Pointwise densities of homogeneous Cantor measure and critical values

Abstract

Let N ⩾ 2 and ρ ∈ (0, 1/N 2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system Let s = dim H E be the Hausdorff dimension of E, and let be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density and upper s-density Θ∗s (μ, x) of μ at x. This extends some early results of Feng et al (2000 J. Math. Anal. Appl. 250 692–705). Furthermore, we determine two critical values a c and b c for the sets respectively, such that dim H E *(a) > 0 if and only if a < a c, and that dim H E*(b) > 0 if and only if b > b c. We emphasize that both values a c and b c are related to the Thue–Morse type sequences, and our strategy to find them relies on ideas from open dynamical systems and techniques from combinatorics on words.