Some examples of affine isometries of Banach spaces arising from 1-D dynamics

Abstract

We provide a large family of examples of affine isometries of the Banach spaces $C^0(S^1)$, $L^1(S^1)$ and $L^2(S^1\times S^1)$ that are fixed-point-free despite being recurrent (in particular, they have zero drift). These come from natural cocycles on the group of circle diffeomorphisms, namely the logarithmic, affine and (a variation of the) Schwarzian derivative. Quite interestingly, they arise from diffeomorphisms that are generic in an appropriate context. We also show how to promote these examples in order to obtain families of commuting isometries satisfying the same properties.

Part of the content of this Note is spread as remarks/exercises in several sources by the author. However, they appear with only few details and no major developments (in particular, on commuting diffeomorphisms). This text arose from the necessity of giving a concise yet clarifying discussion and a slight extension of all of this