JournalsjemsVol. 13 , No. 3DOI 10.4171/jems/266

Differences of random Cantor sets and lower spectral radii

  • F. Michel Dekking

    Delft University of Technology, Netherlands
  • Bram Kuijvenhoven

    ABN AMRO Bank, Amsterdam, Netherlands
Differences of random Cantor sets and lower spectral radii cover

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.