Tangent measure distributions of hyperbolic Cantor sets

Krieg, Daniela; Mörters, Peter

doi:10.1007/bf01299055

Cited by 7 publications

(2 citation statements)

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“…Similar notions for measures and sets have been studied by many authors, mostly with the aim of classifying measures and sets by their limiting local behavior [16,1,2,22,28,27,3]. See [17] for a systematic discussion of CP-chains and their relation to other models of "fractal" measures.…”

Section: 4mentioning

confidence: 99%

Local entropy averages and projections of fractal measures

2012

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We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X, Y ⊆ [0, 1] are closed and invariant, respectively, under ×m mod 1 and ×n mod 1, where m, n are not powers of the same integer, then, for any t = 0,A similar result holds for invariant measures and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products of self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.

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Section: 4mentioning

confidence: 99%

Local entropy averages and projections of fractal measures

2012

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“…This idea has a long history; one may view the density theorems of Lebesgue and Besicovitch as early manifestations of it, and similarly the work of D. Preiss on tangent measures. More recently the dynamical perspective has been taken up by many authors [27,3,18,4,2,5,23,25,26,6,28,12,16,20].…”

Section: Introductionmentioning

confidence: 99%

Dynamics on fractals and fractal distributions

Hochman

2010

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We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often equidistributes for some invariant distribution. The first part of the paper develops basic properties of these limiting distributions and the relations between them and other models of dynamics on fractals, specifically to Zähle distributions and Furstenberg's CP-processes. In the second part of the paper we study the geometric properties of measures arising in these contexts, specifically their behavior under projection and conditioning on subspaces.

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