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 $\log(1-\delta)$, then $\delta$-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.