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(•), introduced by P. Dodos, uses the transfinite Schreier families and has the property that AD(X) < ω1 if and only if X is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order l1 spreading models. We also compute this rank for several Banach spaces. In particular, it is shown that the class of Banach spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {X_0^{{\omega ^\xi }}} \right)\xi < {\omega _1}$$\end{document}, which each admit l1 and c0 spreading models hereditarily, and were introduced by S. A. Argyros, the first and third author, satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AD\left( {X_0^{{\omega ^\xi }}} \right) = {\omega ^\xi } + 1$$\end{document}. This answers some questions of Dodos.