Counterexamples to Ruelle's inequality in the noncompact case

TL;DR

This paper demonstrates that Ruelle's inequality, relating entropy and Lyapunov exponents, can fail in a broad class of smooth dynamical systems on noncompact spaces, challenging previous assumptions.

Contribution

It constructs explicit examples of noncompact smooth dynamical systems where Ruelle's inequality does not hold, highlighting limitations of the inequality in noncompact settings.

Findings

Existence of noncompact systems violating Ruelle's inequality

Construction of a large family of counterexamples

Implications for the applicability of entropy-Lyapunov relations

Abstract

In this paper we show that there exists a large family of smooth dynamical systems defined over noncompact spaces that does not satisfy Ruelle's inequality between entropy and Lyapunov exponents.