7 problems

Conjecture A. The set $\Omega_{P} \cup \Omega_{Q}$ of pairs $(a,b)$ for which $T_{ab}$ is either periodic or has a piecewise conic invariant circle is dense in $\Omega_{SB}$.

Greene's rational-iterate conjecture. Every KAM curve $gamma$ can be constructed as the limit of an appropriate family of $m(n)$-periodic discrete invariant sets:

Let $S^2$ be the two-dimensional sphere, equipped with its area form, and let $F:S^2\to S^2$ be an area-preserving diffeomorphism. A conformal point of $F$ is a point at which the…

MacKay's conjecture. In area-preserving maps, invariant circles with constant-type rotation numbers are locally robust, and every invariant circle isolated from at least one side h…

Greene's conjecture. The locally most robust rotational circles have noble rotation numbers, and among these circles the most robust one has golden mean rotation number.

Numerical Hölder-rigidity conjecture. The conjugacy $h$ exists and its derivative $Dh$ is Hölder continuous with positive Hölder exponent satisfying

Let $F_*$ be the universal area-preserving map, let $\mathcal{D}^\infty$ be its domain of analytic continuation, and let $\tilde{\mathcal{D}}^\infty$ denote the corresponding invar…