Stefan Müller

Suppose ρ(X) is an invariant of a smooth vector field X that depends smoothly on X, and satisfies ρ(φ.X) = ρ(X) for all diffeomorphisms φ. The invariant ρ can be considered as an invariant of the smooth isotopy corresponding to the vector field X, and ρ is invariant under conjugation by diffeomorphisms. A natural question is whether ρ can be extended to an invariant of an isotopy of homeomorphisms, and whether ρ is also invariant under conjugation by homeomorphisms, at least in the presence of an additional geometric structure (for example a volume, symplectic, or contact form) that is preserved by all vector fields, isotopies, and diffeomorphisms and homeomorphisms. These questions were asked by V. I. Arnold in the context of the helicity of a divergence-free vector field on the three-sphere with its standard volume form.

In this paper, we study Arnold's questions for closed three-manifolds equipped with a regular contact form; the isotopies and diffeomorphisms preserving this contact form are those that commute with the Reeb flow, and are precisely the lifts of Hamiltonian isotopies and diffeomorphisms of the quotient of the underlying contact manifold by the Reeb flow. Similarly, we consider suspensions of surface isotopies previously studied by G. G. Gambaudo and É. Ghys. In contrast to the case of a volume form considered by Arnold, it is a priori not clear what the correct generalization is to continuous Hamiltonian and contact isotopies, and homeomorphisms preserving a symplectic or contact form. Another natural question is whether there actually exist smooth vector fields that are topologically conjugate but not smoothly conjugate.

In both cases, we compute explicit formulas for the helicity in terms of the generating Hamiltonian function of the contact or Hamiltonian isotopy. It was previously shown by myself and Y.-G. Oh, and by A. Banyaga and P. Spaeth, respectively, that the concepts of Hamiltonian isotopies and isotopies preserving a contact form, admit natural and nontrivial generalizations to topological dynamical systems, and to homeomorphisms as transformations preserving the underlying geometric structure. Based on these results, we answer Arnold's questions in the affirmative in the present situations.

In the last part of the paper, we prove the existence of smooth (Hamiltonian and strictly contact) dynamical systems that are topologically conjugate but not smoothly conjugate. The proofs use the transformation laws and the uniqueness theorems of topological Hamiltonian and topological (strictly) contact dynamics. We also briefly discuss higher-dimensional helicities.

One of the key ingredients of the proofs is that our explicit formulas for the helicity are invariant under conjugation, and are continuous with respect to an appropriate choice of metric. Thus the ideas we present apply conceptually to other invariants ρ as well.

In this paper, we consider geometrical structures that are defined as conformal classes σ of a tensor field τ on a smooth manifold. For example, the conformal structure σ = [τ] is an orientation if τ is a volume form, a conformal symplectic structure if τ is a symplectic form, and a contact structure if τ is a contact form; these three structures are the classical conformal structures.

The conformal diffeomorphism group of a conformal structure σ is by definition properly essential if there exists a conformal diffeomorphism of σ that does not preserve any of the tensor fields in the class σ. We prove this to be the case for contact manifolds, for symplectic manifolds that are Liouville, and for oriented manifolds. Our arguments are local-to-global, and rely on an obstruction in the form of a cohomological equation.

Moreover, we study the orbit of a given tensor field under the action of the conformal diffeomorphism group; the orbit of τ in the class σ can be identified with the quotient of the groups of diffeomorphisms that preserve σ by those that preserve τ. Among other results, we show that the orbit of a contact form α on a closed manifold is not maximal. That means there exists another contact form, defining the same contact structure, but not diffeomorphic to α. The method of proof is to show the Reeb flows of the two contact forms are not conjugated. Along the way, we demonstrate that every contact structure (not necessarily a given contact form) on a closed manifold admits a closed Reeb orbit.

We also relate our work to a conformal invariant defined by A. Banyaga and a theorem of W. H. Gottschalk and G. A. Hedlund, and define a new contact invariant (the conformal length) of a contact diffeomorphism.

The notion of the Hamiltonian metric plays a central role in topological Hamiltonian dynamics. In this article, I reprove the results of my JSG paper with Y.-G. Oh in the case of the ∞-Hofer norm in place of the (1,∞)-Hofer norm, and compare the two choices of norm giving rise to different definitions of the Hamiltonian metric. See the description of the JSG paper for details.

In the second part of the paper, I demonstrate the following result. Every topological Hamiltonian dynamical system is arbitrarily close (with respect to the (1,∞)-Hamiltonian metric) to a continuous Hamiltonian dynamical system (i.e., one defined with respect to the stronger ∞-Hamiltonian metric) with the same end point; moreover, the latter is smooth everywhere except possibly at time one. In particular, the two groups of Hamiltonian homeomorphisms (by definition, the time-one maps of topological and continuous Hamiltonian isotopies) arising from the different choices of Hofer norm coincide.

The proof involves the observation by L. Polterovich that a generic smooth Hamiltonian isotopy is regular (in the sense that its tangent vector never vanishes), and an adaptation of a reparameterization procedure also due to Polterovich. A combination with further approximation and reparameterization techniques then completes the proof.

