2015/10/27 by Michal Málek
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #CHAOS (operating system) #Chaos control and synchronization #Computer science #Dimension (graph theory) #Invariant (physics) #Limit (mathematics) #Limit set #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Omega #Physics #Pure mathematics #Quantum mechanics #math.DS #msc:37B05 #msc:37B40 #msc:37E25
paper · pdf · doi:10.1080/10236198.2015.1106485
6 pages
arxiv created 2015/10/27 · arxiv updated 2015/10/29 · openalex publication_date 2015/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.