Geodesic
Differences of random Cantor sets and lower spectral radii
Journal of the European Mathematical Society, Tome 13 (2011) no. 3, pp. 733-760

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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​>√2 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 2×2 matrices.
DOI : 10.4171/jems/266
Classification : 28-XX, 60-XX, 00-XX
Keywords: Random fractals, difference of Cantor sets, Palis conjecture, multitype branching processes in varying environment, lower spectral radius 
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     author = {F. Michel Dekking and Bram Kuijvenhoven},
     title = {Differences of random {Cantor} sets and lower spectral radii},
     journal = {Journal of the European Mathematical Society},
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F. Michel Dekking; Bram Kuijvenhoven. Differences of random Cantor sets and lower spectral radii. Journal of the European Mathematical Society, Tome 13 (2011) no. 3, pp. 733-760. doi: 10.4171/jems/266

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