Lyapunov spectrum

The Lyapunov spectrum of a dynamical system is the ordered set of its Lyapunov exponents, typically written λ_1 ≥ λ_2 ≥ ... ≥ λ_n. The spectrum encodes the complete local stability structure of the system: the number of positive exponents gives the dimension of the unstable manifold, the number of negative exponents gives the dimension of the stable manifold, and the number of zero exponents indicates the presence of neutral or center directions. For a system with a smooth invariant measure, the sum of all exponents equals the exponential rate of phase-space volume change. The spectrum is a fundamental invariant in smooth ergodic theory and appears in the Pesin entropy formula as the raw material from which Kolmogorov-Sinai entropy is computed. Not all ordered sets of real numbers can be realized as Lyapunov spectra: the possible spectra are constrained by the dynamics, the dimension, and the measure. \n\nThe Lyapunov spectrum is central to the theory of Non-uniform hyperbolicity, where the spectrum determines the existence and structure of invariant manifolds even when hyperbolicity is not uniform across phase space.