Differences of random Cantor sets and lower spectral radii

Abstract

We investigate the question under which conditions the algebraic difference between two independent random Cantor sets _C_1 and _C_2 almost surely contains an interval, and when not. The natural condition is whether the sum _d_1 + _d_2 of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that generically it should be true that _d_1 + _d_2 > 1 should imply that _C_1 - _C_2 contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities (_p_0, _p_1) the interior of the region where the Palis conjecture does not hold is given by those _p_0, _p_1 which satisfy _p_0 + _p_1 > √2 and _p_0 _p_1(1+_p_02 + _p_12) < 1. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of 2 x 2 matrices.