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December  2007, 19(4): 799-811. doi: 10.3934/dcds.2007.19.799

Rates of convergence towards the boundary of a self-similar set

L. Olsen 1,

1. 

Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, United Kingdom

Received  October 2006 Revised  April 2007 Published  September 2007

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 $\overline U$ is the key technique for obtaining information about the geometry of $K$. In order to obtain a better understanding of the OSC and because of the geometric significance of the boundaries of the iterates of $\overline U$, it is clearly of interest to provide quantitative estimates for the "number" of points close to the boundaries of the iterates of $\overline U$. This motivates a detailed study of the rate at which the distance between a point in $K$ and the boundaries of the iterates of $\overline U$ converge to $0$. In this paper we show that for each $t\in I$ (where $I$ is a certain interval defined below) there is a significant number of points for which the rate of convergence equals $t$. In fact, for each $t\in I$, we show that the set of points whose 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.
Keywords: Fractals, self-similar sets, Hausdorff dimension..
Mathematics Subject Classification: Primary: 28A8.
Citation: L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799
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