Barney Bramham (IAS) on a conjecture of Katok

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Question (A. Katok): In low dimensions, is every conservative dynamical system with $h_\top = 0$ a limit of integrable systems?

Low dimensions = 2 dims for maps and 3 dims for flows;
Conservative = area/volume preserving;
Limit = to be defined;
integrable system = a disk map is integrable if exists a function $f \colon D \to \R$ not constant on any open set, $\phi^*f = f$.

$h_\top \in [0, \infty]$ describes the complexity of the system.

Definition: ($h_\top = 0$) A pseudorotation is an area preserving disk map such that $\phi(0) = 0$ and without any other periodic points.

Franks: a.e. point $z \in D \setminus \{ 0 \}$ has a well defined rotation number about $0$. The lack of other periodic points is eqvt (?) to saying these are the same and equal to an irrational number $\alpha$.

Theorem (Anosov-Katok '70s) There exist ergodic pseudorotations.
In particular a.e. $z \in D$ has a dense orbit.

Theorem Let $\phi$ be a pseudorotation with irrational rotation number $\alpha$. There exists a sequence of smooth $\phi_n$ (fixing $0$) which converge to $\phi$ in $C^0$, and such that for all $n$ exists diffeo $g_n$ (fixing $0$) and $p_n/q_n \to \alpha$ s.t. $\phi_n = g_n^{-1} \circ R_{2\pi p_n/q_n} \circ g_n$.

Two questions: can $\phi_n$ be made symplectic? can we obtain higher regularity of convergence?

Remark: if we consider such a sequence of homeo
$\phi_n$ and $p_n/q_n \to \alpha$, and these $C^0$-converge to an area preserving homeo $\phi$, then $\phi$ is a pseudorotation.

Also remark: this is more-or-less how Anosov-Katok construct their examples.

Pseudoholo curves for disk maps

Assume $\phi$ area preserving diffeo. Suspend this to a ``Hamiltonian'' mapping torus. Obtain symplectization $ \R \times S^1 \times D^2$.
The boundary, $ \R \times S^1 \times \bdy D^2$ is filled by two-tori, called $L_a = \{ a \} \times S^1 \times \bdy D^2$. Totally real.
If $\phi$ non-degenerate then exists a FEF. Some remarks about removal of the condition that the map be a rotation at the boundary, also about existence of many foliations, also existence of FEF with certain prescribed orbits as binding.
Claim: projected map has smooth extension to the boundary!?!?

Approximating pseudorotations

Suppose $\phi \in \Diff^\infty( D, \omega_0)$ is an area preserving disk map.

Take a mapping torus. Lift to universal cover.

Identify each trajectory with a holomorphic curve. Call this a vertical plane. Obtain the vertical foliation.

The claim is that given a vertical (?) foliation, we can construct a map.

Strategy: if $\phi$ is a pseudorotation, find a sequence of foliations so that the $d\lambda$ area is going to $0$. Specifically, we have a sequence of foliations so $d\lambda$ is the fractional part of $n \alpha$, where $\alpha$ is the rot number of $\phi$. Take a subsequence $n_j$ so this goes to $0$.

Each foliation induces a map $\phi_{n_j}$. The claim is that this converges to $\phi$ since the foliations converge to a vertical foliation.

Furthermore, by the construction, $\phi_{n_j}^{(n_j)} = \id$.

Note that we used $| Per(\phi)| = 1$ to know the energies, and also to obtain that each $\mathcal F_n$ is invariant under deck transformations.

somehow, this periodicity tells one that it is ocnjugate to a rational rotation.

Open Question: (Fayad-Katok) Does there exist a strong-mixing area preserving disk map?
Known: suppose that $\phi$ is an irrational pseudorotation. There is some contrast between Diophantine vs Liouville.

Herman showed that Diophantine implies $\phi$ is not strong mixing.
Fayad-Sapryking 2005, Anosov-Katok : any more Liouville than Herman has weak mixing examples.

Barney seems to show that $\alpha$ being sufficiently Liouville gives NOT strong mixing by these foliations. Actually $C^0$ rigid.

Some directions of future work:

Herman asks: If $\alpha$ is diophantine, must $\phi$ be conjugate to a rotation?
2010: Fayad-Kritoriam, answer is yes if $\phi$ is globally close to to a rotation. (Close in $C^k$ norm for a value that depends on the
diophantine order of $\alpha$)

Answers to questions I asked him:
- removing rotation condition is actually delicate. It involves some approximations due to Herman for circle diffeos, and is nontrivial.

