On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds

Abstract

We study Farber's topological complexity for monotone symplectic manifolds. More precisely, we estimate the topological complexity of 4dimensional spherically monotone manifolds whose Kodaira dimension is not −∞.

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

“…So, it is very difficult to know information of non-contractible periodic orbits from Floer homology theory. However, Gürel, Ginzburg-Gürel and Orita proved that non-contractible Floer homology theory is partially applicable and they proved Conjecture 1 (2) for some cases [11,17,23,24]. In this paper, we apply equivariant Floer theory [26,30] and prove Conjecture 1 (2) for very wide classes of symplectic manifolds.…”

On the Hofer-Zehnder conjecture

“…It would be very interesting to understand how to treat this case: while for S 2 there currently exist symplectic proofs of the complete smooth Franks theorem [6,11], in higher dimensions this currently seems to be out of reach. Finally, interpreting the Hofer-Zehnder conjecture more generally, Ginzburg, Gürel [28,30] and Orita [48,47], have shown that in certain settings the existence of a non-contractible periodic orbit implies the existence of an infinite number of periodic points. We expect our results and methods to generalize to settings of this kind, and to yield new cases of the Conley conjecture.…”

On the Hofer–Zehnder conjecture on weighted projective spaces

“…On the tori T 2d , Conley conjectured that every Hamiltonian diffeomorphism has infinitely many periodic points. This statement was proven by Hingston [Hin09] after decades of advances [CZ86, SZ92,FH03,LeC06] and then generalized to a large class of symplectic manifolds by Ginzburg [Gin10], Ginzburg and Gürel [GG12, GG15, GG19] and Orita [Ori19]. However, the Conley conjecture does not hold in CP(q): the Hamiltonian diffeomorphism…”