Hausdorff dimensions of the divergence points of self-similar measures with the open set condition

Abstract

Let μ be the self-similar measure supported on the self-similar set K with the open set condition. For x ∊ K, let A(D(x)) denote the set of accumulation points of as r ↘ 0. In this paper, we show that the set A(D(x)) is either a singleton or a closed subinterval of for any x ∊ K, and for any closed subinterval determines the Hausdorff dimension of the set of points x for which the set A(D(x)) equals I. Our main result solves the conjecture posed by Olsen and Winter (2003 J. Lond. Math. Soc. 67 103–22) positively and generalizes the classical result of Arbeiter and Patzschke (1996 Math. Nachr. 181 5–42).