[Paper Review] Birkhoff sum fluctuations in substitution dynamical systems

Elliot Paquette, Y. G. Son|arXiv (Cornell University)|May 6, 2015

Mathematical Dynamics and Fractals14 references5 citations

TL;DR

This paper establishes central limit theorems for Birkhoff sums in primitive substitution dynamical systems, showing that eigenfunctions of the substitution matrix with unimodular eigenvalues (modulus 1) that are not coboundaries exhibit asymptotically normal fluctuations. For non-coboundary eigenfunctions with unimodular eigenvalues, the normalized Birkhoff sums converge to a normal distribution, while eigenfunctions with non-unimodular eigenvalues converge to singular distributions supported on Cantor sets.

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 noncoboundary 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

Motivation & Objective

• To analyze the distributional fluctuations of Birkhoff sums along fixed orbits in primitive substitution dynamical systems.
• To characterize when eigenfunctions of the substitution matrix are coboundaries, as this affects the limiting distribution of Birkhoff sums.
• To establish a central limit theorem for non-coboundary eigenfunctions with unimodular eigenvalues (|λ| = 1).
• To identify the limiting distribution for eigenfunctions with non-unimodular eigenvalues, showing convergence to singular measures on Cantor sets.
• To provide a new criterion for determining whether an eigenfunction is a coboundary using spectral and renormalization properties.

Proposed method

• Use the renormalization identity $ S_f( heta^k(w)) = \lambda_f^k S_f(w) $ for eigenfunctions $ f $ of the substitution matrix $ M $, which links Birkhoff sums across different scales.
• Apply the martingale central limit theorem to the normalized Birkhoff sum $ Y_N $, decomposed as $ Y_N = Z_N + \lambda h(X_1) - \lambda^{N+1}(Ph)(X_N) $, where $ Z_N $ is a martingale.
• Analyze the variance of the martingale increment $ \mathbb{E}|h(X_{i+1}) - (Ph)(X_i)|^2 $, showing it vanishes if and only if $ (I - P^*P)h = 0 $, which characterizes coboundaries.
• Use Neumann series expansions for $ (I - \overline{\lambda}P^*)^{-1} $ and $ (I - \lambda P)^{-1} $ to derive the limiting variance of the normalized sum.
• Establish that for $ \lambda \notin \mathbb{R} $, the real and imaginary parts of the limiting distribution are independent and identically distributed, implying circular symmetry.
• Use the spectral properties of the substitution matrix $ M $, particularly the Perron-Frobenius eigenvalue $ \lambda $, to define the normalization scaling $ \log_\lambda N $ in the central limit theorem.

Experimental results

Research questions

• RQ1 Under what conditions do Birkhoff sums of eigenfunctions in substitution dynamical systems exhibit asymptotically normal fluctuations?
• RQ2 When is an eigenfunction of the substitution matrix a coboundary, and how does this affect the limiting distribution of Birkhoff sums?
• RQ3 What is the limiting distribution of Birkhoff sums for eigenfunctions with non-unimodular eigenvalues?
• RQ4 How does the renormalization structure of substitution systems lead to self-similar fluctuations in Birkhoff sums?
• RQ5 Can a new criterion be derived to determine whether a given eigenfunction is a coboundary using spectral and Markov chain properties?

Key findings

• For eigenfunctions $ f $ with unimodular eigenvalue $ \lambda_f = 1 $ that are not coboundaries, the normalized Birkhoff sum $ \frac{S_f(u_{\leq n}) - c_1 \log_\lambda n}{c_2 \sqrt{\log_\lambda n}} $ converges in distribution to a standard normal random variable.
• The limiting variance of the normalized Birkhoff sum is given by $ \mathbb{E}|Z|^2 = \left( (I - \overline{\lambda}P^* - \lambda P + \lambda \overline{\lambda} P^*P)g, g \right)_\pi $, derived via Neumann series expansion.
• For eigenfunctions with non-unimodular eigenvalues, the Birkhoff sums converge in distribution to a singular measure supported on a Cantor set, indicating non-Gaussian fluctuations.
• The condition $ \mathbb{E}|h(X_2) - (Ph)(X_1)|^2 = 0 $ holds if and only if $ (I - P^*P)h = 0 $, providing a spectral criterion for coboundaries.
• When $ \lambda \notin \mathbb{R} $, the real and imaginary parts of the limiting distribution are independent and have equal variance, implying circular symmetry.
• The paper provides a new characterization of substitution systems with bounded discrepancy: such systems are precisely those for which all eigenfunctions of modulus 1 are coboundaries.