Approximating \(\omega\)-limit sets with periodic orbits (Q5919883)

Let \((I,f)\) be a discrete dynamical system where \(I=[0,1]\) and \(f:I\to I\) is a continuous mapping. The paper deals with the following question: which \(\omega\)-limit sets are contained in the Hausdorff closure of periodic orbits? More explicitly, if \((R,H)\) represents the Hausdorff metric space, then, for a given \(\omega\)-limit set \(\omega\), the authors study if for any \(\varepsilon>0\) it is possible to find a periodic orbit \(\alpha\) such that \(H (\omega,\alpha)<\varepsilon\). Let denote \(\Lambda(f)\) as \(\bigcup_{x\in I}\omega (x,f)\) where \(\omega (x,f)\) represents the \(\omega\)-limit set of the point \(x\) by \(f\). The statement of the main results is the following: for a map \(f\) such that each wandering interval converges to a periodic orbit, then \(\overline P(f)= \Lambda(f)\). On the other hand, there exists a map \(F\) with zero topological entropy for which the collection of \(\omega\)-limit sets not contained in \(\overline{P(F)}\) is uncountable and contains chains linearly ordered by inclusions of arbitrary countable order type.