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Stability of dynamical structures under perturbation of the generating function. (English) Zbl 1161.37017
Set valued functions \(C\), \(NW\) and \({\mathcal L}\) taking \(f\) in \(C(I,I)\) to its centre \(C(f)\), its set of nonwandering points \(NW(f)\) and its colleciton of \(\omega\)-limit sets \({\mathcal L}(f)= \{\omega(x,f): x\in I\}\) are considered. The problem of how these sets are affected by perturbations of \(f\) is studied. It is proved that either of this maps \(C\) and \(NW\) is continuous at \(g\) if and only if one of the following conditions is satisfied:
(i) The map \(\omega\) which takes a function \(f\) to its set \(\omega(f)\) of \(\omega\)-limit points is continuous at \(g\);
(ii) the periodic orbits of \(g\) which are \(p\)-stable, i.e. stable with respect to small perturbations of \(g\), are dense in the set \(CR(g)\) of chain recurrent points of \(g\);
(iii) \(CR(g)= \omega(g)\) and the \(p\)-stable periodic orbits of \(g\) are dense in the set of periodic points of \(g\).

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37E05 Dynamical systems involving maps of the interval
26A18 Iteration of real functions in one variable
54H20 Topological dynamics (MSC2010)
Full Text: DOI
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