S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry

Thuswaldner, Jörg Μ.

doi:10.1007/978-3-030-57666-0_3

Cited by 9 publications

(6 citation statements)

References 95 publications

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“…Proof. Here we use a known relationship between the continued fraction expansion of α and first-return maps under a rotation by α [2,6,20]. Instead of starting with a circle (−1, 0] modulo 1, it is convenient to use (−1, α] modulo 1 + α, where α is "identified" with 0 by being the first image of 0 under the rotation by α.…”

Section: Further Remarksmentioning

confidence: 99%

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On trigonometric skew-products over irrational circle-rotations

Koch¹

2021

DCDS

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We investigate some asymptotic properties of trigonometric skewproduct maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.

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Section: Further Remarksmentioning

confidence: 99%

“…This follows from the fact that (up to a reflection) the first-return map of a rotation by α 0 = α on a circle (−1, α 0 ] is a rotation by α 0 α 1 on the circle (−α 0 , α 0 α 1 ]; see e.g. Lemma 2.12 in [20]. After repeating this n times, the rotation is byᾱ n α n and the return map is to (−ᾱ n ,ᾱ n α n ].…”

Section: Further Remarksmentioning

confidence: 99%

On trigonometric skew-products over irrational circle-rotations

Koch¹

2021

DCDS

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“…The IETs are the good part since they behave well with induced transformations and admit continued fraction algorithms [8,44,52,54]. The bad part was much improved since then with various recent results using multidimensional continued fraction algorithms including Brun's algorithm [21] which provides measurable-theoretic conjugacy with symbolic systems for almost every toral rotations on T 2 [16,51]. As the authors wrote in [14], the term ugly "refers to some esthetic difficulties in building two-dimensional sequences by iteration of patterns".…”

Section: Introductionmentioning

confidence: 99%

Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

Labbé¹

2021

JMD

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<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{P} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id="M8">\begin{document}$ W $\end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id="M9">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id="M10">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id="M11">\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M12">\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id="M13">\begin{document}$ \widehat{R}|_W $\end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id="M15">\begin{document}$ W $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id="M16">\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id="M17">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr{P}}_0 $\end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id="M19">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M20">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>. It also implies that the subshift <inline-formula><tex-math id="M21">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> is uniquely ergodic and is isomorphic to the toral <inline-formula><tex-math id="M22">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M23">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.</p>

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“…Proof. Here we use a known relationship between the continued fractions expansion of α and first-return maps under a rotation by α [14,15,24]. Instead of starting with a circle (−1, 0] modulo 1, it is convenient to use (−1, α] modulo 1 + α, where α is "identified" with 0 by being the first image of 0 under the rotation by α.…”

mentioning

confidence: 99%

“…This follows from the fact that (up to a reflection) the first-return map of a rotation by α 0 = α on a circle (−1, α 0 ] is a rotation by α 0 α 1 on the circle (−α 0 , α 0 α 1 ]; see e.g. Lemma 2.12 in [24]. After repeating this n times, the rotation is by ᾱn α n and the return map is to (−ᾱ n , ᾱn α n ].…”

mentioning

confidence: 99%

On trigonometric skew-products over irrational circle-rotations

Koch¹

2020

Preprint

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We describe some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. Analogous results are believed to be true for the self-dual almost Mathieu maps, but they seem presently beyond reach.

show abstract

S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry

Cited by 9 publications

References 95 publications

On trigonometric skew-products over irrational circle-rotations

On trigonometric skew-products over irrational circle-rotations

Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

On trigonometric skew-products over irrational circle-rotations

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