## $\omega$-chaos and topological entropy

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- by Shi Hai Li PDF
- Trans. Amer. Math. Soc.
**339**(1993), 243-249 Request permission

## Abstract:

We present a new concept of chaos, $\omega$-chaos, and prove some properties of $\omega$-chaos. Then we prove that $\omega$-chaos is equivalent to positive entropy on the interval. We also prove that $\omega$-chaos is equivalent to the definition of chaos given by Devaney on the interval.## References

- R. L. Adler, A. G. Konheim, and M. H. McAndrew,
*Topological entropy*, Trans. Amer. Math. Soc.**114**(1965), 309–319. MR**175106**, DOI 10.1090/S0002-9947-1965-0175106-9
G. Birkhoff, - 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
W. A. Coppel, - Robert L. Devaney,
*An introduction to chaotic dynamical systems*, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR**811850** - W. H. Gottschalk,
*Orbit-closure decompositions and almost periodic properties*, Bull. Amer. Math. Soc.**50**(1944), 915–919. MR**11436**, DOI 10.1090/S0002-9904-1944-08262-1 - Walter Helbig Gottschalk and Gustav Arnold Hedlund,
*Topological dynamics*, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR**0074810**, DOI 10.1090/coll/036 - Gustav A. Hedlund,
*Sturmian minimal sets*, Amer. J. Math.**66**(1944), 605–620. MR**10792**, DOI 10.2307/2371769 - Shi Hai Li,
*Dynamical properties of the shift maps on the inverse limit spaces*, Ergodic Theory Dynam. Systems**12**(1992), no. 1, 95–108. MR**1162402**, DOI 10.1017/S0143385700006611 - 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**
Y. Oono, - Marston Morse and Gustav A. Hedlund,
*Symbolic dynamics II. Sturmian trajectories*, Amer. J. Math.**62**(1940), 1–42. MR**745**, DOI 10.2307/2371431 - Melvyn B. Nathanson,
*Permutations, periodicity, and chaos*, J. Combinatorial Theory Ser. A**22**(1977), no. 1, 61–68. MR**424573**, DOI 10.1016/0097-3165(77)90063-2
A. N. Sharkovskii, - 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**
—, - J. Smítal,
*Chaotic functions with zero topological entropy*, Trans. Amer. Math. Soc.**297**(1986), no. 1, 269–282. MR**849479**, DOI 10.1090/S0002-9947-1986-0849479-9
J.-C. Xiong, - Jin Cheng Xiong,
*The attracting centre of a continuous self-map of the interval*, Ergodic Theory Dynam. Systems**8**(1988), no. 2, 205–213. MR**951269**, DOI 10.1017/S0143385700004429 - Jin Cheng Xiong,
*A chaotic map with topological entropy*, Acta Math. Sci. (English Ed.)**6**(1986), no. 4, 439–443. MR**924033**, DOI 10.1016/S0252-9602(18)30503-4

*Dynamical systems*, Amer. Math. Soc., Providence, R.I., 1927; reprinted 1990. J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey,

*On Devaney’s definition of chaos*, preprint (1990).

*Continuous maps of an interval*, Lecture Notes, IMA Preprint Series #26, Minneapolis, Minn., 1983.

*Period*$\ne {2^n}$

*implies chaos*, Progr. Theoret. Phys.

**59**(1978), 1028.

*About continuous maps on the set of*$\omega$-

*limit points*, Proc. Acad. Sci. Ukraine

**1965**, 1407-1410.

*On the properties of discrete dynamical systems*, Proc. Internat. Colloq. on Iterative Theory and Appl., Toulouse, 1982.

*A note on minimal sets of interval maps*, preprint (1986).

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**339**(1993), 243-249 - MSC: Primary 58F13; Secondary 58F08
- DOI: https://doi.org/10.1090/S0002-9947-1993-1108612-8
- MathSciNet review: 1108612