Inventiones mathematicae

, Volume 104, Issue 1, pp 601-629

First online:

Intersecting random translates of invariant Cantor sets

  • Richard KenyonAffiliated withIHES
  • , Yuval PeresAffiliated withMathematics Institute, Hebrew University of Jerusalem

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Given two Cantor setsX andY in [0, 1), invariant under the mapxb x mod 1, the Hausdorff dimension of (X+t)∩Y is constant almost everywhere. WhenX,Y are defined by admissible digits in baseb, and more generally by sofic systems, we compute this dimension in terms of the largest Lyapunov exponent of a random product of matrices. The results are extended to higher dimensions and multiple intersections.