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Birkhoff sum fluctuations in substitution dynamical systems

Published online by Cambridge University Press:  07 September 2017

ELLIOT PAQUETTE  and
YOUNGHWAN SON
ELLIOT PAQUETTE
Affiliation:
Department of Mathematics, The Ohio State University, USA email paquette.30@osu.edu
YOUNGHWAN SON
Affiliation:
Department of Mathematics, POSTECH, Korea email yhson@postech.ac.kr
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Abstract

We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For non-coboundary eigenfunctions with eigenvalue of modulus $1$, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1971 - 2005
Copyright
© Cambridge University Press, 2017 

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