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Measures with uniform scaling scenery

Published online by Cambridge University Press:  18 January 2010

MATAN GAVISH
MATAN GAVISH*
Affiliation:
Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel (email: [email protected])
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 33 - 48
Copyright
Copyright © Cambridge University Press 2009

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