Combinatorics of the zeta map on rational Dyck paths

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths onto itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing $\zeta(P)$ and $\zeta(P^c)$ is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We conclude with two possible routes to a proof that $\zeta$ is a bijection. Notably, we provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$. We show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case and provide evidence that $\delta$ may be computable from $\zeta(P)$ in general.