M. M. Peixoto

Structural stability on two-dimensional manifolds☆

THE ABOVE paper appeared in Volume 1, pp. 101-120 of Topology, and the aim of it was to prove that the set of all structurally stable differential equations is open and dense in the space, with the Cl-topology, of all differential equations defined on a compact 12f2. In this note we clarify a point concerning the proof of that theorem that was brought to our attention by E. Lima and S. Schwartzman.

On an approximation theorem of Kupka and Smale.

Proof of Kupka and Smale approximation theorem concerning differential equations defined on compact manifold

On the Classification of Flows on 2-Manifolds†

Publisher Summary This chapter presents the classification of flows on 2-manifolds. The set Σ of all structurally stable flows is open and dense in an approximation problem. The chapter presents a classification of all equivalent classes of Σ modulo Σ, thus, completing the solution of the fundamental problem. This is done by establishing a one-to-one correspondence between these equivalence classes and certain distinguished graphs, that is, graphs together with a distinguished set of edges satisfying some conditions. This results in a precise rule for labeling all equivalence classes of Σ modulo ˜ in such a way that each equivalence class appears exactly once in the labeling process. The chapter also discusses the concept of structurally stable flows, the graph of a gradient-like flow, the nonorientable case, without closed orbit, the orientable case with closed orbits, and the nonorientable case with closed orbits.

On Brillouin Zones

Abstract: Brillouin zones were introduced by Brillouin [Br] in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in ℝn. They play an important role in solid-state physics. It was shown by Bieberbach [Bi] that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin zones to apply to an arbitrary discrete set in a proper metric space, and show that analogs of Bieberbachs results hold in this context.We then use these ideas to discuss focusing of geodesics in spaces of constant curvature. In the particular case of the Riemann surfaces ?2/Γ (k) (k= 2,3, or 5), we explicitly count the number of geodesics of length t that connect the point i to itself.

On Focal stability in dimension two

In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.

Focal stability of Riemann metrics

Abstract Let M be a complete Riemann manifold with dimension m and metric g. For p, q ∈ M and ℓ > 0, let the index I (g, p, q, ℓ) be the number of g-geodesics of length ℓ that join p to q. The following generic bounds for this index are the main results we present here. We denote by ℛ the space of complete Riemann metrics on M. (a) For each p ∈ M, there is a residual 𝒢 (p) ⊂ ℛ such that for all g ∈ 𝒢(p) (b) If M is compact, there is a residual 𝒢 ⊂ ℛ such that for all g ∈ 𝒢 These finiteness results are part of our study of the focal decomposition—i.e., the partition Stability of this focal deomposition (as g varies) has a natural meaning, in analogy with structural stability in the theory of dynamical systems, and here we begin an investigation in that direction. Our methods involve the multi-transversality theory of J. Mather and the Bumpy Metric Theorem of R. Abraham, as proved by D. Anosov.

Focal rigidity of flat tori

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = S i 6 i called the focal decomposition of TM. The sets 6 i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n ≥ 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

Focal decomposition, renormalization and semiclassical physics

We review some recent results concerning a connection between focal decomposition, renormalization and semiclassical physics. The dynamical behaviour of a family of mechanical systems which includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time can be characterized through a renormalization scheme acting on the dynamics of this family. We have proved that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we have obtained an asymptotic universal focal decomposition for this family of mechanical systems which can now be used to compute estimates for propagators in semiclassical physics.

An asymptotic universal focal decomposition for non-isochronous potentials

Galileo, in the XVII century, observed that the small os- cillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi- librium. It is well known that, for small oscillations of the pendulum and small intervals of time, the dynamics of the pendulum can be ap- proximated by the dynamics of the harmonic oscillator. We study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected. We analyze such dynamical behaviour through a renor- malization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we obtain an universal asymptotic focal decomposition for this family of mechanical systems. This paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu- lum type as a natural application of the focal decomposition associated to the two-point boundary value problem.

Renormalization and Focal Decomposition

We introduce a renormalization scheme to study the asymptotic dynamical behaviour of a family of mechanical systems with non-isochronous potentials with an elliptic equilibrium. This renormalization scheme acts on a family of orbits of these mechanical systems, all of which contained on neighbourhoods of the elliptic equilibrium, by rescaling space and shifting time in an appropriate way. We will review some new results regarding properties of this renormalization scheme, as well as the strong connection it has with the focal decomposition for the Euler–Lagrange equation of this family of mechanical systems.