Dimensions of orthogonal projections of typical self-affine sets and measures

2025/02/06 by De‐Jun Feng, Feng, De-Jun, Xie, Yu-Hao
Business, Management and Accounting · Engineering · Computer Science · #Optics and Image Analysis #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Mathematical Control Systems and Analysis

paper · pdf · doi:10.48550/arxiv.2502.04000

Abstract

Let T₁,…, T_m be a family of d× d invertible real matrices with ‖T_i‖<1/2 for 1≤ i≤ m. For \bf a=(a₁,…, a_m)∈ \Bbb R^md, let π^{\bf a}\colon Σ=\1,…, m\^{\Bbb N}→ \Bbb R^d denote the coding map associated with the affine IFS \T_ix+a_i\_i=1^m, and let K^{\bf a} denote the attractor of this IFS. Let W be a linear subspace of \Bbb R^d and P_W the orthogonal projection onto W. We show that for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md, the Hausdorff and box-counting dimensions of P_W(K^{\bf a}) coincide and are determined by the zero point of a certain pressure function associated with T₁,…, T_m and W. Moreover, for every ergodic σ-invariant measure μ on Σ and for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md, the local dimensions of (P_Wπ^{\bf a})_*μ exist almost everywhere, here (P_Wπ^{\bf a})_*μ stands for the push-forward of μ by P_Wπ^{\bf a}. However, as illustrated by examples, (P_Wπ^{\bf a})_*μ may not be exact dimensional for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md. Nevertheless, when μ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic σ-invariant measure, (P_Wπ^{\bf a})_*μ is exact dimensional for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md.

Related

• Dimensions of projected sets and measures on typical self-affine sets
• The dimension of projections of self-affine sets and measures
• On the dimension of orthogonal projections of self-similar measures
• The dimension of projections of planar diagonal self-affine measures
• On self-affine measures associated to strongly irreducible and proximal systems