## Abstract

The Geometry of self-similar sets K has been studied intensively during the past 20 years frequently assuming the so-called Open Set Condition (OSC). The OSC guarantees the existence of an open set U satisfying various natural invariance properties, and is instrumental in the study of self-similar sets for the following reason: a careful analysis of the boundaries of the iterates of (U) over bar is the key technique for obtaining information about the geometry K. In order to obtain a better understanding of the OSC and because of the geometric of the boundaries of the iterates of (U) over bar, it is clearly of interest to provide quantitative estimates for the "number" of points close to the boundaries of the iterates of (U) over bar. This motivates a detailed study of the rate at which the distance between a point of K and the boundaries of the iterates (U) over bar converge to 0. In this paper we show that for each t epsilon I (where I is a certain interval defined below) there is a significant number of points for which the rate of convergence equals t has positive Hausdorff dimension, and we obtain a lower bound for this dimension. Examples show that this bound is, in general, the best possible and cannot be improved.

Original language | English |
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Pages (from-to) | 799-811 |

Number of pages | 13 |

Journal | Discrete and Continuous Dynamical Systems - Series A |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2007 |

## Keywords

- fractals
- self-similar sets
- Hausdorff dimension
- MULTIFRACTAL ANALYSIS