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Alberto Abbondandolo

Publications and source records attributed to Alberto Abbondandolo.

At least 37 records · Page 2Linked to original sources
arXiv2016

The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range $(e_0,e_1)$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, $e_0=0$ is the minimal energy of the system).

math.DS↗
arXiv2015

Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces

The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $Λ$ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on $Λ$ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold $Λ$, under the stronger assumption that the dynamics on $Λ$ is strongly chain transitive.

math.DS↗
arXiv2015

A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.

math.DG↗
arXiv2014

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

math.SG↗
arXiv2014

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

math.CV↗
arXiv2013

How large is the shadow of a symplectic ball?

Consider the image of a 2n-dimensional unit ball by an open symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection into a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

math.SG↗
arXiv2013

Corrigendum: "On the Floer homology of cotangent bundles"

We fix an orientation issue which appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel-Whitney class of the underlying manifold does not vanish on 2-tori, this isomorphism requires the use of a twisted version of the Floer complex.

math.SG↗
arXiv2013

Lectures on the free period Lagrangian action functional

In this expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. We use a variational approach, by studying how the behavior of the free period Lagrangian action functional changes when the energy crosses certain values, known as the Mañé critical values.

math.DS↗
arXiv2013

The role of the Legendre transform in the study of the Floer complex of cotangent bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M.

math.SG↗
arXiv2013

On the existence of three closed magnetic geodesics for subcritical energies

We consider exact magnetic flows on closed orientable surfaces. We show that for almost every energy $κ$ below Mañé's critical value of the universal covering there are always at least three distinct closed magnetic geodesics with energy $κ$. If in addition the energy level is assumed to be non-degenerate we prove existence of infinitely many closed magnetic geodesics.

math.DS↗
arXiv2011

Stable manifolds for holomorphic automorphisms

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are well-defined is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of hte conjugacy operator by a lower triangular matrix on the space of \textit{k}-homogeneous polynomial endomorphisms of \mathbb{C}^{d}.

math.DS↗
arXiv2011

Global stable manifolds in holomorphic dynamics under bunching conditions

We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to a complex vector space, provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.

math.DS↗
arXiv2009

Floer homology of cotangent bundles and the loop product

We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.

math.SG↗
arXiv2009

Estimates and computations in Rabinowitz-Floer homology

The Rabinowitz-Floer homology of a Liouville domain W is the Floer homology of the free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. It has been introduced by K. Cieliebak and U. Frauenfelder. Together with A. Oancea, the same authors have recently computed the Rabinowitz-Floer homology of the cotangent disk bundle D^*M of a closed manifold M, by establishing a long exact sequence. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz-Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T^*M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mane' critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz-Floer equation to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by suitable versions of the Aleksandrov maximum principle.

math.SG↗
arXiv2009

A smooth pseudo-gradient for the Lagrangian action functional

We study the action functional associated to a smooth Lagrangian function on the cotangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H^1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional.

math.DS↗