Hokkaido Mathematical Journal

NISHIZAWA Yusuke,

Existence of horseshoe sets with nondegenerate one-sided homoclinic tangencies in ${\mathbb R}^{3}$.

Hokkaido Mathematical Journal, 37 (2008) pp.133-145

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Abstract

In this paper, we present some class of three dimensional $C^{\infty}$ diffeomorphisms with nondegenerate one-sided homoclinic tangencies $q$ associated with hyperbolic fixed points $p$ each of which exhibits a horseshoe set. A key point in the proof is the existence of a transverse homoclinic point arbitrarily close to $q$. This result together with Birkhoff-Smale Theorem implies the existence of a horseshoe set arbitrarily close to $q$.

MSC(Primary)
MSC(Secondary)37D10(MSC2000), 37C15(MSC2000), 37C05(MSC2000), 37D40(MSC2000)
Uncontrolled KeywordsHorseshoe sets, Homoclinic tangencies, Singular \lambda$-Lemma, Birkhoff-Smale Theorem.