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

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.