arXiv subjects

Josep Fontana-McNally

Publications and source records attributed to Josep Fontana-McNally.

3 recordsLinked to original sources

arXiv2023

A counterexample to the singular Weinstein conjecture

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [23]. Our construction shows that there are b-contact manifolds with no singular periodic orbit and no regular periodic orbit away from Z. We do not know whether there are constructions with no generalized escape orbits whose $α$ and $ω$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.

math.SG↗

arXiv2023

An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow

We prove that the correspondence between Reeb and Beltrami vector fields can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the $n$-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler-Euler flow.

math.SG↗

arXiv2023

From 2N to infinitely many escape orbits

In this short note, we prove that singular Reeb vector fields associated with generic $b$-contact forms have either (at least) $2N$ or an infinite number of escape orbits, where $N$ denotes the number of connected components of the critical set.

math.DS↗