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The decoration conjecture for forcing relations of horseshoe orbits

From papers

Let ww and w′w' be decorations, and let PqwP_q^w denote a periodic or homoclinic horseshoe orbit of height qq and decoration ww. Write w≽w′w\succcurlyeq w' if the homoclinic orbit P0wP_0^w forces P0w′P_0^{w'}, and write w∼w′w\sim w' if these homoclinic orbits have the same homoclinic braid type.

The decoration conjecture. If q<q′q<q' and w≽w′w\succcurlyeq w', then PqwP_q^w forces Pq′w′P_{q'}^{w'}. Moreover, PqwP_q^w and Pq′w′P_{q'}^{w'} have the same braid type if and only if q=q′q=q' and w∼w′w\sim w'.

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.

Progress summary

Partially solved

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 1212, with tables through signature 99, 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
Sources & referencesView supporting material

Primary source

Pieter Collins, “Forcing relations for homoclinic and periodic orbits of the Smale horseshoe map”, arXiv:math/0311188 (2003).

forcing-relationshorseshoe-dynamicssymbolic-dynamics

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