The study of fractional Dehn twist coefficients dates at least from the work of Gabai and Oertel [@GO] in the context of essential laminations of 3-manifolds, where, with different conventions than those used here, it appeared as the slope of the “degenerate curve" [@GO pg. 62]. Honda, Kazez, and Matic [@HKMVeer1; @HKMVeer2] observed a connection with contact topology through open book decompositions, which has been explored by various authors [@johnjohn; @kazezroberts; @ItoKawamuro]. The following proposition summarizes a few key properties of the fractional Dehn twist coefficient.

[@Malyutin; @ItoKawamuro] Let $\tau: {\mathrm{Aut}}(S,\partial S)\to {\mathbb{Q}}$ be the fractional Dehn twist coefficient, and let $t_\partial$ denote the mapping class of a right-handed Dehn twist around a curve parallel to $\partial S$. Then for all $\phi,\psi\in{\mathrm{Aut}}(S, \partial S)$, we have:

1.  [*(Quasimorphism)*]{} $|\tau(\phi\circ\psi) - \tau(\phi) - \tau(\psi)| \leq 1.$

2.  [*(Homogeneity)* ]{}$\tau(\phi^n) = n \tau(\phi).$

3.  [*(Boundary Twisting)*]{}   $\tau(\phi\circ t_\partial) = \tau(\phi) + 1.$

The first two properties easily imply that the fractional Dehn twist is invariant under conjugation (see e.g., [@Ghys Proposition 5.3]), and the third implies that it can be arbitrarily large, either positively or negatively. There are constraints, however, on the possible denominators of $\tau(\phi)$ based on the topology of $S$; 