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I have been learning a bit about stable and unstable manifold theory for a non-uniformly hyperbolic diffeomorphism $f: M \to M$ on a smooth manifold. It seems that there are two completely separate cases, each having its own universe of literature: the case where $f$ is $C^1$ and the case where $f$ is $C^{1 + \alpha}$. I can understand more or less why the cases are so different, but I don't understand why anybody cares about the $C^1$ case (or really anything below $C^\infty$). The important examples with which I am familiar are the geodesic flow on a compact Riemannian manifold and a symplectomorphism on a symplectic manifold. Riemannian geometry is garbage below $C^2$, and I have trouble believing symplectic geometry is any better. So why all the fuss?

Thanks in advance!

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What would you recommend (what did you read?) as introductory reading on symplectomorphisms? What are the basic examples and how flexible they are? Is it easy to perturb them? –  Zarathustra Mar 28 '10 at 19:47
    
I suppose my first introduction to symplectomorphisms came from various lectures (mainly by Sergei Tabachnikov). For some dynamical applications, I have also read parts of Arnold's "Mathematical Methods in Classical Mechanics" which uses symplectic geometry as a setting for the Hamiltonian flow (which also answers your second question about what the basic examples are). I'm honestly not sure how flexible they are; I am far from an expert on this stuff. –  Paul Siegel Mar 29 '10 at 3:22

8 Answers 8

I will probably say a banal thing but the interesting degree of regularity depends on the problem you consider. In the theory of foliations, the foliations themselves are usually considered smooth although the transversal behaviour an be rather bad. In some theories the degree of regularity is already fixed but in most of the theories it really depends on the question.

As a remark, I propose a problem that I find nice and for which I do not know the asnwer. Consider the map of the circle for which there exists an orbit which consists of all rational numbers. What degree of smoothness can this map have?

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I have also asked myself somewhat of the same question as I sat for a couple of days through a smooth dynamics conference. I am by no means an expert (in fact more like clueless!), but I would suggest reading some of the intros from Giovanni Forni's papers. For example, the cohomological equation for nilflows (with Livio Flaminio) and the sobolev reguarlity ones. (both are on the arxiv) They at least mention some of the general philosophies and hopes of answering such regularity questions.

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There is a considerable amount of research in dynamics which aims at the understanding of the generic behaviour of the generic diffeomorphism. I do believe that people care about any class of differentiability, but most of the techniques developed so far do not reach much beyond the $C^1$-topology. Tipically one has to deform a given diffeo to arrive at generic conditions and it is much much harder to make small perturbations in the $C^{\infty}$-topology than in the $C^1$-topology. An archetypical example is the $C^1$-closing lemma which is not known to hold in finer topologies.

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Re: perturbations in $C^\infty$: Anosov systems are structurally stable. en.wikipedia.org/wiki/Structural_stability –  Steve Huntsman Mar 28 '10 at 17:27

There are some fascinating phenomena in dynamical systems and related fields whose existence depends on the degree of differentiability.

Among all of these, my favorite is the fact, proved by Haefliger, that although there exist C codimension-1 foliations of S3, there does not exist any Cω codimension-1 foliation of S3.

A much more basic such result, due to Denjoy, is that a C2 diffeomorphism of the circle having irrational rotation number is topologically conjugate to the rotation of the circle having that same rotation number. But there are counterexamples for diffeomorphisms that are merely C1.

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First, the very fact that things change with regularity is interesting, and therefore worth investigating. In the case of riemannian geometry this is quite well understood, since $C^2$ is what is needed to introduce curvature. Yet, one can isometrically embed a flat torus in $\mathbb{R}^3$ by a $C^1$ map, and this seems worth understanding (and definitely not "garbage").

Second, there are some cases where the "generic" regularity is weak: if you consider the stable and unstable foliation of the geodesic flow on a negatively curved compact surface, then it is in general only $C^{2-\varepsilon}$. A very nice theorem (by E. Ghys, if I remember well) says that if these foliations are $C^2$, then the surface has constant curvature.

