Controllability, Lyapunov Exponents, and Upper Bounds

Abstract

AbstractIn this chapter, we restrict our attention to smooth systems given by differential equations. Under additional controllability assumptions, we derive upper bounds for the invariance entropy in terms of Lyapunov exponents. These numbers measure the exponential rates of divergence for nearby trajectories, and hence are indicators for stability or instability of the system. In the entropy theory of classical dynamical systems, several relations between entropy and Lyapunov exponents are known. A classic result in this direction is Pesin’s formula (Uspehi Mat. Nauk 32(4) (196):55–112, 287, 1977) which says that the metric entropy of a \({\mathcal{C}}^{2}\)-diffeomorphism f : M → M on a compact Riemannian manifold M with respect to a smooth invariant probability measure μ is given by the μ-integral over the sum of the positive Lyapunov exponents which are defined almost everywhere. (If the invariant measure is ergodic, the Lyapunov exponents are constant almost everywhere, and hence the integral in Pesin’s formula can be replaced by the integrand, that is, the sum of those (almost everywhere constant) Lyapunov exponents which are positive. Moreover, the assumption of f being \({\mathcal{C}}^{2}\) can be weakened to \({\mathcal{C}}^{1+\alpha }\).) Liu (Nagoya Math. J. 150:197–209, 1998) generalized this result to the case of (not necessarily invertible) \({\mathcal{C}}^{2}\)-maps. Ruelle (Bol. Soc. Brasil. Mat. 9(1):83–87, 1978) (and independently, Margulis) showed that without the assumption of μ being equivalent to the Riemannian volume and only assuming that f is a \({\mathcal{C}}^{1}\)-map, the expression in Pesin’s formula is still an upper bound for the entropy. The crowning achievement finally is a result by Ledrappier and Young (Ann. Math. (2) 122:509–539, 540–574, 1985) which provides a formula for the metric entropy of a \({\mathcal{C}}^{2}\)-diffeomorphism which involves a weighted sum of positive Lyapunov exponents, where the weights are certain dimension-like characteristics of the conditional measures on unstable manifolds.KeywordsLyapunov ExponentUnstable ManifoldCompact Riemannian ManifoldNonempty InteriorPeriodic TrajectoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.