Premium
Pseudo‐Anosov Braid Types of the Disc or Sphere of Low Cardinality Imply all Periods
Author(s) -
Guaschi John
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/50.3.594
Subject(s) - mathematics , isotopy , braid , invariant (physics) , homeomorphism (graph theory) , cardinality (data modeling) , pure mathematics , rotation number , periodic point , combinatorics , mathematical physics , materials science , computer science , composite material , data mining
We show that any orientation‐preserving homeomorphism of the 2‐disc possessing either a 3‐ or 4‐point invariant set X either possesses periodic orbits of all periods or belongs to one of a small number of periodic or reducible isotopy classes relative to X . We prove also that for any homeomorphism of the annulus isotopic to the identity which is pseudo‐Anosov relative to a finite invariant set, there exist periodic orbits in the interior whose rotation numbers are those of the boundaries. Finally we show that if f is an orientation‐preserving homeomorphism of the 2‐sphere possessing an invariant set Y of cardinality 4 or 5 such that the braid type of Y is pseudo‐Anosov, then f has periodic orbits of all periods.