Abstract

Let $M$ be a smooth compact connected manifold of dimension greaterthan two, on which there exists a free (modulo zero) smooth circleaction that preserves a positive smooth volume. In this article, weconstruct volume-preserving diffeomorphisms on $M$ that are metricallyisomorphic to ergodic translations on the torus of dimension greaterthan two, where one given coordinate of the translation is anarbitrary Liouville number. To obtain this result, we determinesufficient conditions on translation vectors of the torus that allow usto explicitly construct the sequence of successive conjugacies inAnosov--Katok's method, with suitable estimates of their norm.

Highlights

  • In this paper, we construct non-standard smooth realizations of some ergodic translations on the torus, translations with one arbitrary Liouville coordinate

  • We show the following result: for any Liouville number β, the ergodic translation on the torus h of vector (β1, ..., βh−1, β) admits a non-standard smooth realization, where the βi, i = 1, ..., h − 1 are chosen in a dense set of h−1

  • The section is dedicated to the construction of the sequence of diffeomorphisms Bn satisfying the conditions of lemma 3.3

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Summary

Introduction

We construct non-standard smooth realizations of some ergodic translations on the torus, translations with one arbitrary Liouville coordinate. In their paper [1], Anosov and Katok constructed ergodic translations on the torus h, h ≥ 2, that admit non-standard smooth realizations. We show the following result: for any Liouville number β, the ergodic translation on the torus h of vector (β1, ..., βh−1, β) admits a non-standard smooth realization, where the βi, i = 1, ..., h − 1 are chosen in a dense set of h−1. There exists a dense set E(β, d) ⊂ h−1 such that for any (β1, .., βh−1) ∈ E(β, d), there is T ∈ Diff∞(M, μ) metrically isomorphic to the ergodic translation of vector (β1, ..., βh−1, β) To obtain this result, we explicitly construct the sequence of successive conjugacies in Anosov-Katok’s periodic approximation method, with suitable estimates of their norm. To obtain this result, we need to suitably relax one of Anosov-Katok’s original assumptions

Definitions
Basic steps of the proof
Partitions of the torus
The sequence of conjugacies
Construction in dimension 2
2: We let
Convergence
Ergodicity
Extension to more general manifolds

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