Clustering of periodic orbits of a hyperbolic automorphism on the torus T2

This paper deals with clustering of periodic orbits of the hyperbolic toral automorphism induced by matrix A. We prove that Ta satisfies the Axiom A. The clustering of periodic orbits of Ta is ivestigated via the notion of 'p-closeness' of periodic sequences of the respective symbolic dynamical system. We also provide the number of clusters of periodic sequences with given periods in the case of 2-closeness.

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

<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>

Chaos on Discrete Neural Network Loops with Self-Feedback

In this paper, the complex dynamical behaviors in a discrete neural network loop with self-feedback are studied. Specifically, an invariant closed set of the system of neural network loops is built and the subsystem restricted on this invariant closed set is topologically conjugate to a two-sided symbolic dynamical system which has two symbols. In the end, some illustrative numerical examples are given to demonstrate our theoretical results.

Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

For a Denjoy homeomorphism $f$ of the circle $S$, we call a pair of distinct points of the $\omega$-limit set $\omega (\,f)$ whose forward and backward orbits converge together a gap, and call an orbit of gaps a hole. In this paper, we generalize the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Moreover, for any Denjoy homeomorphism $f$ having a finite number of holes and for any transitive orientation-preserving homeomorphism $f_1$ of the circle with the same rotation number $\rho (\,f_1)$ as $\rho (\,f)$, we construct a family $f_\varepsilon$ of Denjoy homeomorphisms of rotation number $\rho (\,f)$ containing $f$ such that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f), f)$ for $0<\varepsilon <\tilde{\varepsilon }<1$, but the number of holes changes at $\varepsilon =\tilde{\varepsilon }$, that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f_{\tilde{\varepsilon }}), f_{\tilde{\varepsilon }})$ for $\tilde{\varepsilon }\leqslant \varepsilon <1$ but $\lim _{\varepsilon \nearrow 1}f_\varepsilon (t)=f_1(t)$ for any $t\in S$, and that $f_\varepsilon$ has a singular limit when $\varepsilon \searrow 0$. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.

On the metrical theory of a non-regular continued fraction expansion

We introduced a new continued fraction expansions in our previous paper. For these expansions, we show the Brodén-Borel-Lévy type formula. Furthermore, we compute the transition probability function from this and the symbolic dynamical system of the natural number with the unilateral shift.

WHAT MAPS CAN ADMIT TWO-SIDED SYMBOLIC DYNAMICAL SYSTEMS?

A continuous map f from a metric space X to itself is said to contain a two-sided symbolic dynamical system if there exists an invariant set X0 of f such that the subsystem f|X0 is topologically conjugate to the shift map on a two-sided sequence space of some symbols. In this paper we show that, for any given integer n ≥ 2, there exists a Lipschitz continuous interval map which contains a two-sided symbolic dynamical system of n symbols. Furthermore, we investigate the effect of differentiability and monotonicity assumptions, and prove that neither piecewise monotonic nor piecewise continuously differentiable graph map can contain a two-sided symbolic dynamical system.

TWO GROUPS ASSOCIATED WITH QUADRATIC PISOT UNITS

In a previous work, we have investigated an automata-theoretic property of numeration systems associated with quadratic Pisot units that yields, for every such number θ, a certain group Gθ. In this paper, we characterize a cross-section of a congruence γθ of ℤ4 that had arisen when constructing Gθ. This allows us to completely describe the quotient Hθ of ℤ4 by γθ, that becomes then a second group associated with θ. Moreover, the cross-section thus described is very similar to the symbolic dynamical system associated, by a theorem of Parry, with the two numeration systems attached to θ. The proof is combinatorial, and based upon rewriting techniques. Résumé: Dans un article précédent, nous avions associé à chaque nombre de Pisot quadratique unitaire θ un certain groupe Gθ par le biais de la construction d'un automate qui réalise le passage entre les représentations des entiers dans deux systèmes de numération naturellement attachés à θ. Dans cet article, nous donnons une caractérisation d'un ensemble de représentants pour une congruence γθ de ℤ4 qui avait été utilisée pour la définition de Gθ. Cette caractérisation permet la description complète du quotient Hθ de ℤ4 par γθ, autre groupe associé à θ. Elle est d'autre part remarquablement similaire à la description, donnée par un théorème de Parry, du système dynamique symbolique associé aux deux systèmes de numération attachés à θ. La preuve est combinatoire et utilise les techniques des systèmes de réécriture.