X-minimal patterns and a generalization of Sharkovskiĭ's theorem

• Autorzy: Jozef Bobok , Milan Kuchta
• Języki publikacji: EN

Abstrakty

EN

We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.

Słowa kluczowe

EN

iteration; periodic orbit; cycle; pattern; minimal; forcing relation; Sharkovskiĭ s theorem

Kategorie tematyczne

• 54H20: Topological dynamics
• 26A18: Iteration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 156
• Numer: 1
• Strony: 33-66

Daty

wydano

1998

otrzymano

1996-10-24

poprawiono

1997-04-22

Twórcy

autor

Jozef Bobok

• KM FSv. ČVUT, Thákurova 7, 166 29 Praha 6, Czech Republic,

autor

Milan Kuchta

• athematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovak Republic,

Bibliografia

• [ALM] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynam. 5, World Sci., Singapore, 1993.
• [ALS] L. Alsedà, J. Llibre and R. Serra, Mimimal periodic orbits for continuous maps of the interval, Trans. Amer. Math. Soc. 286 (1984), 595-627.
• [B] S. Baldwin, Generalizations of a theorem of Sharkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), 111-127.
• [B1] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391-398.
• [B2] L. Block, Periodic orbits of continuous mappings of the circle, ibid. 260 (1980), 553-562.
• [BGMY] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimensional maps, in: Global Theory of Dynamical Systems, Lecture Notes in Math. 819, Springer, Berlin, 1980, 18-34.
• [Bl] A. Blokh, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, Ergodic Theory Dynam. Systems 15 (1995), 1331-1337.
• [BM] A. Blokh and M. Misiurewicz, Entropy of twist interval maps, MSRI Preprint No. 041-94, Math. Sci. Res. Inst., Berkeley, 1994.
• [BK] J. Bobok and M. Kuchta, Invariant measures for maps of the interval that do not have points of some period, Ergodic Theory Dynam. Systems 14 (1994), 9-21.
• [C] W. A. Coppel, Šarkovskii-minimal orbits, Math. Proc. Cambridge Philos. Soc. 93 (1983), 397-408.
• [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, Oxford, 1960.
• [LMPY] T.-Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc. 273 (1982), 191-199.
• [S] A. N. Sharkovskiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Zh. 16 (1964), 61-71 (in Russian).
• [St] P. Štefan, A theorem of Šarkovskiĭ on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248.