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Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory Harmonic analysis in one variable

Published online by Cambridge University Press:  06 October 2020

SILVIUS KLEIN Open the ORCID record for SILVIUS KLEIN [Opens in a new window] ,
XIAO-CHUAN LIU  and
ALINE MELO
SILVIUS KLEIN*
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Brazil (e-mail: alinedemelo.m@gmail.com)
XIAO-CHUAN LIU
Affiliation:
Instituto de Matemática e Estatística da Universidade de São Paulo, R. do Matão, 1010 - Vila Universitaria, São Paulo, Brazil (e-mail: lxc1984@gmail.com)
ALINE MELO
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Brazil (e-mail: alinedemelo.m@gmail.com)
*
e-mail: silviusk@impa.br
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Abstract

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.

Keywords

classical ergodic theoryconvergence of Birkhoff meansDenjoy–Koksma inequalityDiophantine conditionFejér and Jackson kernels

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 42A10: Trigonometric approximation 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series 11K06: General theory of distribution modulo $1$
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3363 - 3388
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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