Rotation number of contracted rotations

Abstract

Let 0 < λ < 1. We consider the one-parameter family of circle λ-affine contractions f δ : x ∈ [0, 1) → λx + δ mod 1, where 0 ≤ δ < 1. Let ρ be the rotation number of the map f δ . We will give some numerical relations between the values of λ, δ and ρ, essentially using Hecke-Mahler series and a tree structure. When both parameters λ and δ are algebraic numbers, we show that ρ is a rational number. Moreover, in the case λ and δ are rational, we give an explicit upper bound for the height of ρ under some assumptions on λ.

Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces

“…The properties of this map provide the basis for several of our results. This map has been studied previously; see for example [5,6,7,8,9,13,18,22]. Some of the results in §3 reproduce parts of those earlier works.…”

Symbolic dynamics of piecewise contractions

“…The following result is a corollary of Theorem 2.1. The particular family of 2-PCs f : I → I defined by f (x) = λx + δ (mod 1) was considered by Bugeaud [4,5], Bugeaud and Conze [6] and, more recently, by Janson andÖberg [16], and also by Laurent and Nogueira [18], by means of a rotation number approach.…”

“…Concerning such family, we provide the following corollary, which turns out to be a special case of [18,Corollary 7]. We recall that an n-PC f :I → I is topologically semiconjugate to an n-IET T :I → I if there exists a continuous, nondecreasing and surjective map h: In Corollary 2.5, we have that δ = 1 4 k≥0 θ k 2 −k , where θ = θ 0 θ 1 .…”

A spectral decomposition of the attractor of piecewise-contracting maps of the interval

“…Associating a rotation number to the map, they proved that the attractor is either a periodic orbit (rational rotation number) or a Cantor set (irrational rotation number) and that the latter case corresponds to a quasi-periodic asymptotic dynamics with Sturmian complexity. It is in particular the case for the half-closed unit interval map x → λx + µ mod 1, for which the properties of the rotation number as a function of λ and µ ∈ [0, 1) have been studied in detail [2,3,6,8]. For injective PCIMs with N 2 contraction pieces, it has been proved that the complexity of the itinerary of any orbit is an eventually affine function [4,13].…”

“…, that is, the attractor consists either of a single X -minimal Cantor set or of one or two periodic orbits. For any of these cases there exist examples of PCIMs with such an attractor [2,3,[6][7][8]. So, the inequality is optimal at least for PCIMs with two contraction pieces.…”