Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

Abstract

We prove that every C ∞ -smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic C ∞ -smooth diffeomorphisms. In particular every smooth irrational pseudo-rotation can be C 0 -approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.Date: April 23, 2012. 1 1.2. Idea of the proof. Pick a closed loop of Hamiltonians H t : D → D, over t ∈ R/Z, which generate a symplectic isotopy whose time-one map is ϕ. Denote the 1-periodic path of Hamiltonian vector fields on the disk by X Ht . For each n ∈ N equip Z n = R/nZ × D with coordinates (τ, z). Then the vector field R n (τ, z) = ∂ τ + X Ht (z) defines a flow on Z n with time-one map ϕ and first return map ϕ n .Consider the 4-manifold W n := R × Z n with the unique almost complex structure satisfyingwhere ∂ R is the vector field dual to the R-coordinate on W n . Then (W n , J n ) is a so called cylindrical, symmetric, almost complex manifold. That is, it is compatible in a precise way with the necessary symplectic structures for the compactness framework from symplectic field theory [4] to apply to J n -holomorphic curves.

Connected sums and finite energy foliations I: Contact connected sums

“…This in turn implies that the map F is injective since double points of F can be seen as either intersections between two distinct C τ 's or a self-intersection of some given C τ . We next claim 10 As with Theorem 3.20, we again caution the reader here that the continuous extension of this diffeomorphism over the punctures is not, in general, smooth.…”

“…Recently, Bramham has introduced the use of finite energy foliations to the study of area-preserving maps of the disk [9,8]. Using the foliations that he constructs in [8], Bramham proves in [10] that every smooth, irrational pseudorotation of the 2-disk is the uniform limit of a sequence of maps which are each conjugate to a rotation about the origin. In [11], these foliations are again used to prove there is a dense subset L * ⊂ L of the Liouville numbers so that a pseudorotation of the disk with rotation number in L * has a sequence of iterates which converge uniformly to the identity map and thus such a pseudorotation can't exhibit strong mixing.…”

“…Lemma 6. 10. The pseudoholomorphic building C ∞,+ is a height-2 building with Z p on the top level, and P + and trivial cylinders over the negative orbits of C p,± on the bottom level, i.e.…”

A finite dimensional approach to Bramham’s approximation theorem

“…Among the applications, one can note the following approximation result (see [Lec3]): every minimal C 1 diffeomorphism F of T 2 that is isotopic to the identity is a limit in the C 0 topology of a sequence of periodic diffeomorphisms. The proofs given in [Br1] and [Lec3] share a thing in common: the construction of a foliation satisfying a certain "dynamical transverse property" on which a finite group acts, the approximating map being naturally related to this action. In [Br1] the foliation is defined on R × D × T 1 and the leaves are either pseudoholomorphic cylinders or pseudoholomorphic half cylinders transverse to the boundary; in [Lec3], the foliation is singular and naturally conjugate to the foliation by orbits of ξ on an invariant torus.…”

“…The proofs given in [Br1] and [Lec3] share a thing in common: the construction of a foliation satisfying a certain "dynamical transverse property" on which a finite group acts, the approximating map being naturally related to this action. In [Br1] the foliation is defined on R × D × T 1 and the leaves are either pseudoholomorphic cylinders or pseudoholomorphic half cylinders transverse to the boundary; in [Lec3], the foliation is singular and naturally conjugate to the foliation by orbits of ξ on an invariant torus. Therefore it is natural to look for a proof of Bramham's theorem by a method close to the one given in [Lec3].…”