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On fast Birkhoff averaging

Fan, AH and Schmeling, Jörg LU (2003) In Mathematical Proceedings of the Cambridge Philosophical Society 135(3). p.443-467
Abstract
We study the pointwise behavior of Birkhoff sums S(n)phi(x) on subshifts of finite type for Holder continuous functions phi. In particular, we show that for a given equilibrium state mu associated to a Holder continuous potential, there are points x such that S(n)phi(x) - nE(mu)phi similar to an(beta) for any a > 0 and 0 < beta < 1. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of mu and it is attained by some maximizing equilibrium state nu such that E(nu)phi = E(mu)phi. On such points the ergodic average n(-1) S(n)phi(x) converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different... (More)
We study the pointwise behavior of Birkhoff sums S(n)phi(x) on subshifts of finite type for Holder continuous functions phi. In particular, we show that for a given equilibrium state mu associated to a Holder continuous potential, there are points x such that S(n)phi(x) - nE(mu)phi similar to an(beta) for any a > 0 and 0 < beta < 1. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of mu and it is attained by some maximizing equilibrium state nu such that E(nu)phi = E(mu)phi. On such points the ergodic average n(-1) S(n)phi(x) converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices (alpha, beta), are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy-Weierstrass functions and lacunary Taylor series. (Less)
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organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematical Proceedings of the Cambridge Philosophical Society
volume
135
issue
3
pages
443 - 467
publisher
Cambridge University Press
external identifiers
  • wos:000186738900006
  • scopus:0242593811
ISSN
1469-8064
DOI
10.1017/S0305004103006819
language
English
LU publication?
yes
id
1db40f4d-3441-4aef-ac19-521cd0d84b37 (old id 294409)
date added to LUP
2016-04-01 16:35:48
date last changed
2025-04-04 14:01:18
@article{1db40f4d-3441-4aef-ac19-521cd0d84b37,
  abstract     = {{We study the pointwise behavior of Birkhoff sums S(n)phi(x) on subshifts of finite type for Holder continuous functions phi. In particular, we show that for a given equilibrium state mu associated to a Holder continuous potential, there are points x such that S(n)phi(x) - nE(mu)phi similar to an(beta) for any a &gt; 0 and 0 &lt; beta &lt; 1. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of mu and it is attained by some maximizing equilibrium state nu such that E(nu)phi = E(mu)phi. On such points the ergodic average n(-1) S(n)phi(x) converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices (alpha, beta), are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy-Weierstrass functions and lacunary Taylor series.}},
  author       = {{Fan, AH and Schmeling, Jörg}},
  issn         = {{1469-8064}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{443--467}},
  publisher    = {{Cambridge University Press}},
  series       = {{Mathematical Proceedings of the Cambridge Philosophical Society}},
  title        = {{On fast Birkhoff averaging}},
  url          = {{http://dx.doi.org/10.1017/S0305004103006819}},
  doi          = {{10.1017/S0305004103006819}},
  volume       = {{135}},
  year         = {{2003}},
}