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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.

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37C35 Orbit growth in dynamical systems
54H20 Topological dynamics (MSC2010)
dense orbit
Full Text: DOI
[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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