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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

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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.

Information

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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