arXiv subjects

Marco Mazzucchelli

Publications and source records attributed to Marco Mazzucchelli.

43 records · Page 3Linked to original sources

arXiv2016

The Morse index of Chaperon's generating families

This is an expository paper devoted to the Morse index of Chaperon's generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott's iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linearized Hamiltonian flow in the symplectic group. We will conclude the survey with a proof that the classical Morse index from Tonelli Lagrangian dynamics coincides with the Maslov index.

math.SG↗

arXiv2014

Isometry-invariant geodesics and the fundamental group

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the celebrated Weinstein conjecture: on a closed contact manifold with a selected contact form, any strict contactomorphism that is contact-isotopic to the identity possesses an invariant Reeb orbit.

math.DG↗

arXiv2013

On the multiplicity of isometry-invariant geodesics on product manifolds

We prove that on any closed Riemannian manifold $(M_1\times M_2,g)$, with $\rank\Hom_1(M_1)\neq0$ and $\dim(M_2)\geq2$, every isometry homotopic to the identity admits infinitely many isometry-invariant geodesics.

math.DG↗

arXiv2012

Symplectically degenerate maxima via generating functions

We provide a simple proof of a theorem due to Nancy Hingston, asserting that symplectically degenerate maxima of any Hamiltonian diffeomorphism of the standard symplectic 2d-torus are non-isolated contractible periodic points or their action is a non-isolated point of the average-action spectrum. Our argument is based on generating functions.

math.SG↗

arXiv2011

On the multiplicity of non-iterated periodic billiard trajectories

We introduce the iteration theory for periodic billiard trajectories in a compact and convex domain of the Euclidean space, and we apply it to establish a multiplicity result for non-iterated trajectories.

math.DS↗

arXiv2010

The Lagrangian Conley Conjecture

We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler-Lagrange system has infinitely many periodic solutions. More precisely, we show that there exist infinitely many contractible integer periodic solutions with a priori bounded mean action and either infinitely many of them are 1-periodic or they have unbounded period.

math.DS↗

arXiv2010

Periodic bounce orbits of prescribed energy

We prove the existence of periodic bounce orbits of prescribed energy on an open bounded domain in Euclidean space. We derive explicit bounds on the period and the number of bounce points.

math.DS↗