Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets

V. Afraimovich; T.R. Young

doi:10.3934/dcds.2000.6.691

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2000, Volume 6, Issue 3: 691-704. Doi: 10.3934/dcds.2000.6.691

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Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets

V. Afraimovich^1, and
T.R. Young^2,

1.
IICO-UASLP, A. Obregón 64, 78000 San Luis Postosí, SLP
2.
Department of Mathematics, Ohio University, Athens, OH 45701

Received: November 30, 1998

Revised: December 31, 1999

Published: June 2000

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Abstract

Suppose that $f$ is a surface diffeomorphism with a hyperbolic fixed point $\mathcal O$ and this fixed point has a transversal homoclinic orbit. It is well known that in a vicinity of this type of homoclinic there are hyperbolic invariants sets. We introduce smooth invariants for the homoclinic orbit which we call the multipliers. As an application, we study the influence of the multipliers on numerical invariants of the hyperbolic invariant sets as the vicinity becomes small.

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Mathematics Subject Classification: 34C35, 58C25.

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