Multiple recurrence theorems for set-valued maps
In [Compos. Math. 1, 177--179 (1934; JFM 60.0359.01)] \textit{A. Khintchine} strengthened the Poincaré recurrence theorem, showing that if \(T\) is a measure-preserving transformation of a probability space \((X,{\mathcal {B}},\mu)\) and if \(A\in {\mathcal {B}}\) with \(\mu (A) > 0\), then for every \(\varepsilon > 0\), the set \( \{n\in {\mathbb {N}}: \mu (A \cap T^{-n}(A)) > \mu (A)^2 - \varepsilon\}\) is syndetic. In this paper the authors extend the Khinchine's recurrence theorem, the Furstenberg's multiple recurrence theorem, and the multiple Birkhoff recurrence theorem from single valued maps to set-valued maps.
scientific article; zbMATH DE number 6756245