A Geometric Model of Mixing Lyapunov Exponents Inside Homoclinic Classes in Dimension Three

Abstract

For C 1 diffeomorphisms of three dimensional closed manifolds, we provide a geometric model of mixing Lyapunov exponents inside a homoclinic class of a periodic saddle p with non-real eigenvalues. Suppose p has stable index two and the sum of the largest two Lyapunov exponents is greater than log(1 − δ), then δ-weak contracting eigenvalues are obtained by an arbitrarily small C 1 perturbation. Using this result, we give a sufficient condition for stabilizing a homoclinic tangency within a given C 1 perturbation range.

Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

“…where the 'old' coefficients are given by (19). The functions ĥ now do not contain constant terms, and the estimates for ĥi (i = 1, 2, 4, 5) remain the same as (21), and those for ĥ3 satisfy (25).…”

“…One of the authors discussed [27] the creation of heterodimensional cycles when there is no domination, which is tightly related to the existence of homoclinic tangencies. This problem is pursued in [6,7,21]. Results in [14] also imply the existence of heterodimensional cycles near homoclinic tangencies.…”