Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet

TL;DR

This paper demonstrates that on specific Gatzouras-Lalley carpets, multiple ergodic measures can have full Hausdorff dimension, challenging previous conjectures in the field.

Contribution

It provides the first known counterexample to the uniqueness of ergodic measures with full Hausdorff dimension on these fractals.

Findings

Existence of multiple ergodic measures with full Hausdorff dimension on certain Gatzouras-Lalley carpets

Negative answer to the conjecture of Gatzouras and Peres

Counterexample to the presumed uniqueness of such measures

Abstract

In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres.