Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-10T17:07:09.787Z Has data issue: false hasContentIssue false
  • English
  • Français

On dynamics of triangular maps of the square

Published online by Cambridge University Press:  19 September 2008

S. F. Kolyada
S. F. Kolyada
Affiliation:
Institute of Mathematics, Ukrainian Academy of Sciences, Repin str. 3, 252601 Kiev-4, Ukraine
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 749 - 768
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AKM]Adler, R., Konheim, A. & McAndrew, M.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[B]Birkhoff, G. D.. Über gewisse Zentralbevegungen dynamischer Systeme. Ges. Wiss. Gottingen Nachr., Math.-Phys. Klasse. (1926), 81–92.Google Scholar
[BS]Birkhoff, G. D. & Smith, P. A.. Structure analysis of surface transformations. J. Math. Pures Appl. 7 (1928), 345–379.Google Scholar
[Bl1]Block, L.. Stability of periodic orbits in the theorem of Sharkovskii. Proc. Amer. Math. Soc. 81 (1981), 333–336.Google Scholar
[Bl2]Block, L.. Homoclinic points of mappings of the interval. Proc Amer. Math. Soc. 72 (1978), 576–580.CrossRefGoogle Scholar
[Bow1]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[Bow2]Bowen, R.. Periodic points and measures for axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[Bow3]Bowen, R.. Topological entropy and axiom A. Global Anal., Proc. Symp. in pure Math., Amer. Math. Soc. 14 (1970), 23–41.CrossRefGoogle Scholar
[BF]Bowen, R. & Franks, J.. The periodic points of maps of the disk and the interval. Topology 15 (1976), 337–342.CrossRefGoogle Scholar
[Br]Bronshtein, I. U.. Non-autonomous dynamical systems. Kishinev: Shtiintsa. (1984) (Russian).Google Scholar
[CE]Collet, P. & Eckmann, J.-P.. Iterated maps on the interval as dynamical systems. Progress in Physics. Birkhauser, 1980.Google Scholar
[CH]Coven, E. M. & Hedlund, G. A.. for maps of the interval. Proc. Amer. Math. Soc. 79 (1980), 316–318.Google Scholar
[D]Dinaburg, E. I.. Connection with various entropy characterizations of dynamical systems. Izvestija AN SSSR 35 (1971), 324–366 (Russian).Google Scholar
[Ef]Efremova, L. S.. On the nonwandering set and the centre of triangular maps with closed set of periodic points in the base, (to appear) (Russian).Google Scholar
[FSh]Fedorenko, V. V. & Sharkovsky, A. N.. On coexistence of periodic and homoclinic trajectories. Kishinev: Shtiintsa (1979), 174–175 (Russian).Google Scholar
[Kl]Kloeden, P. E.. On Sharkovsky's cycle coexistence ordering. Bull Austr. Math. Soc. 20 (1979), 171–177.CrossRefGoogle Scholar
[Ko]Kolyada, S. F.. On triangular maps of type 2∞ with positive entropy. Dynamical systems and turbulence. Inst. Math. Ukrain. Acad. Sci., Kiev (1989), 76–82 (Russian).Google Scholar
[KoSh]Kolyada, S. F. & Sharkovsky, A. N.. On topological dynamics of triangular maps of the plane. ECIT-89, Austria: Batschuns (to appear).Google Scholar
[KoSh1]Kolyada, S. F. & Sharkovsky, A. N.. On depth of a center of dynamical system, (to appear.)Google Scholar
[M]Milnor, J.. On the concept of attractor. Commun. Math. Phys. 99 (1985), 177–195.CrossRefGoogle Scholar
[MT]Milnor, J. & Thurston, W.. On iterated maps of the interval. Springer Lecture Notes in Mathematics 1342 Springer Verlag, Berlin, 1988, pp. 465–563.Google Scholar
[M1]Misiurewicz, M.. Horseshoes for mappings of the interval. Bull. Acad. Pol. Sci. 27 (1979), 167–169.Google Scholar
[M2]Misiurewicz, M.. Structure of mappings of an interval with zero entropy. Publ. Math. IHES 53 (1981), 5–16.CrossRefGoogle Scholar
[MN]Misiurewicz, M. & Nitecki, Z.. Combinatorial patterns for maps of the interval. Mathematica Gottingensis 35 (1989).Google Scholar
[MS]Misiurewicz, M. & Szlenk, W.. Entropy of piecewise monotone mappings. Studia Mathematica 67 (1980), 45–63.CrossRefGoogle Scholar
[Ni]Nitecki, Z.. Topological dynamics on the interval. Ergod. Theory & Dynam. Sys. II (Proc. Special Year, Maryland 1978–80), ed. Katok, A.. Birkhauser, 1982, pp. 1–73.Google Scholar
[Sh]Sharkovsky, A. N.. On ω-limit sets of discrete dynamical systems. Avtor. Dis. Dok. Phys.-Math. Nauk, Kiev, 1967, p. 9.Google Scholar
[Sh1]Sharkovsky, A. N.. Coexistence of cycles of a continuous map of the line into itself. Ukrain. Math. J. 16 (1964), 61–71 (Russian).Google Scholar
[Sh2]Sharkovsky, A. N.. Some problems of ODE theory. Uspehi Mat. Nauk. 38 (1983), 172 (Russian).Google Scholar
[Sh3]Sharkovsky, A. N.. On cycles and structure of a continuous map. Ukrain. Math. J. 17 (1965), 104–111 (Russian).Google Scholar
[Sh4]Sharkovsky, A. N.. Nonwandering points and the center of a continuous mappings of the line into itself. Dopovidi Ukrain. Acad. Sci. 7 (1964), 865–868 (Ukrainian).Google Scholar
[Sh5]Sharkovsky, A. N.. On one theorem of G. Birkhoff. Dopovidi Ukrain. Acad. Sci. 5 (1967), 429–432 (Ukrainian).Google Scholar
[Sh6]Sharkovsky, A. N.. On isomorphism problem of dynamical systems. ICNO-69. Kiev: Naukova Dumka 2 (1970), 541–545 (Russian).Google Scholar
[Sh7]Sharkovsky, A. N.. The partially-ordered system of attracting sets. Dokl. AN SSSR 170 (1966), 1276–1278 (Russian).Google Scholar
[ShKSF]Sharkovsky, A. N., Kolyada, S. F., Fedorenko, V. V. & Sivak, A. G.. Dynamics of one-dimensional maps. Kiev: Naukova Dumka, 1989, (Russian).Google Scholar
[VSh]Vereikina, M. B. & Sharkovsky, A. N.. Returnability in one-dimensional dynamical systems. Approx. and Qualit. Meth. in Theory Q.-F.E. Inst. Math. Ukrain. Acad. Sci., Kiev, 1983, pp. 35–46 (Russian).Google Scholar