Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics
Date
Tuesday, February 11, 2025 14:00 - 15:30
Speaker
Dongchen Li (Imperial College London)
Location
Office Bldg West / Ground floor / Heinzel Seminar Room (I21.EG.101)
Series
Seminar/Talk
Tags
DynamIST
Host
Kaloshin Group
Contact
Url
A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some central subspace has a non-empty interior and thus has a higher topological dimension than the set itself.We show that, for any symplectic Cr-diffeomorphism (where r is sufficiently large and finite, or r=,) of a 2N-dimensional (N>1) symplectic manifold, symplectic blenders can be obtained by an arbitrarily small symplectic perturbation near any one-dimensional whiskered KAM-torus that has a homoclinic orbit. Using this result, we prove that non-transverse homoclinic intersections between invariant manifolds of a saddle-center periodic point (i.e., it has exactly one pair of complex multipliers on the unit circle) are persistent in the following sense: the original map is in the Cr closure of a C1 open set in the space of symplectic Cr-diffeomorphisms, where maps having such saddle-center homoclinic intersections are dense. These results also hold for Hamiltonian flows in the corresponding settings.