Devaney’s chaos implies existence of 𝑠-scrambled sets

DOI10.1090/S0002-9939-04-07514-8zbMath1055.54019OpenAlexW1844563096MaRDI QIDQ4813695

Jie Hua Mai

Publication date: 13 August 2004

Published in: Proceedings of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0002-9939-04-07514-8

zbMATH Keywords

sensitivitytransitivityDevaney's chaosscrambled setsynchronously proximal set

Mathematics Subject Classification ID

Strange attractors, chaotic dynamics of systems with hyperbolic behavior (37D45) Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) (37B05)

Related Items (23)

Invariant scrambled sets, uniform rigidity and weak mixing ⋮ Sensitivity of dendrite maps ⋮ Transitivity, mixing and chaos for a class of set-valued mappings ⋮ The set-valued mapping induced by a non-minimal transitive system is Li-Yorke chaotic ⋮ The nonexistence of sensitive commutative group actions on dendrites ⋮ The simplest erasing substitution ⋮ Cowen-Douglas function and its application on chaos ⋮ Cardinalities of scrambled sets and positive scrambled sets ⋮ Devaney chaos plus shadowing implies distributional chaos ⋮ Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions ⋮ Chaotic behaviour of the map \(x\mapsto\omega(x,f)\) ⋮ A class of primitive substitutions and scrambled sets ⋮ Chaos in a topologically transitive semi-flow ⋮ Chaos for Cowen-Douglas operators ⋮ Chaos among self-maps of the Cantor space ⋮ Transitivities of maps of general topological spaces ⋮ Chaotic and topological properties of \(\beta \)-transformations ⋮ Sufficient conditions under which a transitive system is chaotic ⋮ Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics ⋮ ON INVARIANT ε-SCRAMBLED SETS ⋮ Devaney's chaos implies distributional chaos in a sequence ⋮ Devaney's chaotic on induced maps of hyperspace ⋮ Recent development of chaos theory in topological dynamics

Cites Work

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• Continuous maps with the whole space being a scrambled set
• Homeomorphisms with the whole compacta being scrambled sets
• Recurrent Homeomorphisms on ℝ 2 are Periodic
• On Devaney's Definition of Chaos
• Period Three Implies Chaos
• Recurrent surface homeomorphisms
• Scrambled sets of continuous maps of 1-dimensional polyhedra
• Sensitive dependence on initial conditions
• Topological complexity
• On the set of points with a dense orbit
• CHAOTIC PROPERTIES OF SUBSHIFTS GENERATED BY A NONPERIODIC RECURRENT ORBIT
• Devaney's chaos or 2-scattering implies Li-Yorke's chaos

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