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A geometric model of mixing Lyapunov exponents inside homoclinic classes in dimension three. (English) Zbl 1377.37033
Summary: 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.
##### MSC:
 37C20 Generic properties, structural stability of dynamical systems 37C29 Homoclinic and heteroclinic orbits for dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D30 Partially hyperbolic systems and dominated splittings
##### Keywords:
homoclinic class; Lyapunov exponent; homoclinic tangency
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