Measures with uniform scaling scenery

Matan Gavish*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


We introduce a property of measures on Euclidean space, termed 'uniform scaling scenery'. For these measures, the empirical distribution of the measure-valued time series, obtained by rescaling around a point, is (almost everywhere) independent of the point. This property is related to existing notions of self-similarity: it is satisfied by the occupation measure of a typical Brownian motion (which is 'statistically' self-similar), as well as by the measures associated to attractors of affine iterated function systems (which are 'exactly' self-similar). It is possible that different notions of self-similarity are unified under this property. The proofs trace a connection between uniform scaling scenery and Furstenberg's 'CP processes', a class of natural, discrete-time, measure-valued Markov processes, useful in fractal geometry.

Original languageAmerican English
Pages (from-to)33-48
Number of pages16
JournalErgodic Theory and Dynamical Systems
Issue number1
StatePublished - Feb 2011


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