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A fixed point theorem for bounded dynamical systems. (English) Zbl 1014.54028
Ill. J. Math. 46, No. 2, 491-495 (2002); addendum ibid. 48, No. 3, 1079-1080 (2004).
Suppose that there is given a dynamical (discrete or continuous) system on a topological space \(X\). The authors propose the following terminology: a compact set \(W\) (contained in \(X)\) is said to be a window for the dynamical system if the forward orbit of every point \(x\in X\) intersects \(W\). If a dynamical system has a window, then it is called a bounded dynamical system. The main result of the paper is the following Theorem 1: Every bounded dynamical system on \(\mathbb R^n\) has a fixed point. Certain corollaries and remarks concerning extensions of these for locally compact topological spaces are presented. The proof of the main result is based on the Lefschetz Fixed Point Theorem and on the following Lemma: Every bounded dynamical system has a forward invariant window.
{Editor’s remark: In the Addendum the authors give priority to G. Fournier (for maps) and R. Srzednicki (for flows) of his Theorem 1.}

54H25 Fixed-point and coincidence theorems (topological aspects)
37B30 Index theory for dynamical systems, Morse-Conley indices
37B25 Stability of topological dynamical systems
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