2010/01/13 by Ievgen Bondarenko, Bondarenko, Ievgen, Rostyslav Kravchenko +1
Computer Science · Mathematics · #20F65 #37D40 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Primary 28A80 #Secondary 37B10 #Topological and Geometric Data Analysis #advanced mathematical theories #math.DS #math.GR #msc:20F65 #msc:28A80 #msc:37B10 #msc:37D40
paper · pdf · doi:10.48550/arxiv.1001.2291
25 pages, 2 figures
openalex publication_date 2010/01/13 · arxiv created 2010/10/05 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a group and ϕ:H→ G be a contracting homomorphism from a subgroup H<G of finite index. V.Nekrashevych [25] associated with the pair (G,ϕ) the limit dynamical system (\lims,\si) and the limit G-space \limGs together with the covering ∪g∈ G\tile⋅ g by the tile \tile. We develop the theory of self-similar measures μ on these limit spaces. It is shown that (\lims,\si,μ) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile \tile has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles \tile∩ (\tile⋅ g) for g∈ G. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.