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.