On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C} \mathrm{P}^d$ via generating functions

Abstract

We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus g; it implies that a symplectic diffeomorphism isotopic to the identity with more than 2g − 2 fixed points, counted homologically, has infinitely many periodic points. Furthermore, we present examples of symplectic diffeomorphisms with a prescribed number of periodic points. In particular, we construct symplectic flows on surfaces possessing only one fixed point and no other periodic orbits.

On the Hofer–Zehnder conjecture on weighted projective spaces

“…In this section, we recall definitions and properties already discussed in [All22b, Section 5] and [All22a, Section 3.2] in the case of unweighted projective space and that generalize directly to our ‘weighted’ case. Let us fix once for all the weights , the -action of will always refer to the action induced by defined in (1).…”

“…This homology group can be naturally identified to the homology of sublevel sets of a map (see [All22b, Section 5.4]) for some -map defined on a manifold and that is smooth in the neighborhood of its critical points. The function is some kind of finite-dimensional action: critical points of are in one-to-one correspondence with capped fixed points of with action value inside .…”

“…Let us denote by the local homology of the critical point of a map : We can define up to isomorphism where is the critical point of associated with the fixed point of action . Let us briefly justify the independence on , we refer to [All22b, Section 5.5] for the technical details. In § 3.3, we define an isomorphism between and .…”

“…As is a generating function of the identity, its kernel as a quadratic form has dimension and (see [All22a, Proposition 4.1]). The variation of index is governed by the Maslov index of so that (see [All22b, Section 3 and Lemma 5.5]). Similarly to the unweighted case, we deduce that the persistence module is isomorphic to the persistence module , , induced by the family of non-decreasing projective subspaces of complex dimension .…”