On fast Birkhoff averaging
(2003) In Mathematical Proceedings of the Cambridge Philosophical Society 135(3). p.443467 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, HardyWeierstrass functions and lacunary Taylor series. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/294409
 author
 Fan, AH and Schmeling, Jörg ^{LU}
 organization
 publishing date
 2003
 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
 14698064
 DOI
 10.1017/S0305004103006819
 language
 English
 LU publication?
 yes
 id
 1db40f4d34414aefac19521cd0d84b37 (old id 294409)
 date added to LUP
 20070903 07:26:46
 date last changed
 20181003 11:42:42
@article{1db40f4d34414aefac19521cd0d84b37, 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 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, HardyWeierstrass functions and lacunary Taylor series.}, author = {Fan, AH and Schmeling, Jörg}, issn = {14698064}, language = {eng}, number = {3}, pages = {443467}, 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}, volume = {135}, year = {2003}, }