## Institute for Mathematical Sciences

## Preprint ims01-02

** A. de Carvalho and T. Hall**
* The Forcing Relation for Horseshoe Braid Types.*

Abstract: This paper presents evidence for a conjecture concerning the structure of the set of braid types of periodic orbits of
Smale's horseshoe map, partially ordered by Boyland's forcing
order. The braid types are partitioned into totally ordered
subsets, which are defined by parsing the symbolic code of a
periodic orbit into two segments, the {\em prefix} and the
{\em decoration}: the set of braid types of orbits with each
given decoration is totally ordered, the order being given by
the unimodal order on symbol sequences. The conjecture is
supported by computer experiment, by proofs of special cases,
and by intuitive argument in terms of pruning theory.

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