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

• [AJS] L. Alsedà, V. Jiménez López and L'. Snoha, On 1-difactors of Markov graphs and the prevalence of simple solenoids, preprint, 1995.
• [BJ] F. Balibrea and V. Jiménez López, A structure theorem for C² functions verifying the Misiurewicz condition, in: Proceedings of the European Conference on Iteration Theory (ECIT 91), Lisbon, 1991, World Sci., Singapore, 1992, 12-21.
• [BH] A. M. Bruckner and T. Hu, On scrambled sets and chaotic functions, Trans. Amer. Math. Soc. 301 (1987), 289-297.
• [BKNS] H. Bruin, G. Keller, T. Nowicki and S. van Strien, Absorbing Cantor sets in dynamical systems: Fibonacci maps, preprint Stony Brook 1994/2.
• [CE] P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progr. Phys. 1, Birkhäuser, Boston, 1980.
• [Ge1] T. Gedeon, There are no chaotic mappings with residual scrambled sets, Bull. Austral. Math. Soc. 36 (1987), 411-416.
• [Ge2] T. Gedeon, Generic chaos can be large, Acta Math. Univ. Comenian. 54/55 (1988), 237-241.
• [Gu] J. Guckenheimer, Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), 133-160.
• [JaS] K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283-292.
• [Ji1] V. Jiménez López, C¹ weakly chaotic functions with zero topological entropy and non-flat critical points, Acta Math. Univ. Comenian. 60 (1991), 195-209.
• [Ji2] V. Jiménez López, Large chaos in smooth functions of zero topological entropy, Bull. Austral. Math. Soc. 46 (1992), 271-285.
• [Ji3] V. Jiménez López, Paradoxical functions on the interval, Proc. Amer. Math. Soc. 120 (1994), 465-473.
• [Ji4] V. Jiménez López, Order and chaos for a class of piecewise linear maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1379-1394.
• [JiS] V. Jiménez López and L'. Snoha, There are no piecewise linear maps of type $2^∞$, preprint, 1994.
• [Ka] I. Kan, A chaotic function possessing a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 45-49.
• [Ku] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
• [LY] T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.
• [Li] G.-F. Liao, A note on generic chaos, Ann. Polon. Math. 59 (1994), 99-105.
• [MT] M. Martens and C. Tresser, Forcing of periodic orbits and renormalization of piecewise affine maps, preprint Stony Brook 1994/17.
• [Mi] M. Misiurewicz, Chaos almost everywhere, in: Iteration Theory and its Functional Equations, Lecture Notes in Math. 1163, Springer, Berlin, 1985, 125-130.
• [Pi1] J. Piórek, On the generic chaos in dynamical systems, Univ. Iagell. Acta Math. 25 (1985), 293-298.
• [Pi2] J. Piórek, On generic chaos of shifts in function spaces, Ann. Polon. Math. 52 (1990), 139-146.
• [Pi3] J. Piórek, On weakly mixing and generic chaos, Univ. Iagell. Acta Math. 28 (1991), 245-250.
• [Sm1] J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), 54-56.
• [Sm2] J. Smítal, A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 50-54.
• [Sm3] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-282.
• [Sn1] L'. Snoha, Generic chaos, Comment. Math. Univ. Carolin. 31 (1990), 793-810.
• [Sn2] L'. Snoha, Dense chaos, Comment. Math. Univ. Carolin. 33 (1992), 747-752.
• [Sn3] L'. Snoha, Two-parameter chaos, Acta Univ. M. Belii 1 (1993), 3-6.