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

2022/09/01 by De‐Jun Feng, Feng, De-Jun, Chiu-Hong Lo +3
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms

paper · pdf · doi:10.48550/arxiv.2209.00228

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}: Σ=\1,…, m\^{\Bbb N}→ \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 μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π^{\bf a}_*μ is constant for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md, where π^{\bf a}_*μ stands for the push-forward of μ by π^{\bf a}. In particular, we give a necessary and sufficient condition on μ so that π^{\bf a}_*μ is exact dimensional for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md. Moreover, for every analytic set E⊂ Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of π^{\bf a}(E) is constant for \mathcal L^md-a.e.~\bf a∈ \Bbb R^md. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

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