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

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 L >= 2 be an integer and let alpha is an element of (0,1/L). Let C alpha,L be the uniform Cantor set defined by the following set equation: C-alpha,(L) = boolean OR(L-1)(j=0)alpha(C-alpha,C-L + j). We show that for any alpha, beta is an element of (0,1/L-2), C-alpha,C-L and C-beta,C-L essentially have the same self-similar subsets. Precisely, E is a self-similar subset of C-alpha,C-L if and only if pi(beta) o pi(-1)(alpha)(E) is a self-similar subset of C-beta,C-L, where pi(alpha) (similarly pi(beta)) is the coding map from the symbolic space {0, 1,..., L - 1}(N) to C-alpha,C-L.