- Liouville condition comes in only when he proves the lack of strong
mixing. This is somehow related to showing the foliation is translation invt.

David Farris (Indian Institute of Science/UC Berkeley) The embedded contact homology of circle bundles over Riemann surfaces

The abstract:

Embedded contact homology is a topological invariant of three-manifolds defined by choosing a contact structure on the manifold and studying pseudoholomorphic curves in the four-dimensional symplectization. We compute this invariant for circle bundles over Riemann surfaces (prequantization spaces).


David gave a brief introduction to ECH and then sketched the proof of his thesis result. He computed the ECH invariant for non-trivial circle bundles over surfaces by taking the prequantization contact form. Note that since ECH is a smooth invariant, one is free to take any contact structure on this.

The main result is that, with $Y$ denoting the circle bundle and $\Sigma$ the base,
\[
ECH(Y) = \oplus_{d \ge 1} Sym^d( H_*(\Sigma) )
\]
Furthermore, recall that ECH has a filtration by $H_1(Y)$, representing the homology class of the generators in question. We note that $H_1(Y) = \mathbb{Z}/|e| \oplus \mathbb{Z}^{2g+2}$, where $g$ is the genus of $\Sigma$. The torsion factor is the only relevant one. Then,
\[
ECH(Y, l) = ???see notes??
\]

The key difficulty is that for an $S^1$-invariant complex structure, one can show that the index $1$ moduli spaces are empty... however, these $S^1$-invariant acs are not generic. David solves this problem by first showing the moduli spaces are empty for a domain dependent perturbation. He is able to carry out the Cieliebak-Mohnke/Fabert methods, even though he has genus, essentially because the formation of a ghost bubble with genus is codimension at least 2. (This uses the fact we are in dimension 4 quite crucially). Then, the next tricky point is to show that the count of embedded curves remains $0$ as he deforms the $S^1$-invariant domain-dependent almost complex structure to a generic domain-independent complex structure. This relies on a theorem of Hutchings and Taubes about bifurcations, since David can rule out connectors in the compactness argument.

In thinking about this, I wonder if there is a way of making sense of this as some statement about moduli spaces with an $S^1$ action. This is an idea Jian He explained to me, and is the heart of the geometric picture behind his work on descendants. The general principle (stolen from localization in algebraic geometry) is that if you have a non-generic $J$ and a moduli space of which a connected component has a free $S^1$ action, then after perturbation, this connected component still can be arranged to have an $S^1$ action, in particular then, not contribute to the count of isolated solutions. If one could understand this principle in dimension 4 for embedded curves, it might then be possible to extend David's result to the case of a trivial circle bundle. (Ironically, that case seems harder.)

Welcome

This is a blog I am starting to keep track of the notes I take during talks. Usually, I won't include a long discussion, but will instead mention an interesting idea or two.

The latex rendering is done by mathjax. I followed the instructions I found at irreducible representations.

Masahiro Futaki (Universiteit Antwerpen) On the Sebastiani-Thom theorem for directed Fukaya categories

The abstract:

The directed Fukaya category is an A-model counterpart for the derived category of toric Fano B-model under the Fano/Landau-Ginzburg mirror correspondence. Auroux-Katzarkov-Orlov conjectured that the directed Fukaya category of the direct sum of two potentials splits as tensor product (up to derived equivalence) and pointed out that this is the case for the mirror of $\mathbb{P}^1 \times \mathbb{P}^1$. We show that the conjecture holds if one of the potentials is of complex dimension 1, partially generalizing Seidel's suspension theorem for directed Fukaya categories.


If I understood the talk correctly, part of the challenge is to make sense of the tensor product for $A_\infty$ categories. Alas, most of the aspects of this I understand superficially at best.

This conjecture of AKO is related to some things I do understand, however. Oancea, in his thesis, proved a Künneth formula for the symplectic homology of the product of two Liouville manifolds. By the appendix to Bourgeois-Ekholm-Eliashberg's surgery exact sequence written by Ganatra and Maydanski (and conjectured by Seidel), symplectic homology of the total space of an exact Lefschetz fibration is the Hochschild [Co?]homology of the derived Fukaya category. In this sense, the AKO conjecture is saying that Oancea's theorem actually lives at the level of the Fukaya category.

I wonder if it would be possible to prove this theorem by carefully using the BEE surgery construction.