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If I understand you correctly, the stable and unstable foliation is generally not $C^2$ in the transverse direction, but unless there are techniques that nobody has mentioned to me the geodesic flow itself has to be $C^2$. Your example (which does sound very nice) definitely motivates the study of foliations with various degrees of smoothness, but not necessarily the maps that generate them. Ultimately I think I'll have to just accept that $C^1$ maps are mainly interesting because they are different, but it would be nice to have some natural examples to entertain tourists like myself. –  Paul Siegel Mar 29 '10 at 3:33
    
Beware that the regularity of a foliation boils down to the regularity of maps (monodromy). I am not a specialist in this kind of dynamics, if you want details on the example above you can take a look at the 1990 adress of Etienne Ghys to the ICM (available here umpa.ens-lyon.fr/~ghys/articles/cercle-infini.pdf). –  Benoît Kloeckner Mar 30 '10 at 18:42
    
I had double thoughts and checked: in the case discussed the regularity of the stable and unstable foliations (weak foliations, to be precise) is directly equivalent to the regularity of the natural maps $\pi_{pq}$ from the unit tangent circle of a point $p$ to the unit tangent circle of another point $q$. So it really happens that one meets a $C^{2-\varepsilon}$ diffeomorphism that is not $C^2$. –  Benoît Kloeckner Mar 31 '10 at 15:52

I can't answer your question directly, but I want to disagree very strongly with your statement "Riemannian geometry is garbage below $C^2$". This has not been true at least since Cheeger and Gromov introduced the study of convergence and collapse of Riemannian manifolds. This shows that many aspects of Riemannian geometry are well worth studying, even if the metric is only $C^1$ and the curvature is less than $C^0$. Moreover, even if your ultimate interest is $C^2$ Riemannian geometry, a powerful approach is to study limits of Riemannian metrics (obtained either via Cheeger-Gromov theory or solving a PDE) that are initially less than $C^2$ but eventually shown to be $C^2$ or better.

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You are of course correct - my blanket statement about Riemannian geometry below $C^2$ is no good. However as I understand it the key tool for studying the dynamics of the geodesic flow is the Jacobi equation, so in the context of my question I still suspect we want at least two derivatives. –  Paul Siegel Mar 29 '10 at 3:15
    
If you have $C^2$ control of the metric, then Jacobi fields are definitely the main tool for studying the geodesic flow. However, even if you are studying only smooth Riemannian metrics with some type of curvature bound, you sometimes want to study what can happen under a limit (weaker than convergence in $C^2$) in order to establish a conclusion about the smooth metrics themselves. Understanding what happens to the geodesic dynamics under such a limit could conceivably be very useful. –  Deane Yang Mar 29 '10 at 20:57

My impression always was that this is because of ingrained mathematical culture of seeking the most precise requirements for a particular theorem to hold. That way, when a non-smooth situation eventually does emerge, the theorems are already in place to deal with it. It's one of the differences in culture between pure mathematics, applied mathematics and physics.

I always wondered about looking at the problem the other way around: if you assume that stronger and stronger continuity holds, then what extra properties hold? What about $C^{\infty}$ cases and analytic, is there a gap in between?

The same phenomenon appears in a lot of the optimization literature, where weaker and weaker continuity requirements are made on the functions being optimized. I find this weird -- why not instead create faster and faster optimizers that leverage the smoothness properties present in most applications?

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I have a somewhat tangential answer that I nevertheless hope you'll find helpful. One reason from physics would be that systems with collisions (e.g., a hard core classical gas) are trivially not smooth. The chaotic hypothesis of Gallavotti and Cohen is in effect that this makes little practical difference (and continuing the example, that a hard core gas should be "effectively" Anosov). Obviously this requires some mathematical attention to sort out the mess that the physicists have made.

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Systems with collisions are not even continuous. However if a system (e.g. a hard ball system) is continuous on some domain then it is smooth on the domain. –  Zarathustra Mar 28 '10 at 19:50

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