engleski

Stability of Nonuniformly Hyperbolic Dynamics and Lyapunov Functions

Our main objective is to obtain characterizations in terms of Lyapunov functions of several classes of nonuniformly hyperbolic dynamics and to study their persistence under sufficiently small linear perturbations. In Part I, dedicated to the case of almost all trajectories, we describe systematically the relation between nonuniform hyperbolicity and Lyapunov functions. In particular, we describe criteria for nonvanishing Lyapunov exponents of linear cocycles over measure-preserving maps and flows. We also establish converse results, with the explicit construction of Lyapunov functions for any cocycle with nonzero Lyapunov exponents. In Part II, dedicated to the case of a single trajectory, we characterize completely strong nonuniform exponential contractions and dichotomies in terms of quadratic Lyapunov functions, both for maps and flows. We also show that any sufficiently small linear perturbation of a strong nonuniform exponential dichotomy admits strong nonuniform exponential dichotomy.

cone families; ergodic theory; linear perturbations; Lyapunov exponents; Lyapunov functions; Lyapunov sequences; nonuniform exponential contractions; nonuniform exponential dichotomies; nonuniform hyperbolicity; robustness.

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