Dynamics on fractals and fractal distributions

Abstract

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.

arXiv commentv2: 62 pages. v2: Corrections to the statement of Prop 1.9 and Thms 1.27 and 1.31. Added Thm 1.39 and more detailed constructions in Sec 8. Expanded and debugged proofs. The numbering in the introduction is consistent with version 1, but in later section some numbering has changed

Cited by in corpus (15)

• Weak separation condition, Assouad dimension, and Furstenberg homogeneity
• On the Assouad dimension of projections
• Hausdorff dimension of limit sets
• Dynamics of the scenery flow and geometry of measures
• Slicing Theorems and rigidity phenomena for self affine carpets
• On the Hausdorff dimension of microsets
• Self-affine sets with fibered tangents
• Locally rich compact sets
• The Scenery Flow for Self-Affine Measures
• Dynamics of the scenery flow and conical density theorems
• Projections of Patterson-Sullivan measures and the Mohammadi-Oh dichotomy
• Radial projections along chains
• Scenery flow, conical densities, and rectifiability
• Dimension conservation for self-similar sets and fractal percolation
• On the asymptotics of the scenery flow