4 problems

Time-reversal braid-type conjecture. Horseshoe orbits which are time-reversals have the same braid type.

The decoration conjecture. If $q<q'$ and $w\succcurlyeq w'$, then $P_q^w$ forces $P_{q'}^{w'}$. Moreover, $P_q^w$ and $P_{q'}^{w'}$ have the same braid type if and only if $q=q'$ a…

de Carvalho–Hall conjecture. (a) If $q\ne q_w$, then $P$ has pseudo-Anosov braid type. (b) If $P'$ is another periodic orbit with height $q'$ and the same decoration $w$, then

Decoration conjecture. The set of horseshoe braid types is partitioned into families $\mathcal D^w=\{\beta^w_q\}$, each parameterized by a rational number $q$, which are totally or…