[Paper Review] Lyapunov exponents of probability distributions with non-compact support

Adriana Cristina Sánchez Chavarría, Marcelo Viana|arXiv (Cornell University)|Oct 6, 2018

Mathematical Dynamics and Fractals6 references4 citations

TL;DR

This paper investigates the continuity properties of Lyapunov exponents for probability distributions with non-compact support in $GL(2,\mathbb{R})$. It establishes that the top Lyapunov exponent is upper semi-continuous under the Wasserstein topology but not under the weak* topology, and shows that continuity fails even in the Wasserstein topology, providing a counterexample via a perturbation of matrix products with controlled growth rates.

ABSTRACT

We prove that the Lyapunov exponents, cosidered as functions of measures with non compact support, are semicontinuous with respect to the Wasserstein topology but not with respect to the weak* topology. Moreover, we prove that they are not continuous in the Wasserstein topology.

Motivation & Objective

• To extend the understanding of Lyapunov exponent continuity beyond compactly supported distributions in $GL(2,\mathbb{R})$.
• To analyze the behavior of Lyapunov exponents under non-compact support, particularly their continuity properties.
• To compare the convergence behavior of Lyapunov exponents under the Wasserstein and weak* topologies.
• To construct explicit counterexamples demonstrating discontinuity of Lyapunov exponents in the Wasserstein topology.

Proposed method

• Uses the Wasserstein topology on probability measures with finite first moments to analyze convergence of distributions.
• Applies the Furstenberg-Kesten theorem to define Lyapunov exponents as almost sure limits of matrix norm growth rates.
• Constructs a sequence of measures $q_n$ converging to $q$ in Wasserstein distance via a diagonal coupling with controlled $L^1$-distance.
• Employs a perturbation of matrix products using shear matrices to create a system where the top Lyapunov exponent drops to zero despite small Wasserstein distance.
• Uses the first return map argument from Bocker-Viana to show that $\lambda_+(B, q_n) = 0$ for all $n$, while $\lambda_+(B, q) > 0$.
• Analyzes the dynamics on invariant subspaces $H_x = \mathbb{R}(1,0)$ and $V_x = \mathbb{R}(0,1)$ to verify the action of the cocycle.

Experimental results

Research questions

• RQ1 Is the top Lyapunov exponent upper semi-continuous with respect to the Wasserstein topology for non-compact support distributions in $GL(2,\mathbb{R})$?
• RQ2 Does the weak* topology preserve semi-continuity of Lyapunov exponents in the non-compact case?
• RQ3 Can Lyapunov exponents fail to be continuous even under the Wasserstein topology when support is non-compact?
• RQ4 What structural conditions on matrix products lead to discontinuity of Lyapunov exponents under Wasserstein convergence?
• RQ5 Can a small Wasserstein perturbation of a measure drastically alter the Lyapunov exponent?

Key findings

• The top Lyapunov exponent $\lambda_+(p)$ is upper semi-continuous with respect to the Wasserstein topology.
• The bottom Lyapunov exponent $\lambda_-(p)$ is lower semi-continuous with respect to the Wasserstein topology.
• Lyapunov exponents are not continuous with respect to the Wasserstein topology, as demonstrated by a constructed counterexample.
• The counterexample involves a sequence of measures $q_n$ converging to $q$ in Wasserstein distance, yet $\lambda_+(B, q_n) = 0$ for all $n$ while $\lambda_+(B, q) = 2p_3\log 2 > 0$.
• The discontinuity arises from a perturbation of matrix products that preserves the $L^1$-distance in the Wasserstein metric but alters the asymptotic growth rate of matrix norms.
• The construction relies on a diagonal coupling with $W(q_n, q) \leq 2n^{-l}$ for $l = \min\{\gamma, 1-\gamma\} > 0$, ensuring convergence in Wasserstein distance.