Absolute continuity of self-similar measures on the plane

2023/01/25 by Boris Solomyak, Solomyak, Boris, Adam Śpiewak +1 · 1 citation
Computer Science · Mathematics · #28A78 #28A80 #37A46 #42A38 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2301.10620

openalex publication_date 2023/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider an iterated function system consisting of similarities on the complex plane of the form g_i(z) = λ_i z + t_i, λ_i, t_i ∈ ℂ, |λ_i|<1, i=1,…, k. We prove that for almost every choice of (λ₁, …, λ_k) in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.

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• Fourier decay and absolute continuity for typical homogeneous self-similar measures in \mathbb R^d for d≥ 3

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