## 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.

### 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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