Pseudo-rotations and holomorphic curves

Pseudo-rotations and Steenrod squares

“…REMARK 5. The author was made aware that new relations between pseudorotations and holomorphic curves were also found in recent work of Çineli, Ginzburg, and Gürel [8]. REMARK 6.…”

“…In the recent seminal paper [15] by Ginzburg and Gürel, the C 0 -rigidity result of Bramham, as well as other results regarding the dynamics of Hamiltonian pseudo-rotations, were established for complex projective spaces of all dimensions. This paper, as well as [8], takes a different point of view, considering pseudorotations to be strong counter-examples to the Conley conjecture. From this perspective, a conjecture of Chance and McDuff, arising from [18], asserts that the existence of such counter-examples, and hence that of pseudo-rotations, must imply the existence of non-trivial algebraic counts of pseudo-holomorphic spheres in the manifold.…”

On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics

“…Conley conjecture implies a symplectic manifold with pseudo-rotation is very rare. In [3,4,15,16,28,29], the importance of the existence of non-trivial pseudo-holomorphic curve was pointed out. They proved that the quantum Steenrod square is deformed if the symplectic manifold is monotone and has a pseudo-rotation.…”

“…(1) (M, ω) is monotone (2) c 1 (A) = 0 for every A ∈ π 2 (M ) (3) The minimum Chern number N > 0 is greater than or equal to n − 2 Note that weakly monotone symplectic manifolds cover wide classes of symplectic manifolds. For example, every symplectic manifold whose dimension is less than or equal to 6 is a weakly monotone symplectic manifold.…”

Pseudo-rotations and Steenrod squares

“…Remark 2. The author was made aware that new relations between pseudorotations and holomorphic curves were also found in forthcoming work of Cineli, Ginzburg, and Gürel [4].…”