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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: matan.gavish@mail.huji.ac.il)
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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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References

[1]Falconer, K.. Fractal Geometry—Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester, 2003.CrossRefGoogle Scholar
[2]Fisher, A. M.. Convex-invariant means and a pathwise central limit theorem. Adv. Math. 63 (1987), 213–246.CrossRefGoogle Scholar
[3]Fisher, A. M.. Small-scale structure via flows. Progr. Probab. 57 (2004), 59–78.Google Scholar
[4]Furstenberg, H.. Private communication.Google Scholar
[5]Furstenberg, H.. Intersections of Cantor sets and transversality of semigroups. Problems in Analysis (Princeton Mathematical Series, 31). Ed. Gunning, R. C.. 1970, pp. 41–59.Google Scholar
[6]Furstenberg, H.. Ergodic fractal measures and dimension conservation. Ergod. Th. & Dynam. Sys. 28 (2008), 405–422.CrossRefGoogle Scholar
[7]Furstenberg, H. and Weiss, B.. Markov processes and Ramsey theory for trees, Special issue on Ramsey theory. Combin. Probab. Comput. 12 (2003), 548–563.CrossRefGoogle Scholar
[8]Gavish, M.. Measures with uniform scaling scenery, M.Sc. Thesis, Einstein Institute of Mathematics, Hebrew University of Jerusalem, 2008. Available online at http://math.huji.ac.il/∼matang02/GavishUniformScalingSceneryThesis.pdf.Google Scholar
[9]Hochman, M.. Dynamics on fractals and fractal distributions, in preparation.Google Scholar
[10]Lyons, R., Pemantle, R. and Peres, Y.. Conceptual proofs of Llog L criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), 1125–1138.CrossRefGoogle Scholar
[11]Lyons, R., Pemantle, R. and Peres, Y.. Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergod. Th. & Dynam. Sys. 15 (1995), 593–619.CrossRefGoogle Scholar
[12]Lyons, R., Pemantle, R. and Peres, Y.. Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 (1996), 249–264.CrossRefGoogle Scholar
[13]Lyons, R. and Peres, Y.. Probability on trees and networks. Cambridge University Press, 2009, in preparation. Current version available at http://my.page.iu.edu/∼rdlyons.Google Scholar
[14]Mörters, P. and Preiss, D.. Tangent measure distributions of fractal measures. Math. Ann. 312 (1998), 53–93.Google Scholar