arXiv research

Proposes a contact dynamics framework using generalized geometries.

Quantizes contact structures using dynamical methods.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…

New neural network designs learn contact dynamics efficiently.

Introduces GFC for learning complex dynamical systems with geometric constraints.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched cover…

Physics-informed GCRL tackles sparse feedback learning with hybrid dynamics.

The study of Reeb dynamics on contact manifolds without periodic orbits.

Study of spectral invariants on CR contact manifolds with circle action.

The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.

Characterizes Anosov flows via contact geometry.

Invites study of contact structures and Reeb flows dynamics.

A new metriplectic system on contact manifolds is introduced for thermodynamic consistency.

Recent research on accelerated gradient methods of use in optimization has demonstrated that these methods can be derived as discretizations of dynamical systems. This, in turn, has provided a basis for more systematic investigations, especially into the geometric structure of those dynamical systems and their structur…

The study connects contact forms and Ruelle invariant in convex domains.

New techniques reveal tight contact manifolds with vanishing contact homology.

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…

Characterizes Anosov flows in 3D using symplectic and contact geometry.

New surgery method preserves Anosov flow properties using bi-contact geometry.

Paper bridges quantum and classical mechanics for open systems.

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $S^n$ carries at l…

Bi-contact surgery operations can be applied to Anosov flows.

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satis…

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among con…

In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstraine…

Our focus is on realistically modeling and forecasting dynamic networks of face-to-face contacts among individuals. Important aspects of such data that lead to problems with current methods include the tendency of the contacts to move between periods of slow and rapid changes, and the dynamic heterogeneity in the actor…

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers…

Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.

Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dy…

Paper introduces Eden bracket for nonholonomic systems.

Study develops curvature for contact-sequence networks, revealing temporal dynamics.

The paper proves rigidity results for Anosov flows and their orbit equivalences.

We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation …

Study properties of contact structures on symplectic disk bundles with concave boundaries.

In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conse…

Lecture notes introduce contact complete integrability for odd-dimensional manifolds.

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a c…

Proves Weinstein's and Arnold's conjectures using contact instantons.

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…

Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.

The paper provides a geometric framework for understanding non-equilibrium thermodynamics.

We give a dynamical characterisation of odd-dimensional balls within the class of all contact manifolds whose boundary is a standard even-dimensional sphere. The characterisation is in terms of the non-existence of short periodic Reeb orbits.

A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solution…

In the spirit of Sullivan's paper "Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds", existence of a contact structure on a closed manifold $M$ is shown to be equivalent to existence of an ample $S^1$-invariant cone structure with no nontrivial exact structure cycles on the manifold $S^1 \time…