Characterizations of weakly chaotic maps of the interval

Authors:
V. V. Fedorenko, A. N. Šarkovskii and J. Smítal

Journal:
Proc. Amer. Math. Soc. **110** (1990), 141-148

MSC:
Primary 58F20; Secondary 28D20, 54H20, 58F08, 58F13

DOI:
https://doi.org/10.1090/S0002-9939-1990-1017846-5

MathSciNet review:
1017846

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Abstract: We prove, among others, the following relations between notions of chaos for continuous maps of the interval: (i) A map is not chaotic in the sense of Li and Yorke iff restricted to the set of its -limit points is stable in the sense of Ljapunov. (ii) The topological entropy of is zero iff restricted to the set of chain recurrent points is not chaotic in the sense of Li and Yorke, and this is iff every trajectory is approximable by trajectories of periodic intervals.

**[1]**A. M. Blokh,*Limit behavior of one-dimensional dynamic systems*, Uspekhi Mat. Nauk**37**(1982), no. 1(223), 137–138 (Russian). MR**643772****[2]**Louis Block,*Simple periodic orbits of mappings of the interval*, Trans. Amer. Math. Soc.**254**(1979), 391–398. MR**539925**, https://doi.org/10.1090/S0002-9947-1979-0539925-9**[3]**Rufus Bowen,*Topological entropy and axiom 𝐴*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR**0262459****[4]**K. Janková and J. Smítal,*A characterization of chaos*, Bull. Austral. Math. Soc.**34**(1986), no. 2, 283–292. MR**854575**, https://doi.org/10.1017/S0004972700010157**[5]**M. Kuchta and J. Smítal,*Two-point scrambled set implies chaos*, European Conference on Iteration Theory (Caldes de Malavella, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 427–430. MR**1085314****[6]**M. Misiurewicz and J. Smítal,*Smooth chaotic maps with zero topological entropy*, Ergodic Theory Dynam. Systems**8**(1988), no. 3, 421–424. MR**961740**, https://doi.org/10.1017/S0143385700004557**[7]**D. Preiss and J. Smítal,*A characterization of nonchaotic continuous maps of the interval stable under small perturbations*, Trans. Amer. Math. Soc.**313**(1989), no. 2, 687–696. MR**997677**, https://doi.org/10.1090/S0002-9947-1989-0997677-9**[8]**O. M. Šarkovs′kiĭ,*Fixed points and the center of a continuous mapping of the line into itself*, Dopovidi Akad. Nauk Ukraïn. RSR**1964**(1964), 865–868 (Ukrainian, with Russian and English summaries). MR**0165178****[9]**-,*The partially ordered system of attracting sets*, Soviet Math. Dokl.7 (1966), 1384-1386.**[10]**A. N. Sharkovskiĭ, Yu. L. Maĭstrenko, and E. Yu. Romanenko,*\cyr Raznostnye uravneniya i ikh prilozheniya*, “Naukova Dumka”, Kiev, 1986 (Russian). MR**895825****[11]**A. N. Sharkovskiĭ, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko,*\cyr Dinamika odnomernykh otobrazheniĭ*, “Naukova Dumka”, Kiev, 1989 (Russian). MR**1036027****[12]**J. Smítal,*Chaotic maps with zero topological entropy*, Trans. Amer. Math. Soc.**297**(1986), 269-282.**[13]**Jin Cheng Xiong,*Set of almost periodic points of a continuous self-map of an interval*, Acta Math. Sinica (N.S.)**2**(1986), no. 1, 73–77. MR**877371**, https://doi.org/10.1007/BF02568524**[14]**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**

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1017846-5

Article copyright:
© Copyright 1990
American Mathematical Society