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New phenomena associated with homoclinic tangencies

Published online by Cambridge University Press:  18 October 2004

SHELDON E. NEWHOUSE
SHELDON E. NEWHOUSE
Affiliation:
Mathematics Department, Michigan State University, E. Lansing, MI 48824-1027, USA (e-mail: sen@math.msu.edu)
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Abstract

We survey some recently obtained generic consequences of the existence of homoclinic tangencies in diffeomorphisms of surfaces. Among other things, it has been shown that they give rise to invariant topologically transitive sets with maximal Hausdorff dimension, that they prohibit the existence of various kinds of symbolic extensions and that they form an impediment to the existence of Sinai–Ruelle–Bowen (SRB) measures. The main new result described here, together with a positive answer to an as yet unproved conjecture of Palis, would prove that generically on surfaces, SRB measures only exist on uniformly hyperbolic attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 5 , October 2004 , pp. 1725 - 1738
Copyright
© 2004 Cambridge University Press

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