Arbitrarily distortable Banach spaces of higher order

We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank $AD(\cdot )$, introduced by P. Dodos, uses the transfinite Schreier familes and has the property that $AD(X)<{\omega }_1$ if and only if $X$ is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order ${\mathrm{\ell }}_1$ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces ${\mathfrak{X}}_{0,1}^{{\omega }^{\xi }}$ , which each admit ${\mathrm{\ell }}_1$ and $c_0$ spreading models hereditarily, and were introduced by S.A. Argyros, the first and third author, satisfy $AD(\mathfrak{X}^{\omega^\xi}_{0,1}) = \omega^\xi + 1$. This answers some questions of Dodos.