Downloads provided by UsageCountsGeometric properties of the scattering map of a normally hyperbolic invariant manifold
Copyright policy ) Delshams Valdés, Amadeu; Llave Canosa, Rafael de la; Seara, Tere M.;
Geometric properties of the scattering map of a normally hyperbolic invariant manifold
Given a normally hyperbolic invariant manifold Λ for a map f , whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future. We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions. We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type. We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64–188] using methods of fast–slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.
Peer Reviewed
- The University of Texas at Austin United States
- Universitat Polite`cnica de Catalunya Spain
- Universitat Politècnica de Catalunya Spain
Mathematics(all), 46Lxx, Classificació AMS::53 Differential geometry::53D Symplectic geometry, Normally hyperbolic manifolds, 70-xx]::37J Finite-dimensional Hamiltonian, Geometria simplèctica, Scattering map, 34Cxx, Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry, 70Hxx], Arnold diffuson, Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics, and nonholonomic systems [See also 53Dxx, Hamilton, :70H Hamiltonian and Lagrangian mechanics [See also 37Jxx] [for statistical mechanics, see 82-xx}], Hamiltonian dynamical systems, and nonholonomic systems, Lagrangian, Classificació AMS::37 Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-xx]::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx], for statistical mechanics, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10, Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 57Rxx. For foundational questions of differentiable manifolds, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], :Matemàtiques i estadística [Àrees temàtiques de la UPC], :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], 70Gxx, Classificació AMS::53 Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}::53D Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx], Sistemes dinàmics diferenciables, 58Jxx, Sistemes de, Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], :53 Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}::53D Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx] [Classificació AMS], for statistical mechanics, see 83A05 and 83C10, Classificació AMS::37 Dynamical systems and ergodic theory [See also 26A18, Classificació AMS::53 Differential geometry {For differential topology, contact, Geometria diferencial, contact geometry [See also 37Jxx, 34Dxx, contact geometry, Symplectic deformation, :70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics [Classificació AMS], :37 Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-xx]::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx] [Classificació AMS], Differential geometry, Hamiltonian systems, 28Dxx, Classical scattering theory, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Àrees temàtiques de la UPC::Matemàtiques i estadística, Scattering (Mathematics), :70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10 [Classificació AMS], 35Bxx, 70Fxx, :53 Differential geometry::53D Symplectic geometry, contact geometry [Classificació AMS], Hamilton, Sistemes de, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, see 58Axx}::53D Symplectic geometry
Mathematics(all), 46Lxx, Classificació AMS::53 Differential geometry::53D Symplectic geometry, Normally hyperbolic manifolds, 70-xx]::37J Finite-dimensional Hamiltonian, Geometria simplèctica, Scattering map, 34Cxx, Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry, 70Hxx], Arnold diffuson, Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics, and nonholonomic systems [See also 53Dxx, Hamilton, :70H Hamiltonian and Lagrangian mechanics [See also 37Jxx] [for statistical mechanics, see 82-xx}], Hamiltonian dynamical systems, and nonholonomic systems, Lagrangian, Classificació AMS::37 Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-xx]::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx], for statistical mechanics, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10, Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 57Rxx. For foundational questions of differentiable manifolds, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], :Matemàtiques i estadística [Àrees temàtiques de la UPC], :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], 70Gxx, Classificació AMS::53 Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}::53D Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx], Sistemes dinàmics diferenciables, 58Jxx, Sistemes de, Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-xx}::70H Hamiltonian and Lagrangian mechanics [See also 37Jxx], :53 Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}::53D Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx] [Classificació AMS], for statistical mechanics, see 83A05 and 83C10, Classificació AMS::37 Dynamical systems and ergodic theory [See also 26A18, Classificació AMS::53 Differential geometry {For differential topology, contact, Geometria diferencial, contact geometry [See also 37Jxx, 34Dxx, contact geometry, Symplectic deformation, :70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics [Classificació AMS], :37 Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-xx]::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx] [Classificació AMS], Differential geometry, Hamiltonian systems, 28Dxx, Classical scattering theory, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Àrees temàtiques de la UPC::Matemàtiques i estadística, Scattering (Mathematics), :70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10 [Classificació AMS], 35Bxx, 70Fxx, :53 Differential geometry::53D Symplectic geometry, contact geometry [Classificació AMS], Hamilton, Sistemes de, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, see 58Axx}::53D Symplectic geometry
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).81 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 10% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10% views 86 downloads 68 - 86views68downloads
selected citations
Citations provided by BIP!
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
Citations provided by BIP!popularityPopularity provided by BIP!
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
Popularity provided by BIP!influenceInfluence provided by BIP!
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
Influence provided by BIP!impulseImpulse provided by BIP!
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
Impulse provided by BIP!viewsViews provided by UsageCounts
Views provided by UsageCountsdownloadsDownloads provided by UsageCounts
Downloads provided by UsageCounts 81
Top 10%
Top 10%
Top 10%
86
68
Green
hybrid
Fields of Science (3) View all