Dynamical covering sets in self-similar sets

TL;DR

This paper investigates the size and measure of dynamical covering sets on self-similar sets, providing sharp conditions for full coverage and computing Hausdorff dimension across different regimes using pressure functions.

Contribution

It introduces a unified approach to characterize dynamical covering sets on self-similar sets via a single pressure function, extending previous work with new sharp conditions.

Findings

Determines conditions for full Dvoretzky-type covering.

Computes Hausdorff dimension in various regimes.

Shows behavior characterized by a single pressure function.

Abstract

We study the size of \emph{dynamical covering sets} on a self-similar set. Dynamical covering sets are limsup sets generated by placing shrinking target sets around points along an orbit in a dynamical system. In the case when the target sets are balls with sizes depending on the centre, we determine the size of the dynamical covering set as a function of the shrinking rate. In particular, we find sharp conditions guaranteeing when full Dvoretzky-type covering, and full measure occur. We also compute the Hausdorff dimension in the remaining cases. The proofs apply in the cases of targets centred at typical points of the self-similar set, with respect to any Bernoulli measure on it. Unlike in existing work on dynamical coverings, and despite the dimension value featuring phase transitions, we demonstrate that the behaviour can be characterised by a single pressure function over the full range of parameters. The techniques are a combination of classical dimension theoretical estimates and intricate martingale arguments.