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### Open Access Journals

DCDS

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.

DCDS-S

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.

DCDS

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 $\rho$.
DCDS

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*.
DCDS

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.

CPAA

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.

CPAA

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.
DCDS

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

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.

DCDS

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.

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