Finally it is shown that the two a priori different Hofer norms on the group of Hamiltonian homeomorphisms are equal.

A topological Hamiltonian dynamical system consists of a topological Hamiltonian isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying symplectic manifold. By definition, this Hamiltonian function is the limit of a sequence of (normalized time-dependent) smooth Hamiltonian functions with respect to the usual Hofer metric; the corresponding sequence of smooth Hamiltonian isotopies converges uniformly to the above continuous isotopy. The key notion is the Hamiltonian metric, which is precisely the above combination of the topological C^0-metric with the dynamical Hofer metric.

A topological Hamiltonian isotopy is determined uniquely by its topological Hamiltonian function. In other words, given two topological Hamiltonian dynamical systems with the same topological Hamiltonian function, their topological Hamiltonian isotopies must coincide as well. This follows from the important energy-capacity inequality of F. Lalonde and D. McDuff, and can be formulated in three equivalent ways (cf. my joint papers on topological contact dynamics with P. Spaeth). The proof is thus already contained in the present paper (although somewhat in disguise); an earlier version for standard Euclidean space can be found in the monograph by H. Hofer and E. Zehnder.

As a consequence of the above uniqueness theorem, composition and inversion of topological Hamiltonian dynamical systems, topological Hamiltonian functions, and topological Hamiltonian isotopies and their time-one maps, can be defined as in the smooth case. Thus topological Hamiltonian dynamics is a natural extension of the smooth dynamics of a Hamiltonian vector field (or function) to topological dynamics. We show by example that a topological Hamiltonian isotopy need not even be Lipschitz continuous (in the space or time variable). In other words, the extension to topological dynamics is a genuine extension (on any symplectic manifold).

A Hamiltonian homeomorphism is by definition the time-one map of a topological Hamiltonian isotopy. Much attention in this article is focused on this group of Hamiltonian homeomorphisms and its topological properties. By continuity of the mass flow homomorphism (the topological dual to the volume flux homomorphism, which in turn is related to the symplectic flux homomorphism by multiplication in deRham cohomology), the mass flow of a topological Hamiltonian isotopy vanishes. This fact is related to the question of (non-)simpleness of the (kernel of the mass flow in the) group of area-preserving homeomorphisms of a surface, and thus yields an instance where symplectic methods enter topological dynamics.

A symplectic homeomorphism is by definition the C^0-limit of a sequence of symplectic diffeomorphisms. In other words, the group of symplectic homeomorphisms of a symplectic structure is the uniform closure of the group of symplectic diffeomorphisms in the group of homeomorphisms. This definition is motivated by Eliashberg-Gromov's celebrated C^0-symplectic rigidity theorem; a symplectic homeomorphism that is in addition smooth preserves the symplectic structure, and is thus a symplectic diffeomorphism in the usual sense. Moreover, the well known transformation law extends to topological Hamiltonian dynamical systems and symplectic homeomorphisms. Thus symplectic homeomorphisms can be considered as the topological automorphisms of the symplectic structure, and the group of Hamiltonian homeomorphisms forms a normal subgroup of the group of symplectic homeomorphisms. A symplectic homeomorphism preserves the measure obtained by integrating the canonical volume form induced by the symplectic form. As in the smooth case, the inclusion of symplectic homeomorphisms as a closed subgroup of the group of measure-preserving homeomorphisms is proper if and only if the dimension of the underlying manifold is greater than two. The properness can be deduced from Gromov's seminal non-squeezing theorem.

We repeatedly demonstrate that our choice of Hamiltonian metric gives the objects of topological Hamiltonian dynamics the correct dynamical, topological, and algebraic properties. For example, the set of topological Hamiltonian dynamical systems forms a topological group with respect to the Hamiltonian topology (the topology induced by the Hamiltonian metric), and smooth Hamiltonian dynamical systems form a topological subgroup. This article marks the beginning of the study of topological Hamiltonian dynamics in the sense explained here. The notation and language used to describe it have (in my opinion) improved significantly over the past few years. The present vocabulary is taken from my joint papers with P. Spaeth on topological contact dynamics.

My thesis contains an improved presentation and elaborate development of topological Hamiltonian dynamics. Among other things, the results of my joint JSG paper with Y.-G. Oh and my JKMS paper are explained in greater detail. In particular, the case of open manifolds is developed rigorously, and the various choices in the definitions are discussed and justified at great length.

This sequel to our previous paper continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems, and study the topological properties of the groups of contact and strictly contact homeomorphisms.

