### Bifurcation of periodic orbits of time dependent Hamiltonian systems on symplectic manifolds.

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We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.

It is well-known that the Fundamental Identity (FI) implies that Nambu brackets are decomposable, i.e. given by a determinantal formula. We find a weaker alternative to the FI that allows for non-decomposable Nambu brackets, but still yields a Darboux-like Theorem via a Nambu-type generalization of Weinstein’s splitting principle for Poisson manifolds.

This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of $(0,-2)$-curves on threefolds, or deforming $\mathbb{P}$-objects introduced by D.Huybrechts and R.Thomas.

In this paper we firstly define a tangential Lichnerowicz cohomology on foliated manifolds. Next, we define tangential locally conformal symplectic forms on a foliated manifold and we formulate and prove some results concerning their stability.