Mixed divergence points of self-similar measures

Abstract

For j = 1, ..., k, let K and μ j be the self-similar set and the self-similar measure associated with an IFS with probabilities (S i , p j,i ) i=1,...,N satisfying the Open Set Condition. Let Σ = {1,..., N} N denote the full shift space and let π: Σ → K denote the natural projection. The (symbolic) local dimension of μ j at ω ∈ Σ is defined by lim n (log μ j K ω|n /log diam K ω|n ), where K ω|n = S ω1 o... o S ωn (K) for ω = ω 1 ω 2 ... ∈ Σ. A point ω for which the limit lim n (log μ j K ω|n / log diam K ω|n ) does not exist is called a divergence point for μ j . Previously only divergence points of a single measure have been investigated. In this paper we perform a detailed analysis of sets of points that are divergence points for all the measures μ 1 ,..., μ k simultaneously, and show that these points have a surprisingly rich structure. For a sequence (x n ) n , let A (x n ) denote the set of accumulation points of (x n ) n . For an arbitrary subset C of R k , we compute the Hausdorff and packing dimensions of the set π{ω∈Σ|A(log μ 1 K ω|n /log diam Kω|n) ... log μ k K ω|n ) = C} and related sets. An interesting and surprising corollary to our result is that the set of simultaneous divergence points is extremely visible, namely, (typically) it has full Hausdorff dimension, i.e., dim π({ω ∈ Σ|lim n log μ j K ω|n /log diam K ω|n does not exist}) = dim K.