Mathematical theory of Lyapunov exponents

This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider ( f , μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer a way to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be Rd or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by m, and the dynamics are generated by iterating a self-map of M, written f : M . For flows, the reviewed results are applicable to time-t maps and Poincaré return maps to cross-sections. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. PACS number: 05.45.−a 1. Nonuniformly hyperbolic systems We begin with a quick review of the setting of nonuniform hyperbolic theory, mostly to fix notation but we will also take the opportunity to bring up some issues. 1751-8113/13/254001+17$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 46 (2013) 254001 Review Given a differentiable map f : M , a point x ∈ M and a tangent vector v at x, we define λ(x, v) = lim n→∞ 1 n log |D f n x (v)| if this limit exists; (1) and if it does not, then the limit is replaced by lim inf or lim sup, and we write λ(x, v) and λ(x, v), respectively. Thus, λ(x, v) > 0 means that |D f n x (v)| grows exponentially, and that is interpreted to mean exponential divergence of nearby orbits. Such an interpretation is valid for as long as the orbits in question remain very close to one another; once they move apart, λ(x, v) offers no information. While limits of the type in (1) need not exist at every x, they do exist almost everywhere under stationarity assumptions: the multiplicative ergodic theorem [O] tells us that given an f -invariant Borel probability measure μ, the following hold at μ-a.e. x: there are numbers λ1(x) > λ2(x) > . . . > λr(x)(x) with multiplicities m1(x), . . . ,mr(x)(x), respectively, such that (i) for every tangent vector v at x, λ(x, v) = λi(x) for some i, (ii) ∑ i mi(x) = dim(M) and (iii) ∑ i λi(x)mi(x) = limn→∞ 1 n log | det(D f n x )|. If f is a diffeomorphism, i.e. if it is invertible, then there is a decomposition of the tangent space TxM into TxM = E1(x)⊕ . . .⊕ Er(x)(x), where dim Ei(x) = mi(x) and λ(x, v) = λi(x) for v ∈ Ei(x). The numbers {λi,mi} are called the Lyapunov exponents of the system ( f , μ). If ( f , μ) is ergodic, then λi and mi are constant μ-a.e. Given a differentiable map f and an f -invariant Borel probability measureμ, many general facts about the system ( f , μ) have been proved. The most basic of these facts translates the infinitesimal information given by Lyapunov exponents to local information along orbits for the nonlinear map f . In the conservative case, i.e. where μ is equivalent to m, the results in the next paragraph were first proved by Pesin [Pe]; in the generality discussed here, they are due to Ruelle [R2]. Let us assume for definiteness that f is a C2 (or C1+α, α > 0) diffeomorphism of M. Then, at μ-a.e. x for which E(x) := ⊕i:λi(x)>0 Ei(x) and E(x) := ⊕i:λi(x)<0 Ei(x) are nontrivial, there is a local stable disc W s loc(x) and a local unstable disc W u loc(x) tangent to Es(x) and Eu(x), respectively. These discs are defined μ-a.e. and are invariant, meaning f (W s loc(x)) ⊂ W s loc( f x) and f−1(W u loc(x)) ⊂ W u loc( f−1x). The sizes and directions of W u loc(x) and W s loc(x) vary measurably with x; so outside a small positive μ-measure set, they vary continuously, the two families of discs forming a kind of local coordinate system for f . The global unstable manifold at x, denoted W u(x), is defined as W (x) := { y ∈ M : lim inf n→∞ 1 n log d( f−nx, f−ny) < 0 } and is equal to ∪n 0 f n(W u loc( f−nx)). Global stable manifolds are defined similarly. A number of other results have been proved for ( f , μ) under the assumption that some or all of λi are nonzero; some of them are reviewed in the next sections. I think it is fair to say that this theory, often referred to as nonuniform hyperbolic theory, has worked out quite well; the class of dynamical systems to which it applies is considerably broader than Axiom A systems [Sm] or the systems studied by Anosov [An].