2024/09/27 by Samuel Kittle, Kittle, Samuel, Constantin Kogler +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2409.18936
openalex publication_date 2024/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for d≥ 1 and any given rotations in O(d) acting irreducibly on ℝd as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension ≥ 3 improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.