Pointwise densities of homogeneous Cantor measure and critical values

Let N ≥2 and ρ ∈ (0, 1/N2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system fi(x) = ρx + i(1-ρ) N-1 : I = 0, 1, . . . , N - 1 } . Let s = dimHE be the Hausdorff dimension of E, and let μ = Hs|E be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density Θs∗(μ, x) 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 ac and bc for the sets E∗(a) = { x ∈ E : Θs∗(μ, x) ≥ a } and E∗(b) = {x ∈ E : Θ∗s(μ, x) ≤ b} respectively, such that dimHE∗(a) > 0 if and only if a < ac, and that dimHE∗(b) > 0 if and only if b > bc. We emphasize that both values ac and bc 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.