Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians.     

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.     

Local dynamics of a spatially distributed logistic equation with delay and large transport coefficient

We study the behavior of all solutions in some sufficiently small neighborhood of the positive equilibrium of the spatially distributed Hutchinson equation with diffusion and advection. On the basis of the method of invariant integral manifolds and the method of normal forms, we consider the dynamics for the case critical in the problem on the stability of the stationary solution. We show that, for a sufficiently large value of the transport (advection) coefficient, the critical case has infinite dimension. We construct a quasinormal form, which is a nonlinear parabolic boundary value problem with a deviation in the space variable and which plays the role of a normal form; i.e., its nonlocal dynamics defines the local dynamics of the original equation. Secondary bifurcations in the quasinormal form are considered for the case close to the critical case in the problem on the stability of the stationary solution.     

Sagitta,Lenses, and Maximal Volume

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen by showing any such M that has volume sufficiently close to this upper bound is homeomorphic to the standard sphere \(S^{n}\) or a standard lens space \(S^n/{\mathbb {Z}}_m\) where \(m\in \{2,3,\ldots \}\) is no larger than an a priori constant.     

FRACTAL DIMENSION ESTIMATES FOR INVARIANT SETS OF NON-INJECTIVE MAPS

For a special class of non-injective maps on Riemannian manifolds an upper bound for the fractal dimension of invariant set in terms of singular values of the tangent map and degree of non-injectivity is given     

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.     

Witten Deformation and Polynomial Differential Forms

As is well known, the Witten deformation d_h of the De Rham complex computes the De Rham cohomology. In this paper, we study the Witten deformation on noncompact manifolds and restrict it on differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with the cylindrical structure. We prove that the cohomology of the Witten deformation d_h acting on the complex of the polynomially growing forms (depends on h and) can be computed as the cohomology of the negative remote fiber of h. We show that the assumptions of our main theorem are satisfied in a number of interesting special cases, including generic real polynomials on R_n.     

Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

Normal form theory for volume preserving maps

We discuss the theory of normal forms for volume preserving maps in the non-resonant case. The concept of normal forms is introduced by using the commutative properties with respect to some symmetry groups related to the resonances of the linear eigenvalues. The existence of formal solutions to the conjugation equation, which reduces a map to normal form, is discussed. We also analyze the asymptotic character of the normal forms series and we prove a general theorem which shows that the error between the normal form dynamics and the true dynamics can be exponentially small as a function of the radius of the chosen polydisk centered at the fixed point. As a consequence it is possible to construct analytic manifolds which are approximately invariant under the action of the initial map up to an error which is exponentially small. The connection with a possible KAM theory for volume-preserving maps is suggested. We also show that Noether's theorem can be generalized to volume preserving maps.     

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

On the structure of hyperbolic manifolds

Forn≥2, we quantify the Margulis constant ε(n) giving rise to a thick and thin decomposition of hyperbolicn-manifolds of finite volume. As a consequence, we obtain new universal lower bounds for the volume and Gromov's invariant as well as a geometrical inequality between injectivity radius and diameter for compact manifolds. Finally, we concretise the upper bound for the counting function of hyperbolic manifolds of dimension >4 as described by Burger, Gelander, Lubotzky and Mozes. Partially supported by Schweizerischer Nationalfonds No. 20-61379.00 and 20-67619.02.     

A Harnack-type inequality for non-symmetric Markov semigroups

For a strong Feller and irreducible Markov semigroup on a locally compact Polish space, the Harnack-type inequality (1.1) holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive. Moreover, new sufficient conditions for this inequality to hold, as well as upper bound estimates of the underlying constant, are presented for diffusion semigroups on Riemannian manifolds.     

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsváth–Szabó contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S³ with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsváth–Szabó invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.     

Morse�CBott functions and the Lusternik�CSchnirelmann category

Lusternik–Schnirelmann category of a manifold gives a lower bound of the number of critical points of a differentiable map on it. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C ¹ and their gradient flows, where cone-decompositions are used to give an upper bound for the Lusternik–Schnirelmann category which is a homotopy invariant of a topological space. In particular, the Morse–Bott functions on the Stiefel manifolds considered by Frankel (1965) are effectively used to construct the conedecompositions of Stiefel manifolds and symmetric Riemannian spaces to determine their Lusternik–Schnirelmann categories.     

A note on the dichotomy spectrum

In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques.

In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space.     

Invariant manifolds for parabolic equations under perturbation of the domain

We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the Dirichlet boundary condition. Under the Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium. The continuity results are obtained by keeping track of the construction of invariant manifolds in [P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, in: Dynamics Reported, Vol. 2, in: Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38].     

Positivity of direct images of positively curved volume forms

We discuss Kiselman–Berndtsson’s minimum principle for plurisubharmonic functions in terms of the positivity of direct images of certain positively curved volume forms, and generalize it to holomorphically convex manifolds with compact group actions. With this generalization and other techniques, we establish a minimum principle for positively curved volume forms from the point of view of geometric invariant theory on Stein manifolds. Minimum principle with some noncompact group actions is also considered.     

On invariant subspaces for polynomially bounded operators

We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C.Ambrozie and V.Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).