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Topological and dynamical classification of the unstable manifolds of one-rectangle systems

Published online by Cambridge University Press:  02 October 2001

GIOIA M. VAGO
GIOIA M. VAGO
Affiliation:
Université de Bourgogne, U.F.R. des Sciences et Techniques, Laboratoire de Topologie, U.M.R. 5584 du C.N.R.S., B.P. 47 870, 21078 Dijon Cedex, France (e-mail: [email protected])
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Abstract

The unstable manifolds of hyperbolic systems admitting a one-rectangle Markov partition are here characterized up to homeomorphism and up to conjugacy of the underlying dynamics in a very easy-to-compute way. We associate to each one-rectangle system a word over the alphabet \{+,-\} describing the bending of the image rectangle with respect to the initial rectangle. We endow the set \{+,-\}^{\ast} of such words with a product \wedge having a dynamical origin. The structure of the non-commutative semigroup (\{+,-\}^{\ast},\wedge) is made completely explicit. The topological and dynamical properties of the unstable manifolds of one-rectangle systems are translated in terms of the decomposition of the associated words in such a semigroup.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 21 , Issue 5 , October 2001 , pp. 1563 - 1596
Copyright
2001 Cambridge University Press

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