Uniform scaling limits for ergodic measures
Jonathan M. Fraser
The University of Manchester, UKMark Pollicott
University of Warwick, Coventry, UK
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Abstract
We provide an elementary proof that ergodic measures on one-sided shift spaces are ‘uniformly scaling’ in the following sense: at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we show how the limiting distribution can be expressed in terms of, and derived from, a 'reverse Jacobian’ function associated with the corresponding measure on the space of left infinite sequences. Finally we specialise to the setting of Gibbs measures, discuss some statistical properties, and prove a Central Limit Theorem for ergodic Markov measures.
Cite this article
Jonathan M. Fraser, Mark Pollicott, Uniform scaling limits for ergodic measures. J. Fractal Geom. 4 (2017), no. 1, pp. 1–19
DOI 10.4171/JFG/42