## 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 ∪_{gεG}T.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 language | English (US) |
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Pages (from-to) | 2169-2191 |

Number of pages | 23 |

Journal | Advances in Mathematics |

Volume | 226 |

Issue number | 3 |

DOIs | |

State | Published - 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)