TAILIEUCHUNG - Li–Yorke chaos for invertible mappings on noncompact spaces
In this paper, we give two examples to show that an invertible mapping being Li–Yorke chaotic does not imply its inverse being Li–Yorke chaotic, in which one is an invertible bounded linear operator on an infinite dimensional Hilbert space and the other is a homeomorphism on the unit open disk. | Turk J Math (2016) 40: 411 – 416 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Li–Yorke chaos for invertible mappings on noncompact spaces Bingzhe HOU∗, Lvlin LUO College of Mathematics, Jilin University, Changchun, P. R. China • Received: Accepted/Published Online: • Final Version: Abstract: In this paper, we give two examples to show that an invertible mapping being Li–Yorke chaotic does not imply its inverse being Li–Yorke chaotic, in which one is an invertible bounded linear operator on an infinite dimensional Hilbert space and the other is a homeomorphism on the unit open disk. Moreover, we use the last example to prove that Li–Yorke chaos is not preserved under topological conjugacy. Key words: Invertible dynamical systems, Li–Yorke chaos, noncompact spaces, topological conjugacy 1. Introduction In this paper, we are interested in the invertible dynamical system (X, f ) , where X is a metric space and f : X → X is a homeomorphism. There is a natural problem in invertible dynamical systems as follows. Question Let (X, f ) be an invertible dynamical system. If f has a dynamical property P, does its inverse f −1 also have property P? It is not difficult to see that the answer is positive for many properties such as transitivity, mixing, and Devaney chaos. However, the conclusion for Li–Yorke chaos, which was defined by Li and Yorke in [2] in 1975, is not known. Definition Let (X, f ) be a dynamical system. {x, y} ⊆ X is said to be a Li–Yorke chaotic pair if lim sup d(f n (x), f n (y)) > 0 and lim inf d(f n (x), f n (y)) = 0. n→+∞ n→+∞ Furthermore, f is called Li–Yorke chaotic if there exists an uncountable subset Γ ⊆ X such that each pair of two distinct points in Γ is a Li–Yorke chaotic pair. In the present article, we focus on invertible dynamical systems on noncompact metric spaces, and then we study Li–Yorke chaos for invertible
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