## Abstract

Let N be an integer with N >= 2 and let X be a compact subset of R-d. If S = (S-1, ..., S-N) is a list of contracting similarities S-i : X -> X, then we will write K-S for the self-similar set associated with S, and we will write M for the family of all lists S satisfying the strong separation condition. In this paper we show that the maps

M -> R

S -> H-dimH(KS)(KS) (1)

and

M -> R

S -> S-dimH(KS)(KS) (2)

are continuous; here dim(H) denotes the Hausdorff dimension, H-s denotes the s-dimensional Hausdorff measure and S-s denotes the s-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.

Original language | English |
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Pages (from-to) | 1635-1655 |

Number of pages | 21 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 28 |

Issue number | 5 |

Early online date | 20 May 2008 |

DOIs | |

Publication status | Published - Oct 2008 |

## Keywords

- DIMENSION
- FRACTALS
- MAPS