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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Markov neighborhoods for zero-dimensional basic sets

Author: Dennis Pixton
Journal: Trans. Amer. Math. Soc. 279 (1983), 431-462
MSC: Primary 58F10; Secondary 58F18
MathSciNet review: 709562
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Abstract: We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.

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Keywords: Structural stability, Axiom $ A$, basic set, lamination, Markov partition
Article copyright: © Copyright 1983 American Mathematical Society