Covering numbers: Arithmetics and dynamics for rotations and interval exchanges

Berthé, Valérie; Chekhova, Nataliya; Ferenczi, Sébastien

doi:10.1007/bf02788235

Cited by 15 publications

(10 citation statements)

References 31 publications

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“…Similarly, a large gap for N = q i is split up into a i+1 small gaps for Ñ = q i+1 and 1 large gap for Ñ = q i+1 . These observations imply four fundamental but important facts for disjoint orbits of sets of the form described in (1) with n B = 2. Lemma 2.2.…”

Section: And Only If α Has Unbounded Partial Quotientsmentioning

confidence: 80%

“…The properties of f α and also of the structure of its Rokhlin towers have been comprehensively studied, see e.g. [1,3,[6][7][8][9] to name only a few references. Here, we ask how to find nice sets B for f α given ε > 0, h ∈ N. In [3], a method based on the three gap theorem (Theorem 2.1) was introduced how to calculate the minimal admissible value ε for f α if we choose B as an interval, see Theorem 1.5.…”

Section: Introductionmentioning

confidence: 99%

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

Weiß

2022

Arch. Math.

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Rotations $$f_\alpha $$ f α of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle $$\alpha $$ α are known to be dynamical systems of rank one. This is equivalent to the property that the covering number $$F^*(f_\alpha )$$ F ∗ ( f α ) 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 $$(f_\alpha ^kB)_{k=0}^{h-1}$$ ( f α k B ) k = 0 h - 1 . Although B can be chosen with diameter smaller than any fixed $$\varepsilon > 0$$ ε > 0 , it is not always possible to take an interval for B but this can only be done when the partial quotients of $$\alpha $$ α are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when B is the union of $$n_B \in {\mathbb {N}}$$ n B ∈ N disjoint intervals. This question has been answered in the case $$n_B =1$$ n B = 1 by Checkhova, and here we address the general situation. If $$n_B = 2$$ n B = 2 , we give a precise formula for the maximum proportion. Furthermore, we show that for fixed $$\alpha $$ α , the maximum proportion converges to 1 when $$n_B \rightarrow \infty $$ n B → ∞ . Explicit lower bounds can be given if $$\alpha $$ α 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.

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Section: And Only If α Has Unbounded Partial Quotientsmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

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

Weiß

2022

Arch. Math.

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“…The computation of B in Thorem 2.1 below, and the subsequent Theorem 2.2, which was the main motivation for the present paper, were known to P. Hubert and T. Monteil (private communications), but never written to our knowledge. The quantity 1 B ′ was indeed computed in [11] (see also [5]) as, for irrational rotations, it is equal to another invariant of topological conjugacy, the covering number by intervals [11], which involves covering the space by Rokhlin towers; the spectrum of its possible values is the object of a question in [11] and in [9], to which Theorem 2.3 below gives a first (to our knowledge), though belated and partial, answer.…”

Section: Rotations and The Dynamical Definition Of The Lagrange Spectrummentioning

confidence: 99%

“…38196... and 3 − √ 3 = 1, 26794..., and there is no other element above5 It contains an accumulation point equal to√ 5 − 1 = 1, 2360... It contains the interval [1, 1 + 4 83+18 03688...].The third highest number in this spectrum is 16−2404..., as can be seen with longer computations; the point √ 5 − 1 is the highest accumulation point, but to prove it requires a machinery similar to the one used to prove Theorem 5 in Chapter 1 of[12].…”

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confidence: 99%

Dynamical generalizations of the Lagrange spectrum

Ferenczi¹

2011

Electron. Proc. Theor. Comput. Sci.

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We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne_n, where e_n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1/ne_n

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“…that these brilliant examples, built on purpose, are a little complicated and not very explicit as interval exchange transformations. We know of only one family of interval exchange transformations which have simple spectrum but not rank one, these were built in [5] but only for 3 intervals.…”

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confidence: 99%

Rigidity of square-tiled interval exchange transformations

Ferenczi

Hubert

2019

Journal of Modern Dynamics

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We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction θ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan θ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan θ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

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Covering numbers: Arithmetics and dynamics for rotations and interval exchanges

Cited by 15 publications

References 31 publications

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

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

Dynamical generalizations of the Lagrange spectrum

Rigidity of square-tiled interval exchange transformations

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