Cofinal elements and fractional Dehn twist coefficients

We show that for a surface $S$ with positive genus and one boundary component, the mapping class of a Dehn twist along a curve parallel to the boundary is cofinal in every left ordering of the mapping class group $\mathrm{Mod}(S)$. We apply this result to show that one of the usual definitions of the fractional Dehn twist coefficient – via translation numbers of a particular action of $\mathrm{Mod}(S)$ on $\mathbb{R}$ – is in fact independent of the underlying action when $S$ has genus larger than one. As an algebraic counterpart to this, we provide a formula that recovers the fractional Dehn twist coefficient of a homeomorphism of $S$ from an arbitrary left ordering of $\mathrm{Mod}(S)$.