Defining complete and observable chaos

• Autorzy: Víctor Jiménez López
• Języki publikacji: EN

Abstrakty

EN

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $lim inf_{n→∞} |f^n(x) - f^n(y)| = 0$ and $lim sup_{n→∞} |f^n(x) - f^n(y)| > 0$. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions of "complete" and "observable" chaos.

Słowa kluczowe

EN

chaos in the sense of Li and Yorke   dense chaos   generic chaos   full chaos   scrambled set  

Kategorie tematyczne

• 54H20: Topological dynamics
• 26A18: Iteration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1996
• Tom: 64
• Numer: 2
• Strony: 139-151

Daty

wydano

1996

otrzymano

1995-07-12

poprawiono

1995-12-20

Twórcy

autor

Víctor Jiménez López

• Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Aptdo. de Correos 4021, 30100 Murcia, Spain

Bibliografia

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