Topological Chaos for Differential Inclusions with Multivalued Impulses on Tori

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

Reeb orbits that force topological entropy

We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.

Chaos for Differential Equations with Multivalued Impulses

The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.

Forward triplets and topological entropy on trees

<p style='text-indent:20px;'>We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive entropy if and only if some iterate <inline-formula><tex-math id="M2">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> has a periodic orbit with three aligned points consecutive in time, that is, a triplet <inline-formula><tex-math id="M3">\begin{document}$ (a,b,c) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ f^k(a) = b $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ f^k(b) = c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> belongs to the interior of the unique interval connecting <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ c $\end{document}</tex-math></inline-formula> (a <i>forward triplet</i> of <inline-formula><tex-math id="M9">\begin{document}$ f^k $\end{document}</tex-math></inline-formula>). We also prove a new criterion of entropy zero for simplicial <inline-formula><tex-math id="M10">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M11">\begin{document}$ P $\end{document}</tex-math></inline-formula> based on the non existence of forward triplets of <inline-formula><tex-math id="M12">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M13">\begin{document}$ 1\le k&lt;n $\end{document}</tex-math></inline-formula> inside <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula>. Finally, we study the set <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> of all <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M17">\begin{document}$ P $\end{document}</tex-math></inline-formula> that have a forward triplet inside <inline-formula><tex-math id="M18">\begin{document}$ P $\end{document}</tex-math></inline-formula>. For any <inline-formula><tex-math id="M19">\begin{document}$ n $\end{document}</tex-math></inline-formula>, we define a pattern that attains the minimum entropy in <inline-formula><tex-math id="M20">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> and prove that this entropy is the unique real root in <inline-formula><tex-math id="M21">\begin{document}$ (1,\infty) $\end{document}</tex-math></inline-formula> of the polynomial <inline-formula><tex-math id="M22">\begin{document}$ x^n-2x-1 $\end{document}</tex-math></inline-formula>.</p>

Number-theoretic positive entropy shifts with small centralizer and large normalizer

Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$ .

A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.