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Multifractal analysis of some multiple ergodic averages

Fan, Ai Hua; Schmeling, Jörg LU and Wu, Meng (2016) In Advances in Mathematics 295. p.271-333
Abstract

In this paper we study the multiple ergodic averages, on the symbolic space σm = (0,1,...,m-1}N* where m≥ 2, ℓ ≥ 2, q≥ 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical... (More)

In this paper we study the multiple ergodic averages, on the symbolic space σm = (0,1,...,m-1}N* where m≥ 2, ℓ ≥ 2, q≥ 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns variational principle, pressure function and Legendre transform in this new setting.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Hausdorff dimension, Multifractal, Multiple ergodic average
in
Advances in Mathematics
volume
295
pages
63 pages
publisher
Elsevier
external identifiers
  • Scopus:84962228302
ISSN
0001-8708
DOI
10.1016/j.aim.2016.03.012
language
English
LU publication?
yes
id
b8135123-1799-44ab-b2c5-44e0a702e245
date added to LUP
2016-04-27 16:19:53
date last changed
2016-04-27 16:19:53
@misc{b8135123-1799-44ab-b2c5-44e0a702e245,
  abstract     = {<p>In this paper we study the multiple ergodic averages, on the symbolic space σ<sub>m</sub> = (0,1,...,m-1}<sup>N</sup>* where m≥ 2, ℓ ≥ 2, q≥ 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns variational principle, pressure function and Legendre transform in this new setting.</p>},
  author       = {Fan, Ai Hua and Schmeling, Jörg and Wu, Meng},
  issn         = {0001-8708},
  keyword      = {Hausdorff dimension,Multifractal,Multiple ergodic average},
  language     = {eng},
  month        = {06},
  pages        = {271--333},
  publisher    = {ARRAY(0x7b31110)},
  series       = {Advances in Mathematics},
  title        = {Multifractal analysis of some multiple ergodic averages},
  url          = {http://dx.doi.org/10.1016/j.aim.2016.03.012},
  volume       = {295},
  year         = {2016},
}