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 \(\left( {X_0^{{\omega ^\xi }}} \right)\xi < {\omega _1}\), which each admit l₁ and c₀ spreading models hereditarily, and were introduced by S. A. Argyros, the first and third author, satisfy \(AD\left( {X_0^{{\omega ^\xi }}} \right) = {\omega ^\xi } + 1\). This answers some questions of Dodos.