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On the structure of the space of continuous maps with zero topological entropy. (English) Zbl 0859.54026
Let \(C(I,I)\) be the space of all \(I\to I\) continuous functions with the uniform metric and \(Z(I,I)\) be the set of all continuous \(I\to I\) functions with zero topological entropy. Functions from \(Z(I,I)\) are nonchaotic in the following sense; any trajectory is approximable by cycles of \(f\). The paper concerns the stability of nonchaotic functions with zero entropy. Recall that a nonchaotic function is stable if for any \(\varepsilon>0\), any \(g\in C(I,I)\) sufficiently near to \(f\) has each trajectory approximable by cycles of \(f\). It is shown that the set of stable functions \(S(I,I)\subset Z(I,I)\) is dense in \(Z(I,I)\), and other results concerning the topological structure of the space \(Z(I,I)\) are given, too.
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
28D20 Entropy and other invariants
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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