# Dynamics of the scenery flow and geometry of measures

@article{Kaenmaki2015DynamicsOT, title={Dynamics of the scenery flow and geometry of measures}, author={Antti Kaenmaki and Tuomas Sahlsten and Pablo Shmerkin}, journal={Proceedings of The London Mathematical Society}, year={2015}, volume={110}, pages={1248-1280} }

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous… Expand

#### 12 Citations

Dynamics of the scenery flow and conical density theorems

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Conical density theorems are used in the geometric measure theory to derive geometric information from given metric information. The idea is to examine how a measure is distributed in small balls.… Expand

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We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the… Expand

Scenery flow, conical densities, and rectifiability

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We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the… Expand

Structure of distributions generated by the scenery flow

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It is proved that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery ow. Expand

On the Hausdorff dimension of microsets

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- Proceedings of the American Mathematical Society
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We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set.… Expand

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We consider dimensional properties of limit sets of Moran constructions satisfy- ing the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the… Expand

Uniform scaling limits for ergodic measures

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We prove that ergodic measures on one-sided shift spaces are uniformly scaling in the sense of Gavish. That is, given a shift ergodic measure we prove that at almost every point the scenery… Expand

Spatial recurrence for ergodic fractal measures

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We discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies… Expand

Micromeasure distributions and applications for conformally generated fractals

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- Mathematical Proceedings of the Cambridge Philosophical Society
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Abstract We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our… Expand

Self-affine sets with fibred tangents

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We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then… Expand

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It is proved that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery ow. Expand

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