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

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 |

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\textit{A. Pyörälä}, Ergodic Theory Dyn. Syst. 42, No. 10, 3167--3190 (2022; Zbl 1507.37002)

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