Measures of full dimension on self-affine sets

TL;DR

This paper establishes the existence of ergodic invariant measures with full Hausdorff dimension on typical self-affine sets, using subadditive potentials and equilibrium states in symbolic dynamics.

Contribution

It introduces a method to prove the existence of ergodic measures with maximal dimension on self-affine sets under subadditive potentials.

Findings

Existence of ergodic invariant measures with full Hausdorff dimension on typical self-affine sets.

Application of subadditive thermodynamic formalism to self-affine fractals.

Connection between symbolic measures and geometric properties of self-affine sets.

Abstract

Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we show that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.