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Generic homeomorphisms have the pseudo-orbit tracing property. (English) Zbl 0713.58025
Let M be a compact differentiable manifold, dim $$M\leq 3$$, Homeo(M) be the space of all homeomorphisms of M with $$C^ 0$$-topology. The author shows that the following properties of $$f\in Homeo(M)$$ are generic: f has the pseudo-orbit tracing property; f is $$C^ 0$$ tolerance stable; f is not topologically stable.
These results for the case $$M=S^ 1$$ were previously obtained by K. Yano [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 51-55 (1987; Zbl 0643.58002)].
Reviewer: Yu.E.Gliklikh

##### MSC:
 37C75 Stability theory for smooth dynamical systems 54H20 Topological dynamics (MSC2010)
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##### References:
 [1] Mike Hurley, Consequences of topological stability, J. Differential Equations 54 (1984), no. 1, 60 – 72. · Zbl 0493.58013 · doi:10.1016/0022-0396(84)90142-6 · doi.org [2] James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521 – 554. · Zbl 0108.18101 · doi:10.2307/1970228 · doi.org [3] Zbigniew Nitecki, On semi-stability for diffeomorphisms, Invent. Math. 14 (1971), 83 – 122. · Zbl 0218.58007 · doi:10.1007/BF01405359 · doi.org [4] Kenzi Odani, Generic homeomorphisms have the POTP, Bifurcation phenomena in nonlinear systems and theory of dynamical systems (Kyoto, 1989) World Sci. Adv. Ser. Dynam. Systems, vol. 8, World Sci. Publ., Teaneck, NJ, 1990, pp. 23 – 36. · Zbl 0713.58025 [5] J. Palis, C. Pugh, M. Shub, and D. Sullivan, Genericity theorems in topological dynamics, Dynamical systems — Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 241 – 250. Lecture Notes in Math., Vol. 468. · Zbl 0341.54053 [6] Michael Shub, Structurally stable diffeomorphisms are dense, Bull. Amer. Math. Soc. 78 (1972), 817 – 818. · Zbl 0265.58003 [7] M. Shub and D. Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109 – 132. · Zbl 0408.58023 · doi:10.1016/0040-9383(75)90022-1 · doi.org [8] Floris Takens, On Zeeman’s tolerance stability conjecture, Manifolds — Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 209 – 219. [9] Floris Takens, Tolerance stability, Dynamical systems — Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 293 – 304. Lecture Notes in Math., Vol. 468. · Zbl 0321.54022 [10] Peter Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 231 – 244. · Zbl 0403.58019 [11] J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math. (2) 73 (1961), 154 – 212. · Zbl 0096.37802 · doi:10.2307/1970286 · doi.org [12] Koichi Yano, A remark on the topological entropy of homeomorphisms, Invent. Math. 59 (1980), no. 3, 215 – 220. · Zbl 0434.54010 · doi:10.1007/BF01453235 · doi.org [13] Koichi Yano, Generic homeomorphisms of \?\textonesuperior have the pseudo-orbit tracing property, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 1, 51 – 55. · Zbl 0643.58002
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