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Gabriel P. Paternain

Publications and source records attributed to Gabriel P. Paternain.

At least 55 records · Page 3Linked to original sources
arXiv2013

Symplectic and contact properties of the Mañé critical value of the universal cover

We discuss several symplectic aspects related to the Mañé critical value c_u of the universal cover of a Tonelli Hamiltonian. In particular we show that the critical energy level is never of virtual contact type for manifolds of dimension greater than or equal to three. We also show the symplectic invariance of the finiteness of the Peierls barrier and the Aubry set of the universal cover. We also provide an example where c_u coincides with the infimum of Mather's α-function but the Aubry set of the universal cover is empty and the Peierls barrier is finite. A second example exhibits all the ergodic invariant minimizing measures with zero homotopy, showing that, quite surprisingly, the union of their supports is not a graph, in contrast with Mather's celebrated graph theorem.

math.DS↗
arXiv2013

Floer homology for non-resonant magnetic fields on flat tori

In this article we define and compute the Novikov Floer homology associated to a non-resonant magnetic field and a mechanical Hamiltonian on a flat torus T^{2N}. As a result, we deduce that this Hamiltonian system always has 2N+1 contractible solutions, and generically even 2^{2N} contractible solutions. Moreover if there exists a non-degenerate non-contractible solution then there necessarily exists another.

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↗
arXiv2012

Tensor tomography on surfaces

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

math.DG↗
arXiv2012

The attenuated ray transform for connections and Higgs fields

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.

math.DG↗
arXiv2012

Equivariant symplectic homology of Anosov contact structures

We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.

math.SG↗
arXiv2012

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with sublinear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small T>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period T belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

math.SG↗
arXiv2011

Transparent pairs

Let $M$ be a closed orientable Riemannian surface. Consider an SO(3)-connection $A$ and a Higgs field $Φ:M\to so(3)$. The pair $(A,Φ)$ naturally induces a cocycle over the geodesic flow of $M$. We classify (up to gauge transformations) cohomologically trivial pairs $(A,Φ)$ with finite Fourier series in terms of a suitable Bäcklund transformation. In particular, if $M$ is negatively curved we obtain a full classification of SO(3)-transparent pairs.

math.DG↗
arXiv2011

On the growth rate of leaf-wise intersections

We define a new variant of Rabinowitz Floer homology that is particularly well suited to studying the growth rate of leaf-wise intersections. We prove that for closed manifolds $M$ whose loop space is "complicated", if $Σ$ is a non-degenerate fibrewise starshaped hypersurface in $T^*M$ and $ϕ$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $ϕ$ in $Σ$ grows exponentially in time. Concrete examples of such manifolds $M$ are the connected sum of two copies of $S^2 \times S^2$, the connected sum of $T^4$ and $CP^2$, or any surface of genus greater than one.

math.SG↗
arXiv2010

Bäcklund transformations for transparent connections

Let $M$ be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU(2)-connections and we show that they can be built up from suitable Bäcklund transformations.

math.DG↗
arXiv2010

Index computations in Rabinowitz Floer homology

In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a non-degenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one moves up the orbit cylinder. We also provide an example to show that, even above the Mañé critical value, the periods may be increasing thus producing a jump in the Morse index of the free time action functional in relation to the Morse index of the fixed time action functional.

math.SG↗
arXiv2010

The contact geometry of the restricted 3-body problem

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dynamical system.

math.SG↗
arXiv2010

Stability of Anosov Hamiltonian Structures

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $Σ_{k}:=H^{-1}(k)$ of $H$. Suppose $Σ_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $Σ_{k}$ is Anosov with $C^{1}$ weak bundles then the Hamiltonian structure $(Σ_{k}$ is stable if and only if it is contact. If $n$ is even and in addition the flow is assumed to be 1/2-pinched then the same conclusion holds. As a corollary we deduce that if $g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the twisted symplectic structure is not exact then the Hamiltonian structure $(Σ_{k}$ is never stable for all sufficiently large $k$.

math.DS↗
arXiv2010

Transparent connections over negatively curved surfaces

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent SU(2)-connection associated to each meromorphic function on $M$.

math.DG↗
arXiv2009

On the stability of Mañé critical hypersurfaces

We construct examples of Tonelli Hamiltonians on $\T^n$ (for any $n\geq 2$) such that the hypersurfaces corresponding to the Mañé critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must contain a closed orbit.

math.DS↗