Threshold and Hausdorff spectrum of discontinuous measures
نویسندگان
چکیده
Let χ be a finite Borel measure on [0, 1]. Consider the L-spectrum of χ: τχ(q) = lim infn→∞−n logb P Q∈Gn, χ(Q) 6=0 χ(Q) , where Gn is the set of b-adic cubes of generation n. Let qτ = inf{q : τχ(q) = 0} and Hτ = τ ′ χ(q − τ ). When χ is a mono-dimensional continuous measure of information dimension D, (qτ , Hτ ) = (1, D). When χ is purely discontinuous, its information dimension is D = 0, but the non-trivial pair (qτ , Hτ ) may contain relevant information on the distribution of χ. The connection between (qτ , Hτ ) and the large deviations spectrum of χ is studied in a companion paper. This paper shows that when a discontinuous measure χ possesses self-similarity properties, the pair (qτ , Hτ ) may store the main multifractal properties of χ, in particular the Hausdorff spectrum. This is observed thanks to a threshold performed on χ.
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