The scenery flow of self-similar measures with weak separation condition. (English) Zbl 1507.37002

Pyörälä, Aleksi

Ergodic Theory Dyn. Syst. 42, No. 10, 3167-3190 (2022).

For a Radon measure \(\mu\) on \(\mathbb{R}^d\) the scenery flow \((\mu_{x,t})_{t\geqslant0}\) at a point \(x\) is given by \(\mu_{x,t}(A)= \mu({\mathrm{e}}^{-t}A+x)/\mu(B(x,{\mathrm{e}}^{-t}))\) for a measurable set \(A\subset B(0,1)\), where \(B(a,r)\) denotes the closed metric ball at \(a\) of radius \(r\). Tangent measures are defined to be the weak*-accumulation points of the scenery; these reflect the local structure of \(\mu\). There are Radon measures with the property that every non-zero Radon measure arises as a tangent measure at almost every point. Here this unmanageable complexity is constrained in a natural way for dynamics, by proving that self-similar measures satisfying a separation property have a regularity property called uniformly scaling. This is done using ideas from ergodic theory alongside a geometric analysis arising from the separation property.

Reviewer: Thomas B. Ward (Durham)

MSC:

37A10	Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37A05	Dynamical aspects of measure-preserving transformations
28A80	Fractals
28D05	Measure-preserving transformations
28D10	One-parameter continuous families of measure-preserving transformations

Keywords:

scenery flow; self-similar measure; weak separation condition; uniformly scaling measure

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