On the relation between action and linking

2021

We consider the classical problem of area-preserving maps on annulus A = S × [0, 1] . Let Mf be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on A. Given any μ1 and μ2 in Mf , Franks [2][3], generalizing Poincaré’s last geometric theorem (Birkhoff [1]), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ1 and μ2 have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.

Communications in Mathematical Physics, 1995

Consider a smooth Hamiltonian system in IR 2yv , x = JH'(x\ the energy surface Σ = {x/H(x) = //(O)} being compact, and 0 being a hyperbolic equilibrium. We assume, moreover, that £\{0} is of restricted contact type. These conditions are symplectically invariant. By a variational method, we prove the existence of an orbit homoclinic, i.e. non-constant and doubly asymptotic, to 0.

Let f be a C'lf orientation preserving diffeomorphism of the two-disk with positive topological entropy. We define for f an interval of topological invariants. Each point in this interval describes the way the elements of an infinite sequence of periodic orbits with arbitrarily large periods, are asymptotically linked one around the other, AMS classification scheme numbers: 3405.54EO. 58F15

2010

We define and study a M\"obius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a M\"obius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

Kodai Mathematical Journal, 1991

in a paper of 1985 [5] studied a kind of Dirichlet energy in terms of the torsion τ(τ=Xξg) of a 3-dimensional compact contact manifold and a problem analogous to the Yamabe problem. They raised the question of determining all 3-dimensional contact manifolds with τ-Q (i.e. K-contact). In a long paper of 1989 [8] S. Tanno studied the Dirichlet energy and gauge transformations of contact manifolds. D. E. Blair [2] obtained the critical point condition of I(g)=\ Ric(ξ)dV g over JM(η) (the space of all the as-J M

Journal of Physics A: Mathematical and Theoretical, 2020

We prove that, under some natural conditions, Hamiltonian systems on a contact manifold C can be split into a Reeb dynamics on an open subset of C and a Liouville dynamics on a submanifold of C of codimension 1. For the Reeb dynamics we find an invariant measure. Moreover, we show that, under certain completeness conditions, the existence of an invariant measure for the Liouville dynamics can be characterized using the notion of a symplectic sandwich with contact bread.

Annals of Global Analysis and Geometry, 1993

be an R-contact manifold, then the set of periodic points of the characteristic vector field is a nonempty union of closed, totally geodesic odd-dimensional submanifolds. Moreover, the R-metric cannot have nonpositive sectional curvature. We also prove that no R-contact form can exist on any torus.

Acta Mathematica Vietnamica, 2013

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions, and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.

Communications on Pure and Applied Mathematics, 1985

Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C2(ℝ2N, ℝ). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ϵ ℝ2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ℛ and αℰ ⊂ ℛ ⊂ βℰ for some ellipsoid ℰ ⊂ ℝ2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ℰ and one other geometrical parameter of Σ) such that if furthermore β2/α2 < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S1 -action, from which we derive abstract critical point theorems for S1 -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.

Bulletin of the American Mathematical Society, 1989