On the predictability of discrete dynamical systems
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- by Nilson C. Bernardes Jr. PDF
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Abstract:
Let $X$ be a metric space. A function $f: X \to X$ is said to be non-sensitive at a point $a \in X$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that for any choice of points $a_0 \in B(a;\delta )$, $a_1 \in B(f(a_0);\delta )$, $a_2 \in B(f(a_1);\delta ),\ldots$, we have that $d(a_m,f^m(a)) < \epsilon$ for every $m \geq 0$. Let $H(X)$ be the set of all homeomorphisms from $X$ onto $X$ endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces $X$, “most” functions in $H(X)$ are non-sensitive at “most” points of $X$.References
- N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994. Recent advances. MR 1289410
- 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
- R. Diestel, Graph Theory, Springer-Verlag, 1997.
- James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521–554. MR 121804, DOI 10.2307/1970228
- James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- Kenzi Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc. 110 (1990), no. 1, 281–284. MR 1009998, DOI 10.1090/S0002-9939-1990-1009998-8
- S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), no. 3, 253–266. MR 1711347, DOI 10.1016/S0166-8641(98)00062-5
- Morgan Ward, Note on the general rational solution of the equation $ax^2-by^2=z^3$, Amer. J. Math. 61 (1939), 788–790. MR 23, DOI 10.2307/2371337
- Koichi Yano, Generic homeomorphisms of $S^1$ have the pseudo-orbit tracing property, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 1, 51–55. MR 882123
Additional Information
- Nilson C. Bernardes Jr.
- Affiliation: Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, 24020-140, Niterói, RJ, Brasil
- Email: bernardes@mat.uff.br
- Received by editor(s): August 10, 1999
- Received by editor(s) in revised form: January 16, 2001
- Published electronically: November 21, 2001
- Communicated by: Alan Dow
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1983-1992
- MSC (2000): Primary 37B25, 37B20, 54H20
- DOI: https://doi.org/10.1090/S0002-9939-01-06247-5
- MathSciNet review: 1896031