[Paper Review] Typical Dispersion and Generalized Lyapunov Exponents

Steven R. Finch, Zai-Qiao Bai+1 more|ArXiv.org|Mar 18, 2008

Advanced Differential Equations and Dynamical Systems12 references3 citations

TL;DR

This paper introduces a generalized Lyapunov exponent framework to analyze typical dispersion in stochastic matrix products, extending classical Lyapunov exponents to quantify variance in growth rates of number-theoretic functions like the count of odd binomial coefficients. It derives exact formulas for the variance parameter $\sigma^2$ using series summation and convergence acceleration, confirming that typical dispersion is significantly smaller than average dispersion due to logarithmic damping of rare outliers.

ABSTRACT

Let f(n) denote the number of odd entries in the nth row of Pascal's binomial triangle. We study "average dispersion" and "typical dispersion" of f(n) -- the latter involves computing a generalized Lyapunov exponent -- and then turn to numerical analysis of higher dimensional examples.

Motivation & Objective

• To quantify the typical dispersion of number-theoretic functions such as the number of odd coefficients in binomial and trinomial expansions.
• To extend the classical Lyapunov exponent to a second-order generalized Lyapunov exponent (moment Lyapunov exponent) for measuring variance in growth rates.
• To derive exact analytical expressions for the variance parameter $\sigma^2$ in terms of infinite series over binary words avoiding a forbidden subword.
• To apply convergence acceleration techniques like Wynn’s ε-process to numerically compute $\sigma^2$ for higher-dimensional matrix products.
• To compare typical dispersion ($\sigma^2$) with average dispersion ($L(2)/\ln 2$), revealing that typical dispersion is much smaller due to logarithmic damping of extreme values.

Proposed method

• Define the generalized Lyapunov exponent $L(t) = \lim_{k\to\infty} \frac{1}{k} \ln \mathbb{E}[\|D_z\|^t]$ for real $t$, with $D_z$ being random products of nonnegative matrices $D_0$ and $D_1$.
• Use the identity $\sigma^2 = L''(0) = \lim_{k\to\infty} \frac{1}{k} \mathrm{Var}(\ln \|D_z\|)$ to express the typical dispersion as the variance of the logarithmic norm.
• Exploit the property that $D_0^q$ has rank 1 and can be conjugated to a projection matrix, enabling exact computation of $D_z'(0,0)$ for binary words $z$ avoiding $0^q$.
• Derive closed-form expressions for $\lambda$, $\kappa$, and $\mu$ via infinite series over $\chi(0^q)$, the set of finite binary words ending in 1 and avoiding $0^q$.
• Apply Wynn’s ε-process to accelerate convergence of the slowly converging series for $\sigma^2 = 3\lambda^2 - 2\lambda\kappa + \mu$.
• Verify results on binomial and trinomial coefficient counts, showing that $\sigma^2 = \ln^2(2)/4$ and $L(2)/\ln 2 = 1.3219$ for binomials, with $\sigma^2/\ln 2 = 0.1733$ as the typical dispersion parameter.

Experimental results

Research questions

• RQ1 How does the typical dispersion of the number of odd binomial coefficients in $(1+x)^n$ compare to its average dispersion?
• RQ2 Can the generalized Lyapunov exponent $L(t)$ be used to compute the variance of the logarithmic growth rate of matrix products?
• RQ3 What is the exact analytical form of the variance parameter $\sigma^2$ for matrix products with periodic rank-1 behavior?
• RQ4 How does the logarithmic transformation reduce the impact of rare extreme values in the distribution of $f(n)$?
• RQ5 Is there a central limit theorem for the number of odd coefficients in $(1+x)^n$ or related functions?

Key findings

• For the number of odd coefficients in $(1+x)^n$, the typical dispersion parameter is $\sigma^2 / \ln 2 = \ln(2)/4 \approx 0.1733$, significantly smaller than the average dispersion parameter $L(2)/\ln 2 \approx 1.3219$.
• The exact value of $\sigma^2 = \ln^2(2)/4 \approx 0.1201$ is derived using series summation and verified via convergence acceleration with Wynn’s ε-process.
• The generalized Lyapunov exponent satisfies $L(t) = \ln((2^t + 1)/2)$ for binomials, yielding $\lambda = \ln(2)/2$ and $\sigma^2 = \ln^2(2)/4$, matching known results from Kirschenhofer.
• For trinomials, the average dispersion parameter is $L(2)/\ln 2 \approx 1.4924$, with $e^{L(2)}$ being a root of the cubic $\xi^3 - 2\xi^2 - 3\xi + 2 = 0$.
• The digital sum $\#(n)$, counting 1s in the binary expansion of $n$, is asymptotically normal with mean $\ln n / (2\ln 2)$ and variance $\ln n / (4\ln 2)$, confirming identical distribution across linear forms $aN + b$.
• The paper identifies open problems, including the existence of Fourier expansions for the limiting functions $\Phi(x)$ and $\Psi(x)$ associated with $\#(n)$ and $f(n)$, and the lack of a closed-form expression for $\inf \Psi(x)$.