Birkhoff sum fluctuations in substitution dynamical systems

Abstract

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