Chaotic functions with zero topological entropy

Smítal, J.

doi:10.1090/S0002-9947-1986-0849479-9

Chaotic functions with zero topological entropy
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by J. Smítal

Trans. Amer. Math. Soc. 297 (1986), 269-282

DOI: https://doi.org/10.1090/S0002-9947-1986-0849479-9

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Abstract:

Recently Li and Yorke introduced the notion of chaos for mappings from the class ${C^0}(I,I)$, where $I$ is a compact real interval. In the present paper we give a characterization of the class $M \subset {C^0}(I,I)$ of mappings chaotic in this sense. As is well known, $M$ contains the mappings of positive topological entropy. We show that $M$ contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of $M$ consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) $\delta$-chaos, distinguishable on a certain level $\delta > 0$.

References

Louis Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), no. 3, 576–580. MR 509258, DOI 10.1090/S0002-9939-1978-0509258-X
Louis Block, Stability of periodic orbits in the theorem of Šarkovskii, Proc. Amer. Math. Soc. 81 (1981), no. 2, 333–336. MR 593484, DOI 10.1090/S0002-9939-1981-0593484-8
G. J. Butler and G. Pianigiani, Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc. 18 (1978), no. 2, 255–265. MR 494303, DOI 10.1017/S0004972700008066

Maps of an interval

On the connection between periodicity and chaos of continuous functions and their iterates

19

J. Harrison, Wandering intervals, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 154–163. MR 654888
I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45–49. MR 749887, DOI 10.1090/S0002-9939-1984-0749887-4
A. N. Šarkovskiĭ and H. K. Kenžegulov, On properties of the set of limit points of an iterative sequence of a continuous function, Volž. Mat. Sb. Vyp. 3 (1965), 343–348 (Russian). MR 0199316
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394

Topologie

T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778

Chaos almost everywhere

O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž. 16 (1964), 61–71 (Russian, with English summary). MR 0159905
O. M. Šarkovs′kiĭ, On cycles and the structure of a continuous mapping, Ukrain. Mat. Ž. 17 (1965), no. 3, 104–111 (Russian). MR 0186757
A. N. Šarkovskiĭ, The behavior of the transformation in the neighborhood of an attracting set, Ukrain. Mat. Ž. 18 (1966), no. 2, 60–83 (Russian). MR 0212784

The partially ordered system of attracting sets

7

A. N. Šarkovskiĭ, Attracting sets containing no cycles, Ukrain. Mat. Ž. 20 (1968), no. 1, 136–142 (Russian). MR 0225314

On some properties of discrete dynamical systems

J. Smítal and K. Smítalová, Erratum: “Structural stability of nonchaotic difference equations” [J. Math. Anal. Appl. 90 (1982), no. 1, 1–11; MR0680860 (84d:58046)], J. Math. Anal. Appl. 101 (1984), no. 1, 324. MR 746238, DOI 10.1016/0022-247X(84)90069-6
J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), no. 1, 54–56. MR 677230, DOI 10.1090/S0002-9939-1983-0677230-7
I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45–49. MR 749887, DOI 10.1090/S0002-9939-1984-0749887-4
K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), no. 2, 283–292. MR 854575, DOI 10.1017/S0004972700010157
P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), no. 3, 237–248. MR 445556, DOI 10.1007/BF01614086
M. B. Vereĭkina and A. N. Sharkovskiĭ, Recurrence in one-dimensional dynamical systems, Approximate and qualitative methods of the theory of functional differential equations, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1983, pp. 35–46 (Russian). MR 753681