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convergence of birkhoff means

Ergodic Theory and Dynamical Systems (1)

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Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps
Part of
SILVIUS KLEIN, XIAO-CHUAN LIU, ALINE MELO
Journal:

Ergodic Theory and Dynamical Systems / Volume 41 / Issue 11 / November 2021

Published online by Cambridge University Press:

06 October 2020, pp. 3363-3388

Print publication:

November 2021
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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.