On the latter we construct a bi-invariant metric that resembles the bi-invariant metric on the group of strictly contact diffeomorphisms studied in part I. Among other things, we show that a generic smooth contact isotopy is regular (in the sense that its tangent vector is never stationary). The proof is similar to the one given by L. Polterovich for smooth Hamiltonian isotopies. Generalizing the approximation and reparameterization techniques from my JKMS paper, we prove the following main lemma of part II: every topological contact dynamical system is arbitrarily close (with respect to the (1,∞)-contact metric) to a continuous contact dynamical system (i.e., one defined with respect to the stronger ∞-contact metric) with the same end point; moreover, the latter is smooth everywhere except possibly at time one. In particular, the two groups of contact homeomorphisms (by definition, the time-one maps of topological and continuous contact isotopies) arising from the different choices of a Hofer-like norm (on the space of Hamiltonian functions) coincide. The same holds for the two groups of strictly contact homeomorphisms (by definition, the time-one maps of topological and continuous strictly contact isotopies).

Finally on every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugate by a contact C^1-diffeomorphism.

Some of the most prominent early results in symplectic topology include a profound energy-capacity inequality in Hamiltonian dynamics, the non-degeneracy of the remarkable Hofer metric on the group of Hamiltonian diffeomorphisms, and Eliashberg-Gromov’s fundamental C^0-rigidity of symplectic diffeomorphisms. This article is the first part of a series of papers on topological contact dynamics. We derive an energy-capacity inequality for contact diffeomorphisms, which proves to be an equally powerful tool in contact dynamics. As an immediate consequence, we establish the non-degeneracy of a bi-invariant Hofer-like metric on the group of diffeomorphisms preserving a contact form.

A topological contact dynamical system consists of a topological contact isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying contact manifold, and a topological conformal factor (a continuous function). By definition, this Hamiltonian function is the limit of a sequence of smooth (time-dependent) Hamiltonian functions with respect to a Hofer-like metric. Moreover, the corresponding sequence of smooth contact isotopies converges uniformly to the above continuous isotopy, and their smooth conformal factors converge uniformly to the continuous function associated to the limit isotopy. This definition is explained and justified in detail in this paper.

A topological contact isotopy is not generated by a vector field, and may not even be Lipschitz continuous; nonetheless, as a consequence of the contact energy-capacity inequality, it is uniquely determined by its associated topological Hamiltonian function. Composition and inversion of topological contact dynamical systems, topological Hamiltonian functions, and topological contact isotopies and their time-one maps, can therefore be defined as in the smooth case, and the usual transformation law continues to hold.

We show that the topological automorphism groups of a contact structure and a contact form exhibit surprising rigidity properties, including uniqueness of the topological conformal factor, and C^0-rigidity of contact and strictly contact diffeomorphisms, analogous to the above Eliashberg-Gromov rigidity in the case of a symplectic structure.

The upshot is a natural and genuine extension of the smooth dynamics of a contact vector field to topological dynamics. Consequences and applications to both smooth contact dynamics and topological dynamics, such as the fact that a topological automorphism of a contact structure conjugates the corresponding Reeb flows, are discussed throughout the paper. The uniqueness theorems applied to smooth contact dynamical systems prove rigidity of contact isotopies and their conformal factors in the following sense: if a sequence of contact isotopies and their conformal factors are uniformly Cauchy, and the generating Hamiltonian functions converge (with respect to the Hofer-like metric) to another (continuously differentiable) Hamiltonian function (with uniquely integrable contact vector field), then the limit of the contact isotopies coincides with the contact isotopy generated by the limit Hamiltonian function, and likewise for the conformal factors.

Our general approach to relating topological contact dynamics to topological Hamiltonian dynamics is via symplectization of a contact manifold.

This short note on the flux homomorphism for strictly contact isotopies complements my paper on the helicity written jointly with P. Spaeth. I compute the volume flux homomorphism restricted to symplectic and volume preserving contact isotopies and their C^0-limits for certain classes of symplectic and contact manifolds. A copious number of examples is given.

The above restrictions of the flux homomorphism may fail to be surjective. The flux homomorphism vanishes for an isotopy preserving a regular contact form, but can be non-trivial for non-regular contact forms. Applications are discussed in the article mentioned above. I also find an obstruction to regularizing a strictly contact isotopy that is not present for Hamiltonian isotopies or contact isotopies. This is related to the fact that (for non-regular contact forms) a locally defined strictly contact vector field may not extend to a globally defined strictly contact vector field. Conversely, if the flux of a given isotopy is non-trivial, its generating strictly contact vector field cannot be fragmented into a sum of strictly contact vector fields that are supported in Darboux charts.

In the late '50s and early '60s, J. R. Munkres and M. W. Hirsch independently developed obstruction theories for when a given homeomorphism of a smooth manifold can be approximated uniformly by diffeomorphisms. In the past decade, Y.-G. Oh and J. C. Sikorav independently showed that if a volume preserving homeomorphism can be approximated uniformly by diffeomorphisms, it can also be approximated uniformly by volume preserving diffeomorphisms.

This note is an attempt to give a necessary and sufficient condition for when a volume preserving homeomorphism can be approximated uniformly by (volume preserving) diffeomorphisms. An update should appear shortly.

The principal result of this third part of the series of papers on topological contact dynamics is the converse to the main uniqueness theorem(s) of part I: every topological contact isotopy possesses a unique topological Hamiltonian function. Applications are discussed as well.