On the dimension of orthogonal projections of self-similar measures

2024/07/23 by Algom, Amir, Shmerkin, Pablo
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.16262

Abstract

Let ν be a self similar measure on ℝ^d, d≥ 2, and let π be an orthogonal projection onto a k-dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the IFS on π, and show that it ensures the dimension of πν is preserved; this significantly refines previous results by Hochman-Shmerkin (2012) and Falconer-Jin (2014), and is sharp for projections to lines and hyperplanes. A key ingredient in the proof is an application of a restricted projection theorem of Gan-Guo-Wang (2024).

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