DCDS
Lyapunov exponents for continuous transformations and dimension theory
Luis Barreira César Silva
We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous transformations.
keywords: multiplicative ergodic theorem. Lyapunov exponents
DCDS
Dense area-preserving homeomorphisms have zero Lyapunov exponents
Mário Bessa César M. Silva
We give a new definition (different from the one in [14]) for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a $C^0$-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the $C^0$-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.
keywords: Area-preserving homeomorphisms topological dynamics. Lyapunov exponents
DCDS-B
Integral conditions for nonuniform $μ$-dichotomy on the half-line
António J.G. Bento Nicolae Lupa Mihail Megan César M. Silva

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

keywords: Evolution operators nonuniform dichotomies Datko theorem

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