| Abstrakt: |
Let N ⩾ 2 and ρ ∈ (0, 1/N2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system Let s = dimH 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 ac and bc for the sets respectively, such that dimH E*(a) > 0 if and only if a < ac, and that dimH E*(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. [ABSTRACT FROM AUTHOR] |
