Local Dimensions of Measures on Infinitely Generated Self-Affine Sets

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.

Accepted author manuscript, 10 pages

References in corpus (4)

• Dimension and measures on sub-self-affine sets
• Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets
• Separation conditions on controlled Moran constructions
• Local dimensions of measures on self-affine sets

Cited by in corpus (5)

• Ledrappier-Young formula and exact dimensionality of self-affine measures
• Dimension of invariant measures for affine iterated function systems
• Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension
• Random affine code tree fractals: Hausdorff and affinity dimensions and pressure
• Dimension estimates for $C^1$ iterated function systems and repellers. Part II