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

TL;DR

The paper constructs numerous examples of fixed-point-free affine isometries on certain Banach spaces derived from circle dynamics, which are recurrent and have zero drift, revealing new phenomena in the interplay between dynamics and Banach space geometry.

Contribution

It introduces a broad class of fixed-point-free, recurrent affine isometries on Banach spaces from circle dynamics, using natural cocycles like the logarithmic, affine, and Schwarzian derivatives.

Findings

Examples of fixed-point-free affine isometries with zero drift.

These isometries are recurrent and arise from generic circle diffeomorphisms.

Construction of commuting families of such isometries.

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.