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Choquet simplices as spaces of invariant probability measures on post-critical sets. (English) Zbl 1192.37053

The paper under review deals with topological dynamical systems. In this setting, it is well known that the space of invariant probability measures (i.p.m.’s) is a non-empty metrizable Choquet simplex (respect to the weak* topology). Reciprocally, T. Downarowicz [Isr. J. Math. 74, No. 2–3, 241–256 (1991; Zbl 0746.58047)] proved that every non-empty metrizable Choquet simplex \(\mathcal{C}\) can be realized (up to an affine homeomorphism) as the space of i.p.m.’s of the dynamical system generated by a minimal homeomorphism of a Cantor set.
Now, the authors give a more natural realization of \(\mathcal{C},\) since they prove that it arises as the space of i.p.m.’s associated to some \(\omega\)-limit sets of logistic maps. Namely, if \(f_{\lambda}(x)=\lambda x(1-x),\) \(x\in [0,1],\) and \(X_\lambda\) is the \(\omega\)-limit set of \(x=\frac{1}{2}\) respect to \(f_\lambda,\) it holds:
Main Theorem. For each \(\mathcal{C}\) there is a parameter \(\lambda\in (0,4]\) such that the post-critical set \(X_\lambda\) is a Cantor set, the restriction of \(f_\lambda\) to \(X_\lambda\) is minimal, and such that the space of i.p.m.’s supported by this set, endowed with the weak* topology, is affine homeomorphic to \(\mathcal{C}.\)
The proof of this result relies on the notions of cutting time sequences and kneading maps [see H. Bruin, G. Keller and M. St. Pierre, Ergodic Theory Dyn. Syst. 17, No. 6, 1267–1287 (1997; Zbl 0898.58012)].

MSC:

37E05 Dynamical systems involving maps of the interval
37A99 Ergodic theory
37B10 Symbolic dynamics
54H20 Topological dynamics (MSC2010)
54E99 Topological spaces with richer structures
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