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Author

Victor Bangert

Other affiliations: University of Bern
Bio: Victor Bangert is an academic researcher from University of Freiburg. The author has contributed to research in topics: Riemannian manifold & Geodesic. The author has an hindex of 19, co-authored 50 publications receiving 1532 citations. Previous affiliations of Victor Bangert include University of Bern.


Papers
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Book ChapterDOI
01 Jan 1988
TL;DR: The theory of differential geometry, dynamical systems and solid state physics has attracted growing interest and research activity in the last few years as mentioned in this paper, which is based on independent research in three different fields.
Abstract: The title refers to a theory which is based on independent research in three different fields—differential geometry, dynamical systems and solid state physics—and which has attracted growing interest and research activity in the last few years. The objects of this theory are respectively: (1) Geodesics on a 2-dimensional torus with Riemannian (or symmetric Finsler) metric. (2) The dynamics of monotone twist maps of an annulus. (3) The discrete Frenkel-Kontorova model.

317 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of infinitely many closed geodesics for Riemannian metric on S2 with positive Gaussian curvature has been proved and it is shown that all metrics with positive curvature have this property.
Abstract: In [7] J. Franks proves the existence of infinitely many closed geodesics for every Riemannian metric on S2 which satisfies the following condition: there exists a simple closed geodesic for which Birkhoff's annulus map is defined. In particular, all metrics with positive Gaussian curvature have this property. Here we prove the existence of infinitely many closed geodesics for every Riemannian metric on S2 which has a simple closed geodesic for which Birkhoff's annulus map is not defined. Combining this with J. Franks's result and with the fact that every Riemannian metric on S2 has a simple closed geodesic one obtains the existence of infinitely many closed geodesics for every Riemannian metric on S2.

148 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that for every Finsler metric on S 2 there exist at least two distinct prime closed geodesics, and that these are the same geodesic types as in this paper.
Abstract: In this paper, we prove that for every Finsler metric on S 2 there exist at least two distinct prime closed geodesics.

121 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus and showed that these halforbits are always asymptotic to action minimizing orbits.
Abstract: We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus We show that these so-called rays are always asymptotic to action-minimizing orbits In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray We use this concept to prove that the minimal average action A(α) is differentiable at irrational rotation numbersα while it is generically non-differentiable at rational rotation numbers (cf also [18]) As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus

110 citations

Journal ArticleDOI
TL;DR: The existence of closed geodesics on complete Riemannian manifolds depends to a large extent on the topology of the manifold as discussed by the authors, which is why it is hard to detect if the manifold is simple.
Abstract: The existence of closed geodesics on complete Riemannian manifolds depends to a large extent on the topology of the manifold. Generally, closed geodesics may become rare or at least hard to detect if the topology of the manifold is simple. For complete, non-compact surfaces M the situation is as follows: If M is not homeomorphic to a plane, a cylinder or a M6bius band there always exist infinitely many closed geodesics on M, cf. [-15]. They arise as minima of the energy functional on certain free homotopy classes of loops. Here the possibility that two of these closed geodesics are coverings of each other can be excluded by choosing the free homotopy classes appropriately. On a complete M6bius band these methods yield one closed geodesic, and, indeed, this may be the only one that exists. Obviously there exist complete cylinders and planes without any closed geodesics. In this paper we give conditions on the Riemannian structure which ensure the existence of infinitely many closed geodesics on the three exceptional surfaces as well. For the problem of closed geodesics these three surfaces are the most interesting non-compact ones since their topology is so simple. Combining two more specific theorems we obtain as main result

75 citations


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Book
01 Feb 1993
TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

3,954 citations

Journal ArticleDOI
TL;DR: In this article, a Lagrangian variational formulation of twist maps is proposed to compute the flux escaping from regions bounded by partial barriers formed from minimizing orbits, which form a scaffold in the phase space and constrain the motion of remaining orbits.
Abstract: Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

627 citations

Journal ArticleDOI
TL;DR: In this paper, a generalization of the notion of minimal measure to periodic positive definite Lagrangian systems in more degrees of freedom has been proposed, where the minimal measures can be expressed as Lipschitz sections of the tangent bundle.
Abstract: In recent years, several authors have studied "minimal" orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist diffeomorphisms. Here, "minimal" means action minimizing. This class of orbits has many interesting properties, as may be seen in the survey article of Bangert [4]. It is natural to ask if there is any generalization of this class of orbits to Hamiltonian systems in more degrees of freedom. In this article, we propose a generalization to periodic Hamiltonian systems in more degrees of freedom. However, we generalize not the notion of minimal orbit, but the closely related notion of minimal measure, which we introduced in [18]. We obtain two basic results here: an existence theorem for minimal measures, and a regularity theorem which asserts that the minimal measures can be expressed as (partially defined) Lipschitz sections of the tangent bundle. In the sort of generalization that we do here, a major difficulty is finding the right setting. The setting which we propose here has two important features: the results are valid for periodic positive definite Lagrangian systems, and the results are formulated in terms of invariant measures. I am indebted to J. Moser for pointing out to me several years ago that periodic positive definite Lagrangian systems in one degree of freedom provide a setting in which it is possible to formulate results which generalize both the author's results [17] (and the closely related results of Aubry and Le Dacron [1]) and the results of Hedlund [12] concerning "class A " geodesics on a Riemannian manifold diffeomorphic to the 2-torus. Indeed, Moser has proved [20] that every twist diffeomorphism is the time one map associated to a suitable periodic positive definite Lagrangian system. Denzler [10] has carried out Moser's program in one degree of freedom. This remark of Moser suggested to me that periodic positive definite Lagrangian systems should provide the right setting in more degrees of freedom. There is some earlier work in the direction of this paper. Bernstein and Katok [6] obtained results concerning periodic orbits near invariant tori, using a variational method related to the variational method of this paper.

626 citations

Journal ArticleDOI
TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.org/legal.php) of a fichier do not necessarily imply a mention of copyright.
Abstract: © Annales de l’institut Fourier, 1993, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

356 citations

Book ChapterDOI
01 Jan 1993
TL;DR: This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975, and focus on the uses of homotopy theory in physics, and especially on the calculation of homOTopy groups.
Abstract: This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

335 citations