The Dimension of Integral Self-Affine Sets via Fractal Perturbations: The Box and the Hausdorff Dimensions, Ergodic Measures

TL;DR

This paper introduces a fractal perturbation method to determine the dimensions of integral self-affine sets without separation conditions, and proves the existence of full-dimension ergodic measures on their torus projections.

Contribution

The paper develops a novel perturbation technique for calculating dimensions of self-affine sets, especially with irreducible characteristic polynomials, and establishes the existence of full-dimension ergodic measures.

Findings

Dimension of the self-affine set is obtained as a limit of perturbed fractals.

The method shows the overlap structures of original and perturbed sets are isomorphic.

Existence of a full-dimension ergodic measure on the torus projection is proved.

Abstract

Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original abstract: An integral self-affine set $F=F(T,A)\subseteq \mathbb{R}^n$ is a self-affine set which is generated by an $n\times n$ integer expanding matrix $T$ (not necessarily a similitude) and a finite set $A\subset \mathbb{Z}^n$ of integer vectors so that $F=T^{-1}(F+A)$. The dimension problem of $F$ has not yet been settled fully. For that, we introduce a fractal perturbation method with respect to $T,A$ and get the dimension as the limit of the dimensions of a sequence of better-behaved perturbed fractals, for which a dimension formula already exists. An unexpected feature of this technique is that the overlap structures of $F$ and its perturbations are eventually the same (i.e. the neighbor graphs are isomorphic), which is unlike some known perturbations. Our method has been developed especially for the problematic case of irreducible characteristic polynomial of $T$. Also, we do not impose any separation condition on $F$ (like the open set condition) or any further restriction (such as size, etc.) on $T$ or $A$. As a by-product of the perturbation method, we prove the existence of the box dimension of $F$ too. Further, we consider $F$ as a $T$-invariant subset of the n-torus (i.e, we consider $F \ \rm{mod \ 1}$), and we rather use the perturbation method to show that there is an ergodic $T$-invariant Borel probability measure on $F \ \rm{mod \ 1}$ of full dimension. In contrast to some known results, this is not an almost-sure result.