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

Andrés Navas

TL;DR

The paper exhibits a broad class of affine isometries on infinite-dimensional Banach spaces (namely C^0(S^1), L^1(S^1), and L^2(S^1 × S^1)) that are recurrent yet fixed-point-free, all possessing zero drift along suitable subsequences. The constructions arise from natural cocycles associated to circle diffeomorphisms with irrational rotation numbers, using logarithmic, affine, and Schwarzian-type derivatives to define the translational part. A key methodological theme is that these diffeomorphisms are generic in appropriate spaces, yielding recurrent, fixed-point-free dynamics in the infinite-dimensional setting. The paper also shows how to promote these examples to commuting families isomorphic to Z^∞, highlighting a rich interplay between smooth dynamics on the circle and operator-theoretic actions on Banach spaces, with potential extensions to higher-dimensional tori and related dynamical systems questions.

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.

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

TL;DR

The paper exhibits a broad class of affine isometries on infinite-dimensional Banach spaces (namely C^0(S^1), L^1(S^1), and L^2(S^1 × S^1)) that are recurrent yet fixed-point-free, all possessing zero drift along suitable subsequences. The constructions arise from natural cocycles associated to circle diffeomorphisms with irrational rotation numbers, using logarithmic, affine, and Schwarzian-type derivatives to define the translational part. A key methodological theme is that these diffeomorphisms are generic in appropriate spaces, yielding recurrent, fixed-point-free dynamics in the infinite-dimensional setting. The paper also shows how to promote these examples to commuting families isomorphic to Z^∞, highlighting a rich interplay between smooth dynamics on the circle and operator-theoretic actions on Banach spaces, with potential extensions to higher-dimensional tori and related dynamical systems questions.

Abstract

We provide a large family of examples of affine isometries of the Banach spaces , and 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.
Paper Structure (10 sections, 8 theorems, 61 equations)

This paper contains 10 sections, 8 theorems, 61 equations.

Table of Contents

  1. Some general remarks on affine isometries
  2. Some background on circle diffeomorphisms
  3. Examples of fixed-point-free recurrent isometries
  4. An example in $C^0 (\mathrm{S}^1)$
  5. An example in $L^1 (\mathrm{S}^1)$
  6. An example in $L^2 (\mathrm{S}^1 \times \mathrm{S}^1)$
  7. Adding commutativity
  8. The $C^0$ case.
  9. The $L^1$ case.
  10. The $L^2$ case.

Key Result

Theorem 2.1

[Denjoy]If $f$ is a $C^{1+bv}$ circle diffeomorphism of irrational rotation number $\rho$, then the sequence of iterates $f^{q_n}$ converges to the identity in $C^0$ topology, where $p_n / q_n$ is the sequence of the rational approximations of $\rho$ obtained from its continuous fraction expansion.

Theorems & Definitions (20)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • ...and 10 more