Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Ievgen V. Bondarenko*, Rostyslav V. Kravchenko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let G be a group and Φ:H→G be a contracting homomorphism from a subgroup H<G of finite index. V. Nekrashevych (2005) [25] associated with the pair (G,Φ) the limit dynamical system (JG,s) and the limit G-space XG together with the covering ∪gεGT.g by the tile T. We develop the theory of self-similar measures m on these limit spaces. It is shown that (JG,s,m) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T 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 T∩(T.g) for gεG. 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.

Original languageEnglish (US)
Pages (from-to)2169-2191
Number of pages23
JournalAdvances in Mathematics
Volume226
Issue number3
DOIs
StatePublished - Feb 15 2011

Keywords

  • Bernoulli shift
  • Graph-directed system
  • Limit space
  • Self-affine tile
  • Self-similar group
  • Self-similar measure
  • Tiling

ASJC Scopus subject areas

  • Mathematics(all)

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