps, we prove that these bounds are sharp. These mild assumptions mean that our results significantly extend and complement the existing literature for recurrence on self-affine sets.