×

Stable and non-stable non-chaotic maps of the interval. (English) Zbl 0762.58014

The paper deals with piecewise monotonic functions of the interval on itself. Other than the construction of a non-chaotic non-stable map, the author shows that every piecewise monotonic non-chaotic function is stable and that for any given piecewise monotonic function with zero topological entropy, every infinite \(\omega\)-limit set is perfect.

MSC:

37E99 Low-dimensional dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C75 Stability theory for smooth dynamical systems
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] DEN JOY A.: Sur les courbes definies les equations différentielles a la surface du tore. J. Math. Pures Appl., IX. Ser. 11 (1932), 333-375. · JFM 58.1124.04
[2] GEDEON T.: There are no chaotic mappings with residual scrambled sets. Bull. Austr. Math. Soc. 36 (1987), 411-416. · Zbl 0646.26008 · doi:10.1017/S0004972700003695
[3] HARRISON J.: Wandering intervals. Dynamical Systems and Turbulence. (Warwick 1980), Lecture Notes in Math, vol 898, Springer Berlin, Heidelberg and N. Y., 1981, pp. 154-163.
[4] JANKOVÁ K., SMÍTAL J.: A characterization of chaos. Bull. Austr. Math. Soc. 34 (1986), 283-293. · Zbl 0577.54041 · doi:10.1017/S0004972700010157
[5] JANKOVÁ K., SMÍTAL J.: A Theorem of Sarkovskii characterizing continuous map with zero topological entropy. Math. Slovaca
[6] PREISS D., SMÍTAL J.: A characterization of non-chaotic continuous mappings of the interval stable under small perturbations. Trans. Am. Math. Soc.
[7] SMÍTAL J.: Chaotic functions with zero topological entropy. Trans. Am. Math. Soc. 297 (1986), 269-282. · Zbl 0639.54029 · doi:10.2307/2000468
[8] SMÍTAL J.: A chaotic function with scrambled set of positive Lebesque measure. Proc. Am. Math. Soc. 92 (1984), 50-54. · Zbl 0592.26006 · doi:10.2307/2045151
[9] ŠARKOVSKII A. N.: The behaviour of a map in a neighbourhood of an attracting set. (Russian), Ukrain. Math. Zh. 18 (1966), 60-83.
[10] ŠARKOVSKII A. N.: Attracting sets containing no cycles. (Russian), Ukr. Math. Zh. 20 (1968), 136-142.
[11] ŠARKOVSKII A. N.: On cycles and structure of continuous mappings. (Russian), Ukr. Math. Zh. 17 (1965), 104-111.
[12] ŠARKOVSKII A. N.: A mapping with zero topological entropy having continuum minimal Cantor sets. (Russian). Dynamical Systems and Turbulence. Kiev, 1989, pp. 109-115.
[13] van STRIEN S. J.: Smooth dynamics on the interval. Preprint (1987).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.