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On the set of points with a dense orbit. (English) Zbl 0947.37005
Summary: Under certain conditions on the topological space \(X\) we prove that for every continuous map \(f : X \to X\) the set of all points with a dense orbit has empty interior in \(X\). This result implies a negative answer to two problems proposed by M. Barge and J. Kennedy.

MSC:
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37C35 Orbit growth in dynamical systems
54H20 Topological dynamics (MSC2010)
Keywords:
dense orbit
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References:
[1] M. Barge and J. Kennedy, Continuum theory and topological dynamics. In Open Problems in Topology, J. van Mill and G. M. Reed, editors, pages 633-644. Elsevier Science Publishers B. V. (North-Holland), 1990. CMP 91:03
[2] C. J. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), no. 1, 1 – 40. · Zbl 0782.47002
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