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Birkhoff sum fluctuations in substitution dynamical systems. (English) Zbl 1423.37010
Summary: 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.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
37B10 Symbolic dynamics
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