Periodic points and homoclinic classes

Homoclinic Points

Expansive homoclinic classes

We prove that for C generic diffeomorphisms, every expansive homoclinic class is hyperbolic.

Newhouse Phenomenon and Homoclinic Classes

We show that for a C residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it’s the Hausdorff limit of a family of sources.

Degenerate periodic orbits and homoclinic torus bifurcation.

A one-parameter family of periodic orbits with frequency omega and energy E of an autonomous Hamiltonian system is degenerate when E'(omega) = 0. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of t...

SUPER-EXPONENTIAL GROWTH OF THE NUMBER OF PERIODIC ORBITS INSIDE HOMOCLINIC CLASSES Aim Sciences

We show there is a residual subset S(M) of Diff1(M) such that, for every f ∈ S(M), any homoclinic class of f containing periodic saddles p and q of indices α and β, respectively, where α < β, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown the super-exponential growth occurs for hyperbolic periodic points of index γ inside the h...