Dynamics on fractals and fractal distributions

By Michael Hochman. 2010

58 citations recorded by the source corpus.

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 Z\"ahle 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ff19cad8300cf579f1eb593d7b3da59f31b37391

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “[19] Hochman, Michael, Dynamics on fractals and fractal distributions, (2010) · Zbl 1202.”
Sixty years of fractal projections (2014, methodology): “Their proof uses the CP-chains of Furstenberg [31, 32], see also [33], and has three main ingredients: the lower semicontinuity of the expected Hausdorff dimension of the projection of a measure with respect to its ‘micromeasures’, Marstrand’s projection theorem, and the invariance of the dimension ”
Equidistribution from fractal measures (2013, methodology): “For this reason, in the definition of quasi-palm measure given in [27] there is no need to assume that [x− e−t, x+ e−t] ⊆ [−1, 1], and it has μx in place of μx,t.”
Local entropy averages and projections of fractal measures (2009, background): “or each π, hold almost surely do not a priori hold almost surely for all π. Another issue we do not resolve here is the behavior of the upper Hausdorﬀ dimension. Some of these issues are addressed in [17]. 1.5. Outline of the paper. Section 2 introduces some general notation. Section 3 recalls the n”
WEAK SEPARATION CONDITION, ASSOUAD DIMENSION, AND FURSTENBERG HOMOGENEITY (2015, methodology): “CP-processes were introduced by Furstenberg [11], greatly developed by Hochman [12], and studied by Käenmäki, Sahlsten, and Shmerkin [19].”