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Publication . Article . Preprint . 2018

On ergodic averages for parabolic product flows

Alexander I. Bufetov; Boris Solomyak;
Open Access
English
Published: 01 Jan 2018
Publisher: HAL CCSD
Country: France
Abstract
We consider a direct product of a suspension flow over a substitution dynamical system and an arbitrary ergodic flow and give quantitative estimates for the speed of convergence for ergodic integrals of such systems. Our argument relies on new uniform estimates of the spectral measure for suspension flows over substitution dynamical systems. The paper answers a question by Jon Chaika.
14 pages, minor corrections, to appear in Bulletin de la SMF. arXiv admin note: text overlap with arXiv:1305.7373
Subjects by Vocabulary

Microsoft Academic Graph classification: Suspension (topology) Dynamical systems theory Applied mathematics Dynamical system Mathematics Flow (mathematics) Substitution (logic) Product (mathematics) Ergodic theory Direct product

Subjects

Hoelder continuity, substitution dynamical system, spectral measure, 37A30, 37B10, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Mathematics - Dynamical Systems, General Mathematics, Dynamical Systems (math.DS), FOS: Mathematics

15 references, page 1 of 2

[1] Boris Adamczewski, Symbolic discrepancy and self-similar dynamics Annales Inst. Fourier, 54(7), 2201-2234.

[2] Akiyama, S.; Barge, M.; Berth´e, V.; Lee, J.-Y.; Siegel, A. On the Pisot substitution conjecture. Mathematics of aperiodic order, 33 -72, Progr. Math., 309, Birkhaeuser/Springer, Basel, 2015.

[3] Xavier Bressaud, Alexander I. Bufetov, Pascal Hubert, Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 Proceedings of the London Mathematical Society, 109 (2), 483 -522.

[4] M. Barge, B. Diamond, Coincidence for substitutions of Pisot type, Bulletin SMF 130 (2002), 591 - 626.

[5] Alexander I. Bufetov and Boris Solomyak, On the modulus of continuity for spectral measures in substitution dynamics, Advances in Mathematics 260 (2014), 84-129. [OpenAIRE]

[6] A. Clark and L. Sadun, When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems 23 (2003), 1043-1057. [OpenAIRE]

[7] Erd˝os, Paul. On the smoothness properties of Bernoulli convolutions. Amer. J. Math. 62 (1940), 180-186.

[8] S. Ferenczi, C. Mauduit, A. Nogueira, Substitution dynamical systems: algebraic characterization of eigenvalues, Annales scientifiques de l'E´cole Normale Sup´erieure, S´er. 4, 29 no. 4 (1996), 519-533.

[9] Pytheas N. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Edited by V. Berth´e, S. Ferenczi, C. Mauduit and A. Siegel.

[10] M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory and Dynamical Systems, 23 (02), 533 - 540.

Funded by
NSF| Fractals and Ergodic Theory
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1361424
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
,
EC| IChaos
Project
IChaos
Intermediate Chaos
  • Funder: European Commission (EC)
  • Project Code: 647133
  • Funding stream: H2020 | ERC | ERC-COG
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