DCDS
Admissibility versus nonuniform exponential behavior for noninvertible cocycles
Luis Barreira Claudia Valls
We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.
keywords: nonuniform exponential dichotomies. Admissibility
DCDS-S
Delay equations and nonuniform exponential stability
Luis Barreira Claudia Valls
For nonautonomous linear delay equations $v'=L(t)v_t$ admitting a nonuniform exponential contraction, we establish the nonuniform exponential stability of the equation $v'=L(t) v_t +f(t,v_t)$ for a large class of nonlinear perturbations.
keywords: nonuniform exponential stability. Delay equations
DCDS
Growth rates and nonuniform hyperbolicity
Luis Barreira Claudia Valls
We consider linear equations $v'=A(t)v$ that may exhibit different asymptotic behaviors in different directions. These can be thought of as stable, unstable and central behaviors, although here with respect to arbitrary asymptotic rates $e^{c \rho(t)}$ determined by a function $\rho(t)$, including the usual exponential behavior $\rho(t)=t$ as a very special case. In particular, we consider the notion of $\rho$-nonuniform exponential trichotomy, that combines simultaneously the nonuniformly hyperbolic behavior with arbitrary asymptotic rates. We show that for $\rho$ in a large class of rate functions, any linear equation in block form in a finite-dimensional space, with three blocks having asymptotic rates $e^{c \rho(t)}$ respectively with $c$ negative, zero, and positive, admits a $\rho$-nonuniform exponential trichotomy. We also give explicit examples that cannot be made uniform and for which one cannot take $\rho(t)=t$ without making all Lyapunov exponents infinite. Furthermore, we obtain sharp bounds for the constants that determine the exponential trichotomy. These are expressed in terms of appropriate Lyapunov exponents that measure the growth rate with respect to the function $\rho$.
keywords: Asymptotic behavior growth rates Lyapunov exponents.
DCDS
Reversibility and equivariance in center manifolds of nonautonomous dynamics
Luis Barreira Claudia Valls
We consider reversible and equivariant dynamical systems in Banach spaces, either defined by maps or flows. We show that for a reversible (respectively, equivariant) system, the dynamics on any center manifold in a certain class of graphs (namely $C^1$ graphs with Lipschitz first derivative) is also reversible (respectively, equivariant). We consider the general case of center manifolds for a nonuniformly partially hyperbolic dynamics, corresponding to the existence of a nonuniform exponential trichotomy of the linear variational equation. We also consider the case of nonautonomous dynamics.
keywords: nonuniform exponential trichotomies center manifolds reversibility. equivariance
DCDS
Noninvertible cocycles: Robustness of exponential dichotomies
Luis Barreira Claudia Valls
For the dynamics defined by a sequence of bounded linear operators in a Banach space, we establish the robustness of the notion of exponential dichotomy. This means that an exponential dichotomy persists under sufficiently small linear perturbations. We consider the general cases of a nonuniform exponential dichotomy, which requires much less than a uniform exponential dichotomy, and of a noninvertible dynamics or, more precisely, of a dynamics that may not be invertible in the stable direction.
keywords: Exponential dichotomies linear perturbations noninvertible dynamics robustness. nonuniform exponential dichotomies
CPAA
Topological conjugacies and behavior at infinity
Luis Barreira Claudia Valls
We obtain a version of the Grobman--Hartman theorem in Banach spaces for perturbations of a nonuniform exponential contraction, both for discrete and continuous time. More precisely, we consider the general case of an exponential contraction with an arbitrary nonuniform part and obtained from a nonautonomous dynamics, and we establish the existence of Hölder continuous conjugacies between an exponential contraction and any sufficiently small perturbation. As a nontrivial application, we describe the asymptotic behavior of the topological conjugacies in terms of the perturbations: namely, we show that for perturbations growing in a certain controlled manner the conjugacies approach zero at infinity and that when the perturbations decay exponentially at infinity the conjugacies have the same exponential behavior.
keywords: topological conjugacies. Exponential contractions
CPAA
Center manifolds for nonuniform trichotomies and arbitrary growth rates
Luis Barreira Claudia Valls
We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.
keywords: Center manifolds nonuniform exponential trichotomies.
DCDS
Preface
Luis Barreira
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Yakov Pesin on the occasion of his sixtieth birthday, which took place in December of 2006. Pesin is one of the world leaders in the field of dynamical systems. His work exerted a deep and lasting influence in the area. Subjects of his landmark works include nonuniform hyperbolicity, smooth ergodic theory, partial hyperbolicity, thermodynamic formalism, and dimension theory in dynamics.

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DCDS
Dimension and ergodic decompositions for hyperbolic flows
Luis Barreira Christian Wolf
For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.
keywords: hyperbolic flows pointwise dimension. Ergodic decompositions
DCDS
Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics
Luis Barreira Claudia Valls
The existence of stable manifolds for nonuniformly hyperbolic trajectories is well know in the case of $C^{1+\alpha}$ dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a $C^1$ diffeomorphism that is not of class $C^{1+\alpha}$ for any $\alpha$ and for which there exists no stable manifold. The $C^{1+\alpha}$ hypothesis appears to be crucial in some parts of smooth ergodic theory, such as for the absolute continuity property and thus in the study of the ergodic properties of the dynamics. Nevertheless, we establish the existence of invariant stable manifolds for nonuniformly hyperbolic trajectories of a large family of maps of class at most $C^1$, by providing a condition which is weaker than the $C^{1+\alpha}$ hypothesis but which is still sufficient to establish a stable manifold theorem. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We note that our proof of the stable manifold theorem is new even in the case of $C^{1+\alpha}$ nonuniformly hyperbolic dynamics. In particular, the optimal $C^1$ smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.
keywords: Invariant manifolds hyperbolicity.

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