The decoration conjecture for forcing relations of horseshoe orbits
Let and be decorations, and let denote a periodic or homoclinic horseshoe orbit of height and decoration . Write if the homoclinic orbit forces , and write if these homoclinic orbits have the same homoclinic braid type.
The decoration conjecture. If and , then forces . Moreover, and have the same braid type if and only if and .
This is the part of the decoration conjecture relevant to the paper, concerning how forcing and braid equivalence among periodic orbits are organized by height and decoration. The supplied text does not state whether these assertions are resolved.
References
Primary source
Pieter Collins, “Forcing relations for homoclinic and periodic orbits of the Smale horseshoe map”, arXiv:math/0311188 (2003).
Progress summary
The conjecture remains open in general, with a proof known only for a special class of decorations.
The decoration conjecture, attributed to de Carvalho and Hall, predicts that forcing among horseshoe orbits is governed monotonically by height and decoration, and characterizes when two such orbits have the same braid type. It was presented as unresolved in general in 2003.
Known results
- de Carvalho and Hall (2004) obtained partial results toward the broader forcing-order conjecture.
- A 2003 study computed homoclinic braid equivalence and forcing relations for signatures up to , with tables through signature , but neither proved nor disproved the decoration conjecture.
- A 2008 paper proved the conjecture for lone decorations: the associated braid types are totally ordered by forcing, with an invariant describing which family members an arbitrary horseshoe orbit forces.
Current status (as of August 2026): The full decoration conjecture remains open; only partial results, including the lone-decoration case, are recorded.
Sources
Solutions 0
No solutions have been posted yet.