Non-contractible Periodic Orbits Research Articles

On non-contractible periodic orbits and bounded deviations

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensonal Gorenstein contact lens spaces. As illustrative contact topology applications, we show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism.

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the Biran–Polterovich–Salamon capacities and earlier constructions of Weber (2006 Duke Math. J. 133 527–568) and other authors Biran et al (2003 Duke Math. J. 119 65–118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15 905–936) to the Lie group setting, (2) preservation of minimal Finsler length of closed geodesics in any given free homotopy class by symplectomorphisms, (3) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (4) existence of periodic orbits for Hamiltonians on noncompact domains, (5) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (6) partial solution to a conjecture of Kawasaki (2016 Heavy subsets and non-contractible trajectories (arXiv:1606.01964)), (7) results on squeezing/nonsqueezing theorem on torus cotangent bundles, etc.

Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic orbits, provided that the symplectic form $\omega$ is aspherical and the fundamental group $\pi_1(M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian diffeomorphisms of closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and Gurel. The proof uses the filtered Floer--Novikov homology for non-contractible periodic orbits.

Multiple forced rotational solutions for the planar N-pendulum equation

This paper presents a theorem of existence of \((N - N_{0} + 1)2^{N_{0}}\) noncontractible periodic orbits of any given rotational vector with zero components for the planar forced N-pendulum equation, where N 0 denotes the number of zero components of given rotational vector. We make special arrangements for masses and length instead of the nondegenerate assumption. Moreover, for rotational vectors with identical components, similar results is also obtained.

Non-contractible periodic orbits in Hamiltonian dynamics on tori

We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.

Existence of noncontractible periodic orbits of Hamiltonian systems separating two Lagrangian tori on $T^{*} \mathbb{T}^n$ with application to nonconvex systems

Existence of noncontractible periodic orbits of Hamiltonian systems separating two Lagrangian tori on $T^{*} \mathbb{T}^n$ with application to nonconvex systems

Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

On non-contractible periodic orbits for surface homeomorphisms

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if ${\hat{g}}$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) the set of fixed points for ${\hat{g}}$ projects to a closed subset $F$ which contains an essential continuum; (2) $g$ has non-contractible periodic points of every sufficiently large period; or (3) there exists a uniform bound $M>0$ such that, if $\hat{x}$ projects to a contractible periodic point, then the ${\hat{g}}$ orbit of $\hat{x}$ has diameter less than or equal to $M$. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.

On non-contractible periodic orbits of Hamiltonian diffeomorphisms

We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non-contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a non-degenerate (or even isolated and homologically non-trivial) periodic orbit with non-zero homology class and the set of one-periodic orbits in that class is finite.

An almost existence theorem for non-contractible periodic orbits in cotangent bundles

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H<d} contains the zero section and \alpha is a non-trivial homotopy class of free loops in M. Then for almost every s>=d the level set {H=s} carries a periodic orbit z of the Hamiltonian system (T^*M,\omega_0,H) representing \alpha. Examples show that the condition that {H<d} contains M is necessary and almost existence cannot be improved to everywhere existence.

Propagation in Hamiltonian dynamics and relative symplectic homology

The main result asserts the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

Three approaches towards Floer homology of cotangent bundles

Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and Abbondandolo-Schwarz (2004). The theory is illustrated by calculating Morse and Floer homology in case of the euclidean n-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.