Resumen de Differences of random Cantor sets and lower spectral radii

Michel Dekking, Bram Kuijvenhoven

• We investigate the question under which conditions the algebraic difference between two independent random Cantor sets C1 and C2 almost surely contains an interval, and when not. The natural condition is whether the sum d1 + d2 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 d1 + d2 > 1 should imply that C1 - C2 contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities (p0, p1) the interior of the region where the Palis conjecture does not hold is given by those p0, p1 which satisfy p0 + p1 > v2 and p0 p1(1+p02 + p12) < 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.