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Infinite invariant measures for non-uniformly expanding transformations of 0,1: weak law of large numbers with anomalous scaling

Infinite invariant measures for non-uniformly expanding transformations of 0,1: weak law of large... Abstract. We consider a class of maps of [0,1] with an indifferent fixed point at 0 and expanding everywhere eise. Using the invariant ergodic probability measure of a suitable, everywhere expanding, induced transformation we are able to study the infinite invariant measure of the original map in some detail. Given a continuous function with compact support in ]0,1], we prove that its time averages satisfy a 'weak law of large numbers' with anomalous scaling «/log n and give an upper bound for the decay of correlations. 1991 Mathematics Subject Classification: 60F05; 28D05, 58F11. 1. Introduction We consider a smooth map/of the interval [0,1] into itself with a neutral fixed point, such äs those modelling Pomeau-Manneville type l intermittency [M.P]. When an orbit falls in the vicinity of this fixed point it stays there for a time that can be arbitrarily long before reaching again the 'turbulent region'. Due to this fact, the SRB measure of this dynamical System is simply the Dirac delta measure concentrated at the indifferent fixed point. However, even though the ordinary Cesaro average along a typical orbit would converge to the above trivial measure, the main result of this paper shows that Cesaro averages http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Infinite invariant measures for non-uniformly expanding transformations of 0,1: weak law of large numbers with anomalous scaling

Campanino, Massimo; Isola, Stefano
Forum Mathematicum , Volume 8 (8) – Jan 1, 1996
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References (10)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1996.8.71
Publisher site
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Abstract

Abstract. We consider a class of maps of [0,1] with an indifferent fixed point at 0 and expanding everywhere eise. Using the invariant ergodic probability measure of a suitable, everywhere expanding, induced transformation we are able to study the infinite invariant measure of the original map in some detail. Given a continuous function with compact support in ]0,1], we prove that its time averages satisfy a 'weak law of large numbers' with anomalous scaling «/log n and give an upper bound for the decay of correlations. 1991 Mathematics Subject Classification: 60F05; 28D05, 58F11. 1. Introduction We consider a smooth map/of the interval [0,1] into itself with a neutral fixed point, such äs those modelling Pomeau-Manneville type l intermittency [M.P]. When an orbit falls in the vicinity of this fixed point it stays there for a time that can be arbitrarily long before reaching again the 'turbulent region'. Due to this fact, the SRB measure of this dynamical System is simply the Dirac delta measure concentrated at the indifferent fixed point. However, even though the ordinary Cesaro average along a typical orbit would converge to the above trivial measure, the main result of this paper shows that Cesaro averages

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 1996

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