On the Structure of Orbits from a Neighborhood of a Transversal Homoclinic Orbit to a Nonhyperbolic Fixed Point

Abstract

We consider a one-parameter family \(f_{\mu}\) of multidimensional diffeomorphisms such that for \(\mu=0\) the diffeomorphism \(f_{0}\) has a transversal homoclinic orbit to a nonhyperbolic fixed point of arbitrary finite order \(n\geqslant 1\) of degeneracy, and for \(\mu>0\) the fixed point becomes a hyperbolic saddle. In the paper, we give a complete description of the structure of the set \(N_{\mu}\) of all orbits entirely lying in a sufficiently small fixed neighborhood of the homoclinic orbit. Moreover, we show that for \(\mu\geqslant 0\) the set \(N_{\mu}\) is hyperbolic (for \(\mu=0\) it is nonuniformly hyperbolic) and the dynamical system \(f_{\mu}\bigl{|}_{N_{\mu}}\) (the restriction of \(f_{\mu}\) to \(N_{\mu}\)) is topologically conjugate to a certain nontrivial subsystem of the topological Bernoulli scheme of two symbols.