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### Open Access Journals

DCDS

We prove that the conjugacies in the Grobman-Hartman theorem are
always Hölder continuous, with Hölder exponent determined by the
ratios of Lyapunov exponents with the same sign. We also consider
the case of hyperbolic trajectories of sequences of maps, which
corresponds to a nonautonomous dynamics with discrete time. All the
results are obtained in Banach spaces. It is common knowledge that
some authors claimed that the Hölder regularity of the conjugacies
is well known, however without providing any reference. In fact, to
the best of our knowledge, the proof appears nowhere in the
published literature.

DCDS

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting

*nonuniform exponential contractions*in terms of*strict Lyapunov functions*. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniform exponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works, the persistence of the asymptotic stability under sufficiently small linear and nonlinear perturbations.
DCDS

We construct $C^k$ invariant center manifolds for differential
equations $u'=A(t)u+f(t,u)$ obtained from
sufficiently small perturbations of a

*nonuniform*exponential trichotomy. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. In addition, we can also consider linear perturbations with the same method.
DCDS

We obtain stable invariant manifolds with optimal $C^k$ regularity
for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.

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.

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