On the predictability of discrete dynamical systems II

Bernardes, Nilson, Jr.

doi:10.1090/S0002-9939-05-07924-4

On the predictability of discrete dynamical systems II
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Proc. Amer. Math. Soc. 133 (2005), 3473-3483 Request permission

Abstract:

Given a metrizable compact topological $n$-manifold $X$ with boundary and a metric $d$ compatible with the topology of $X$, we prove that “most” continuous functions $f : X \to X$ are non-sensitive at “most” points of $X$ but are sensitive at every point of an infinite set which is dense in the set of all periodic points of $f$. We also establish some results concerning sets of periodic points and non-wandering points.

References

S. J. Agronsky, A. M. Bruckner, and M. Laczkovich, Dynamics of typical continuous functions, J. London Math. Soc. (2) 40 (1989), no. 2, 227–243. MR 1044271, DOI 10.1112/jlms/s2-40.2.227
Ethan Akin, On chain continuity, Discrete Contin. Dynam. Systems 2 (1996), no. 1, 111–120. MR 1367390, DOI 10.3934/dcds.1996.2.111
Nilson C. Bernardes Jr., On the dynamics of homeomorphisms on the unit ball of $\textbf {R}^n$, Positivity 3 (1999), no. 2, 149–172. MR 1702645, DOI 10.1023/A:1009750622797
Nilson C. Bernardes Jr., On the predictability of discrete dynamical systems, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1983–1992. MR 1896031, DOI 10.1090/S0002-9939-01-06247-5
Reinhard Diestel, Graph theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag, New York, 1997. Translated from the 1996 German original. MR 1448665
James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
K. Simon, On the periodic points of a typical continuous function, Proc. Amer. Math. Soc. 105 (1989), no. 1, 244–249. MR 929418, DOI 10.1090/S0002-9939-1989-0929418-0