Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Analytic invariant manifolds for sequences of diffeomorphisms

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique

We prove that admissible functions for Fubini-Study metric on the complex projective space PmC of complex dimension m, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of PmC. A similar lower bound holds on some projective manifolds. This gives an optimal constant in a Hörmander type inequality on these manifolds, which allows us, using Tian's invariant, to establish the existence of Einstein-Kähler metrics on them.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Center manifolds for difference equations—smooth parameter dependence

For sufficiently small C¹ perturbations of (nonautonomous) linear difference equations with a nonuniform exponential trichotomy, we establish the existence of center manifolds with the optimal C¹ regularity. We also consider the case of parameter-dependent perturbations and we obtain the C¹ dependence of the center manifolds on the parameter. In addition, we consider arbitrary growth rates with the usual exponential estimates of the form in the notion of exponential trichotomy replaced by where ρ is now an arbitrary function. The proof of the regularity, both of the center manifold and of its dependence, on the parameter is based on the fiber contraction principle. The most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument.     

On discretizations of invariant foliations over inertial manifolds

Invariant foliations over inertial manifolds of partial differential equations under numerical discretizations are studied. It is proved that the numerical method considered as a discrete dynamical system has C¹-close invariant foliations. The rate of the C¹-convergence is estimated as well.     

Parameter dependence of stable manifolds under nonuniform hyperbolicity

For a nonautonomous linear equation v^′=A(t)v in a Banach space with a nonuniform exponential dichotomy, we show that the nonlinear equation v^′=A(t)v+f(t,v,λ) has stable invariant manifolds Vλ which are Lipschitz in the parameter λ provided that f is a sufficiently small Lipschitz perturbation. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, the above assumption is very general. We emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach.     

Center manifolds and optimal estimates

For sufficiently small perturbations of a nonuniform exponential trichotomy, we establish the existence of $C^k$ invariant center manifolds. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. In particular, we obtain optimal estimates for the decay of all derivatives along the trajectories on the center manifolds.     

C Invariant Manifolds for Maps of Banach Spaces

In this work we give a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds for a C^k map of a Banach space near a fixed point where k ≥ 1 is an integer.     

Center manifolds depending on a parameter

For differential equations u^′=A(t)u+f(t,u,λ) obtained from sufficiently small C¹ perturbations of a nonuniform exponential trichotomy, we establish the C¹ dependence of the center manifolds on the parameter λ. Our proof uses the fiber contraction principle to establish the regularity property. We note that our argument also applies to linear perturbations, without further changes.     

Invariant manifolds for differential equations

In this paper we discuss the concept ‘generalized exponential dichotomy’ and give the existence ofC ^k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces. Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.     

Integral stable manifolds in Banach spaces

We establish the existence of smooth integral stable manifoldsfor sufficiently small perturbations of nonuniform exponentialdichotomies in Banach spaces. We also consider the case of anonautonomous dynamics given by a sequence of C¹ maps. The optimalsmoothness of the manifolds is obtained at the same time astheir existence, using a convenient lemma of Henry. Furthermore,we obtain not only the exponential decay of the dynamics alongthe stable manifolds, but also of its derivative. In addition,we give a characterization of the stable manifolds in termsof the maximal exponential growth rate that is allowed, we discusshow the manifolds vary with the perturbations, and we discusstheir equivariance with respect to a sequence of linear operators.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces

After a discussion on definability of invariant subdivision rules we discuss rules for sequential data living in Riemannian manifolds and in symmetric spaces, having in mind the space of positive definite matrices as a major example. We show that subdivision rules defined with intrinsic means in Cartan-Hadamard manifolds converge for all input data, which is a much stronger result than those usually available for manifold subdivision rules. We also show weaker convergence results which are true in general but apply only to dense enough input data. Finally we discuss C ¹ and C ² smoothness of limit curves.     

Averaging of Legendrian Submanifolds of Contact Manifolds

We give a procedure to ‘average’ canonically C¹-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby. Mathematics Subject Classification (2000): 53D10.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

Integral manifolds under explicit variable time-step discretization

We study the behavior of the “full hierarchy” of integral manifolds, i.e. in particular those of stable, center-stable, center, center-unstable and unstable type, for nonautonomous ordinary differential equations in Banach spaces under explicit one-step discretization with varying step-sizes. Our main results on C ^m ? 1-closeness under such discretizations are formulated in a quantitative fashion and turn out to be an easy consequence of a general theorem on the existence of invariant fiber bundles within the “calculus on time scales”.