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Journal Article

# Hyperbolicity and the Creation of Homoclinic Orbits

J. Palis and F. Takens
Annals of Mathematics
Second Series, Vol. 125, No. 2 (Mar., 1987), pp. 337-374
DOI: 10.2307/1971313
Stable URL: http://www.jstor.org/stable/1971313
Page Count: 38

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Topics: Eigenvalues, Saddle points, Horseshoes, Coordinate systems, Leaves, Periodic orbits, Zero, Tangents, Quadrants

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## Abstract

We consider one-parameter families $\lbrace \varphi _\mu; \mu \epsilon R \rbrace$ of diffeomorphisms on surfaces which display a homoclinic tangency for $\mu < 0$ and are hyperbolic for $\mu < 0$ (i.e., φ μ has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for μ positive. For many of these families, we prove than φμ is also hyperbolic for most small positive values or μ (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.

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