Decoration invariants for horseshoe braids
Andr\'e de Carvalho, Toby Hall
TL;DR
This paper proves the Decoration Conjecture for lone decorations in horseshoe braids and introduces methods to compute braid invariants, enhancing understanding of the ordering of periodic orbits in dynamical systems.
Contribution
It provides a proof of the Decoration Conjecture for lone decorations and develops techniques to calculate braid conjugacy invariants for horseshoe braids.
Findings
Proof of the Decoration Conjecture for lone decorations
Methods to compute braid conjugacy invariants
Enhanced understanding of periodic orbit forcing in horseshoe maps
Abstract
The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology