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Intersecting random translates of invariant Cantor sets. (English) Zbl 0745.28012
Point of departure of this rich paper is an isolated result of J. Hawkes: the Hausdorff dimension of \(C\cap(C+t)\) equals \({1\over 3}\log 2/\log 3\) for almost all \(t\in(0,1)\), where \(C\) is the classical middle third Cantor set. The authors explain and reprove (in several ways) this result by revealing deep connections with entropy theory in ergodic theory and the theory of products of i.i.d. random matrices. From the many results we cite just one which describes the ‘multifractal’ structure of the set above: \[ \dim\{t: \dim(C\cap(C+t))=\alpha \log 2/\log 3\}=h(\textstyle{{1\over 2}}(1-\alpha), \alpha, \textstyle{{1\over 2}}(1- \alpha))/\log 3, \] for each \(\alpha\in(0,1)\), where \(h(p_ 1,p_ 2,p_ 3)=-p_ 1\log p_ 1-p_ 2\log p_ 2-p_ 3\log p_ 3\).

MSC:
28D99 Measure-theoretic ergodic theory
28A80 Fractals
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
60F10 Large deviations
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