Dimensions of projected sets and measures on typical self-affine sets

TL;DR

This paper investigates the typical dimensional properties of projections of measures and sets on self-affine fractals generated by affine iterated function systems, providing conditions for dimension constancy and explicit formulas.

Contribution

It establishes that various dimensions of projected measures and sets are almost surely constant with respect to parameter variations, and offers necessary and sufficient conditions for exact dimensionality.

Findings

Dimensions are constant for almost every parameter

Provides formulas for projected measure and set dimensions

Estimates Hausdorff dimensions of exceptional parameter sets

Abstract

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$. We show that for every Borel probability measure $\mu$ on $\Sigma$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of $\pi^{\bf a}_*\mu$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$, where $\pi^{\bf a}_*\mu$ stands for the push-forward of $\mu$ by $\pi^{\bf a}$. In particular, we give a necessary and sufficient condition on $\mu$ so that $\pi^{\bf a}_*\mu$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Moreover, for every analytic set $E\subset \Sigma$, each of the Hausdorff, packing, lower and upper box-counting dimensions of $\pi^{{\bf a}}(E)$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.