Commutators of contactomorphisms

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.     

Conley conjecture and local Floer homology

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.     

On the first homology of automorphism groups of manifolds with geometric structures

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category. This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science.     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

Weighted diffeomorphism groups of Riemannian manifolds

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author’s earlier work (Walter, 2012).     

Non-absolutely continuous foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also establish a more general version of this result for general foliations with compact leaves. Partially supported by JSPS. Partially supported by National Science Foundation grant #DMS-0088971 and U.S.-Mexico Collaborative Research grant 0104675.     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

A $${C^\infty}$$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

We prove a \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a \({C^\infty}\) closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.     

Centralizers of area preserving diffeomorphisms on

It has been conjectured that a generic diffeomorphism on a compact manifold will have trivial centralizer. We give some partial results towards proving this conjecture within the class of area preserving diffeomorphisms of the sphere.

     

Bifurcations on contact manifolds

Consider a 1-parameter compactly supported family of Legendrian submanifolds of the 1-jet bundle of a compact manifold with its natural contact structure and a path of intersection points of the Legendrian family with the 1-jet of a constant function. Since the contact distribution is a symplectic vector bundle, it is possible to assign a Maslov-type index to the intersection path. We show that the non-vanishing of the Maslov intersection index implies that there exists at least one point of bifurcation from the given path of intersection points. This result can be viewed as a kind of analogue in bifurcation theory of the Arnold-Sandon conjecture on intersections of Legendrian submanifols. The proof is based on the technique of generating functions that relates the properties of Hamiltonian diffeomorphisms to the Morse theory of the associated functions.     

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:?• The uniform norm of the differential of its n-th iteration;?• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.?We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups. Oblatum 6-XII-2001 & 19-VI-2002?Published online: 5 September 2002 RID="*" ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

The transversal homoclinic point of a saddle-node implies horseshoe

A method of identifying the existence of horseshoe for a two-dimension diffeomorphism is introduced and utilized to generalize the Birkhoff-Smale Theorem to the saddle-node case. Project supported by the National Natural Science Foundation of China (Grant No. 19531070).     

A characterization ofN-dimensional daubechies type tensor product wavelet

In this paper, we consider the problem of the existence of general non-separable variate orthonormal compactly supported wavelet basis when the symbol function has a special form. We prove that the general non-separable variate orthonormal wavelet basis doesn't exist if the symbol function possesses a certain form. This helps us to explicate the difficulty of constructing the non-separable variate wavelet basis and to hint how to construct non-separable variate wavelet basis. This research is supported by the National Natural Science Foundation of China (No.69982002) and the Opening Foundation of National Mobile Communications Research Laboratory in Southeast University.     

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors?? previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.     

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.     

On the dynamics of generic non-Abelian free actions

We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C ¹-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenbergs celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).     

On a Lagrangian Formulation of the Incompressible Euler Equation

In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.     

Flow equivalence,hyperbolic systems and a new zeta function for flows

We analyze the dynamics of diffeomorphisms in terms of their suspension flows. For many Axion A diffeomorphisms we find simplest representatives in their flow equivalence class and so reduce flow equivalence to conjugacy. The zeta functions of maps in a flow equivalence class are correlated with a zeta function ζ _H for their suspended flow. This zeta function is defined for any flow with only finitely many closed orbits in each homology class, and is proven rational for Axiom A flows. The flow equivalence of Anosov diffeomorphisms is used to relate the spectrum of the induced map on first homology to the existence of fixed points. For Morse-Smale maps, we extend a result of Asimov on the geometric index. Partially supported by MCS 76-08795.     

Modelling by a planar map: Lyapunov-type numbers in the presence of spiral points

Lyapunov-type numbers are usually defined for diffeomorphisms with a smooth invariant manifold. We consider here the case of a planar diffeomorphism with an invariant curve that contains spiral points. The limits defining the Lyapunov-type numbers are shown to exist. Numerical results for the delayed logistic map illustrate the analysis.     

Entropy formula for random transformations

Summary We exhibit random strange attractors with random Sinai-Bowen-Ruelle measures for the composition of independent random diffeomorphisms. The research of this author is partially supported by the National Science Foundation and the Sloan Foundation.