Systems of rank one, explicit Rokhlin towers, and covering numbers

Christian Weiss

doi:10.1007/s00013-021-01683-0

Abstract

Rotations f_alpha of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle alpha are known to be dynamical systems of rank one. This is equivalent to the property that the covering number F^*(f_alpha ) of the dynamical system is one. In other words, there exists a basis B such that for arbitrarily high h, an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower (f_alpha ^kB)_{k=0}^{h-1}. Although B can be chosen with diameter smaller than any fixed varepsilon > 0, it is not always possible to take an interval for B but this can only be done when the partial quotients of alpha are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when B is the union of n_B in {mathbb {N}} disjoint intervals. This question has been answered in the case n_B =1 by Checkhova, and here we address the general situation. If n_B = 2, we give a precise formula for the maximum proportion. Furthermore, we show that for fixed alpha , the maximum proportion converges to 1 when n_B rightarrow infty . Explicit lower bounds can be given if alpha has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin lemma and furthermore makes use of the three gap theorem.

Highlights

If a measure-theoretic dynamical system (X, T, μ), with μ being a probability invariant by T, is invertible and ergodic, for arbitrary ε > 0 and h ∈ N, it is always possible to find a measurable set B such that B, T B, . . . , T Bh−1 are disjoint sets and have joint measure greater than 1 − ε. This result is known as the famous Rokhlin lemma
We exclusively consider a special class of transformations, namely rotations fα : x → x + α mod 1 by an irrational angle α of the unit torus T1 = [0, 1) with ends of the interval glued, and aim to contribute to a better understanding of Rokhlin towers when an extra condition on the topological structure of the basis B is imposed
Before we come to the presentation of our results, we mention at first that the dynamical system on T1 defined by fα is uniquely ergodic but even a so-called system of rank one, see [7]

Summary

Introduction

If a measure-theoretic dynamical system (X, T, μ), with μ being a probability invariant by T , is invertible and ergodic, for arbitrary ε > 0 and h ∈ N, it is always possible to find a measurable set B such that B, T B, . . . , T Bh−1 are disjoint sets and have joint measure greater than 1 − ε. Before we come to the presentation of our results, we mention at first that the dynamical system on T1 defined by fα is uniquely ergodic (for the Lebesgue measure) but even a so-called system of rank one, see [7] That means that it does satisfy the properties stated in the Rokhlin lemma, but approximates every partition arbitrarily well by Rokhlin towers, which means that the diameter of B can be chosen arbitrarily small. If we can approximate any partition, in the sense of Definition 1.1, by a tower whose basis B is an interval, we say that (X, T, μ) is of rank one by intervals This idea can be transferred to Definition 1.2 by restricting covering number to unions of nB ∈ N disjoint intervals and considering nB = 1. Let α ∈ R\Q with continued fraction expansion [a0; a1; a2; . . .]

If ai

Then the gaps that can appear have lengths