A REMARK ON SELF-SIMILAR SUBSETS OF UNIFORM CANTOR SETS WITH SMALL HAUSDORFF DIMENSION

Abstract

Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let [Formula: see text] be an integer and let [Formula: see text]. Let [Formula: see text] be the uniform Cantor set defined by the following set equation: [Formula: see text] We show that for any [Formula: see text], [Formula: see text] and [Formula: see text] essentially have the same self-similar subsets. Precisely, [Formula: see text] is a self-similar subset of [Formula: see text] if and only if [Formula: see text] is a self-similar subset of [Formula: see text], where [Formula: see text] (similarly [Formula: see text]) is the coding map from the symbolic space [Formula: see text] to [Formula: see text].