Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

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).     

Lengths of Contact Isotopies and Extensions of the Hofer Metric

Using the Hofer metric, we construct, under a certain condition, a bi-invariant distance on the identity component in the group of strictly contact diffeomorphisms of a compact regular contact manifold. We also show that the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of symplectic diffeomorphisms of certain symplectic manifolds.Mathematics Subject classification (2000): 53C12, 53C15.     

Sur les groupes engendrés par des difféomorphismes proches de l'identité

We consider groups generated by real analytic diffeomorphisms of a compact manifold close to the identity. We show that the dynamics of such a group is recurrent unless the group satisfies a very particular property, similar to solvability. We study in detail the case of diffeomorphisms of the circle and the disc.     

On the geometry of the space of smooth fibrations

We study geometrical aspects of the space of smooth fibrations between two given manifolds M and B, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base B.     

Topological group cohomology with loop contractible coefficients

We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.     

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.     

Normal forms for differentiable maps near a fixed point

We propose in this paper a significant refinement of normal forms for differentiable maps near a fixed point. We give a method to obtain further reduction of classical normal forms with concrete and interesting applications. Our method leads to unique normal forms either with respect to general diffeomorphisms in certain cases or with respect to near identity diffeomorphisms in other cases. Our approach is rational in the sense that if the coefficients of a map are in a field K, so is its normal form. This revised version was published online in June 2006 with corrections to the Cover Date.     

Controllability on the group of diffeomorphisms

Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of the identity of the group of diffeomorphisms of M.     

Polynomial alternatives for the group of affine motions

It is known that every polycyclic-by-finite group – even if it admits no affine structure – allows a polynomial structure of bounded degree. A major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of , in contrast to the group of affine motions, is no longer a finite dimensional Lie group. In this paper we construct a family of (finite dimensional) Lie groups, even linear algebraic groups, of polynomial diffeomorphisms, which we call weighted groups of polynomial diffeomorphisms. It turns out that every polycyclic-by-finite group admits a polynomial structure via these weighted groups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding Mal`cev completion. This, and other properties, provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch. Received November 30, 1998; in final form March 10, 1999     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental’s nonlinear Maslov index. As an application we prove a Bott–Samelson type obstruction theorem for positive loops of contactomorphisms.     

A Variational Approach to Givental’s Nonlinear Maslov Index

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental’s nonlinear Maslov index. As an application we prove a Bott–Samelson type obstruction theorem for positive loops of contactomorphisms.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

On commutators of equivariant homeomorphisms

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing H₁(HG(M)) for more complicated G-manifolds.     

Reduced 1-cohomology and relative property (T)

Shalom characterized property (T) in terms of the vanishing of all reduced first cohomology for compactly generated groups. We characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes. We show that, in a strong sense, this is inequivalent to relative property (T), even for compactly generated group-pairs.     

Expanding maps of solenoids

We prove that compact metric groups which admit expanding maps must be solenoidal groups, and that every expanding map on a solenoidal group is topologically conjugate to a positively expansive group endomorphism. This first was studied by Shub for expanding differentiable maps of tori and by Manning for Anosov diffeomorphisms of tori.     

The identity component of the leaf preserving diffeomorphism group is perfect

It is shown that for any smooth foliated manifold the identity component of the group of all leaf preserving diffeomorphisms is perfect. This result generalizes in a sense a well-known theorem of Thurston.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

Contact cuts

We describe a contact analog of the symplectic cut construction [L]. As an application we show that the group of contactomorphisms for certain overtwisted contact structures on lens spaces contains countably many non-conjugate two tori. Supported by the NSF grant DMS-980305.