Dimension of homogeneous iterated function systems with algebraic translations

2024/05/06 by Feng, De-Jun, Feng, Zhou · 2 citations
#28A80 #42A85 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.03124

Abstract

Let μ be the self-similar measure associated with a homogeneous iterated function system Φ= \ λx + t_j \_j=1^m on \Bbb R and a probability vector (p_j)_j=1^m, where 0≠ λ∈ (-1,1) and t_j∈ \Bbb R. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if t₁,…, t_m are rational numbers and 0<λ<1, then dim μ=min \ 1, \frac∑_j=1^m p_jlog p_j log |λ| \ unless Φ has exact overlaps. In this paper, we further show that the above equality holds in the case when t₁,…, t_m are algebraic numbers and 0<|λ|<1. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú.

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• Entropy rates in the dimension theory of self-similar measures
• On exponential separation of analytic self-conformal sets on the real line

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