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Isotopy Stable Dynamics Relative to Compact Invariant Sets
Author(s) -
Boyland Philip,
Hall Toby
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599012009
Subject(s) - mathematics , isotopy , invariant (physics) , periodic orbits , cantor set , compact space , pure mathematics , uncountable set , mathematical analysis , countable set , mathematical physics
Let f be an orientation‐preserving homeomorphism of a compact orientable manifold. Sufficient conditions are given for the persistence of a collection of periodic points under isotopy of f relative to a compact invariant set A . Two main applications are described. In the first, A is the closure of a single discrete orbit of f , and f has a Smale horseshoe, all of whose periodic orbits persist; in the second, A is a minimal invariant Cantor set obtained as the limit of a sequence of nested periodic orbits, all of which are shown to persist under isotopy relative to A . 1991 Mathematics Subject Classification : 58F20, 58F15.