Local Dimensions of Measures on Infinitely Generated Self-Affine Sets

TL;DR

This paper proves the existence of local dimensions for invariant measures on infinitely generated self-affine sets, establishing their exact dimensionality and relating it to Lyapunov dimensions, with broad applicability under contractive conditions.

Contribution

It demonstrates the existence and exact dimensionality of local measures on complex self-affine sets and relates these dimensions to Lyapunov exponents, extending previous results.

Findings

Local dimension exists for almost all translations.

Invariant measures are exactly dimensional.

Local dimension equals the minimum of Lyapunov and ambient space dimensions.

Abstract

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space. We also give an estimate, that holds for all translation vectors, with only assuming the affine maps to be contractive.