On Li-Yorke pairs

By François Blanchard, Eli Glasner, Sergii Kolyada, Alejandro Maass. Journal für die Reine und Angewandte Mathematik, 2002

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The Li–Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand sufficient conditions for Li–Yorke chaos in a topological dynamical system are given. We solve a long–standing open question by proving that positive entropy implies Li–Yorke chaos. On the other hand properties of dynamical systems without Li–Yorke pairs are investigated; in addition to having entropy 0, they are minimal when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally it is proved that minimal systems without Li–Yorke pairs are disjoint from scattering systems.

Subfields: Computer Networks and Communications, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fb9ffbfab741f51bd326407165005d2e176ae06f

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Citations in the corpus

Stable sets and mean Li–Yorke chaos in positive entropy systems (2012, background): “For every n ≥ 2, deﬁne a measure l n ( m ) on ( X ( n ) , T ( n ) ) by letting where m It is well known that when m is ergodic with positive entropy (see for example [4, 18]), m x is non-atomic for m -a.e x ∈ X and l n ( m ) is a T ( n ) -invariant ergodic measure on X ( n ) .”
Local entropy theory (2009, methodology): “This was done by Blanchard et al in [9].”
A note on sensitivity of semigroup actions (2006, background): “Sensitive dependence on initial conditions is one of the basic ideas in several deflnitions of chaos (see, for example, [2, 5, 8, 4 ]).”
Homoclinic groups, IE groups, and expansive algebraic actions (2011, neutral): “[8] in the case of = Z.”
Stronger forms of sensitivity for dynamical systems (2007, background): “For a dynamical system, Li-Yorke chaos is known to be weaker than having positive topological entropy [ 7 ].”
Combinatorial Independence and Sofic Entropy (2012, background): “In that setting Blanchard, Glasner, Kolyada, and Maass showed that positive entropy implies Li-Yorke chaos [4].”
Independence in topological and C*-dynamics (2006, neutral): “1”
Recent development of chaos theory in topological dynamics (2015, methodology): “t was shownby Huang and Ye [46] that Devaneychaos implies Li-Yorke one by proving that a non-periodic transitive system with a periodic point is LiYorke chaotic; Blanchard, Glasner, Kolyada and Maass [15] proved that positive entropy also implies Li-Yorke chaos (we remark that the authors obtained t”
DISTRIBUTIONAL CHAOS REVISITED (2009, methodology): “This result gave a tool to prove that positive topological entropy implies the existence of a Cantor set which is a scrambled set at the same time [30] (it is remarkable that this statement holds in general compact metric spaces [10]).”
Positive topological entropy implies chaos DC2 (2011, background): “ e.g. [O1] for UPE systems or [S2] for some other cases). In this paper we solve the problem posed by Sm´ıtal by proving his conjecture in the general case, in particular, we strengthen the result of [BGKM]: Theorem 1.1. Assume that a topological dynamical system (X,T) has positive topological entro”
Hereditarily non-sensitive dynamical systems and linear representations (2004, background): “3. It is interesting to compare some of the current deflnitions of chaos and the corresponding classes of dynamical systems (see, for example, [15, 25, 10 ]) with the class of G-systems X such that Asp(X) = fconstantsg.”
Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations (2004, background): “In the qualitative theory of di$erential and di$erence equations nonlocal problems play the important role (See, for instance, [1–16] and the bibliography therein).”
Sufficient conditions under which a transitive system is chaotic (2009, methodology): “(This definition was introduced in [ BGKM02 ].”
On tame dynamical systems (2006, methodology): “For this we need the following (simplifled version of a) theorem of Blanchard, Glasner, Kolyada and Maass [ 5, Theorem 2.3 ].”
Graph theoretic structure of maps of the Cantor space (2012, background): “As is well-known, Li-Yorke chaos is the weakest of all notions of chaos [7].”