We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology, that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+\infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary.

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

The dynamics of moving solids with unilateral contacts are often modelled by assuming rigidity, point contacts, and Coulomb friction. The canonical example of a rigid rod with one endpoint slipping in two dimensions along a fixed surface (sometimes referred to as Painlevé rod) has been investigated thoroughly by many authors. The generic transitions of that system include three classical transitions (slip-stick, slip reversal, and liftoff) as well as a singularity called dynamic jamming, i.e., convergence to a codimension 2 manifold in state space, where rigid body theory breaks down. The goal of this paper is to identify similar singularities arising in systems with multiple point contacts, and in a broader setting to make initial steps towards a comprehensive list of generic transitions from slip motion to other types of dynamics. We show that – in addition to the classical transitions – dynamic jamming remains a generic phenomenon. We also find new forms of singularity and solution indeterminacy, as well as generic routes from sliding to self-excited microscopic or macroscopic oscillations.

In this article we prove that the Weinstein conjecture holds for contact manifolds $({\rm\Sigma},{\it\xi})$ for which $\text{Cont}_{0}({\rm\Sigma},{\it\xi})$ is non-orderable in the sense of Eliashberg and Polterovich [Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal. 10 (2000), 1448–1476]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata 165 (2013), 95–110] on the existence of translated points in the non-degenerate case.