homoclinic class

Let M be a compactPlanetmathPlanetmath smooth manifoldMathworldPlanetmath and f:MM a diffeomorphism. The homoclinic class of a hyperbolic periodic pointPlanetmathPlanetmath p of f, denoted H(p,f), is the closureMathworldPlanetmathPlanetmath of the set of transverse intersectionsMathworldPlanetmath between the stable and unstable manifolds all points in the orbit of p; i.e.


Homoclinic classes are topologically transitive, and the number of homoclinic classes is at most countableMathworldPlanetmath. Moreover, generically (in the 𝒞1 topologyMathworldPlanetmath of Diff(M)), they are pairwise disjoint and maximally transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Title homoclinic class
Canonical name HomoclinicClass
Date of creation 2013-03-22 14:07:30
Last modified on 2013-03-22 14:07:30
Owner Koro (127)
Last modified by Koro (127)
Numerical id 6
Author Koro (127)
Entry type Definition
Classification msc 37C29