Clément Hyvrier (Uppsala) Weinstein conjecture in Hamiltonian fibrations

\[ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\e}{\operatorname{e}} \newcommand{\pt}{\mathrm{pt}} \newcommand{\tensor}{\otimes} \] Defined Weinstein conjecture. Interested in contact-type hypersurfaces in $(M,\omega)$.
Examples in which known: Viterbo for $(\R^{2n}, \omega_0)$. Hofer-Viterbo 1992 for symplectically rationally connected manifolds under semi-positivity assumption. (Through any 2 points of $M$, there is a holomorphic sphere.)
The actual definition of symplectically rationally connected: exists a non-vanishing GW invariant $GW_0(\pt, \pt, A, B, \dots)$.
Liu-Tian result with separating hypersurface and classes $A_0$, $A_1$ with support on either side.
Lu: If $<\pt, \beta_1, \dots, \beta_n>^{M, A}_{0, n+1} \ne 0$ then result holds. (symplectically uniruled)
proof sketch: This non-vanishing implies $\star = <\pt, PD(\omega), \beta_1, \dots, \beta_n>^{M, A}_{0, n+2} \ne 0$. Now the trick is to find $\hat \omega = \omega - d(\beta \lambda)$ where $\lambda$ is a local primitive of $\omega$ near $\Sigma$ and $\beta$ is a cut-off function. Then, this has support away from $\Sigma$. Now obtain \[ \star = < \pt, \gamma_+, \beta> + <\pt, \gamma_-, \beta> \] since $\star \ne 0$, one of these terms $\ne 0$. The result now follows by Liu-Tian.
Corollary: shWC true for $M \times N$ with product form and $M$ symp uniruled.
Question: can we generalize to Hamiltonian fibrations? \[ (F, \omega) \hookrightarrow P \to (B, \omega_B) \] s.t.

• $P|_{B_1}$ is sympl trivial, where $B_1$ is the 1-skeleton
• exists a connection $P$ such that the holonomy is hamiltonian i.e.~exists $\tau \in \Omega^2(P)$ such that $\tau|_{F} = \omega$ and $d\tau = 0$ with a normalization condition.

Then, $(P, \epsilon \tau + \pi^* \omega_B)$ is symplectic for $\epsilon > 0$ small enough.
THEOREM: Suppose

1. $(P, \pi)$ is cohomologically split over $\Q$, i.e.~ \\[ H^*(P) \sim H^*(B) \otimes H^*(F) \\] as vector spaces. And $i_* : H^*(P) \to H^*(F)$ is surjective.
2. $(B, \omega_B)$ is symplectically uniruled for some primitive class $\sigma_B \in H_2(B)$ for spherical class $B$.
3. $(F, \omega)$ satisfies semi-positivity relative to $P$.

then, $(P, \omega_P)$ is symplectically uniruled and thus shWC applies there.
THEOREM: We can drop condition (1) by asking that $(B, \omega_B)$ is symp rationally connected.
PROOF:
Case 1: the fibre is symplectically uniruled. Result done by Ruan, Tian and Liu. \\ Basic idea: \[ 0 \ne < \pt, \beta>^{F, \sigma}_{0, n} = <\pt, i_* \beta>_{0,n}^{P, i_* \sigma} \] where $\sigma \in H_2(F)$.
Case 2: Suppose $(F, \omega)$ not symp uni. consider: $C$ the image of a curve counted in $<\pt, \beta_B>_{0,n}^{\beta, \sigma_B}$ Now restrict $P|_C$. Gives a fibration over $S^2$. this is then described by a loop in $Ham(F, \omega)$. Since $(F, \omega)$ NOT symp uni, for any symplectic loop, exists $\sigma' \in H_2^{sph}(P|_C)$ such that $<\pt_{0,1}^{P|_C, \sigma'} \ne 0$, where $\sigma'$ is a section class. (i.e.~projects to the fundamental class of the base) Then, $=<\pt, [F], \dots, [F]>_{0, n+1}^{P|_C, \sigma'}$.
THEOREM: \[ < \pt, \pi^{-1} \beta_B >_{0,n}^{P, i_* \sigma'} = < \pt, \beta_B>_{0, n}^{B, \sigma_B} < \pt, [F], \dots, [F]>_{0,n}^{P|_C, \sigma'} \]
Remarkable fact: the Seidel element, when we are in a NOT uniruled case, looks like $ S(\gamma) = [F] \tensor \lambda + x $.