\(\mathrm{IP}_r^\ast\)-recurrence and nilsystems
The paper deals with the notion of \(\mathrm{IP}^{*}\)-recurrence introduced by \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)]. The authors show that translations on compact nilmanifolds (which are known to be distal) are characterized by a stronger property of \(\mathrm{IP}^{*}_r\)-recurrence. It is also proved that the property of \(\mathrm{IP}^{*}_r\)-recurrence is equivalent to ostensibly, a much stronger property of polynomial \(\mathrm{IP}^{*}_r\)-recurrence. This should be juxtaposed with the fact that for general distal transformations the polynomial \(\mathrm{IP}^{*}\)-recurrence is strictly stronger than the \(\mathrm{IP}^{*}\)-recurrence.
scientific article; zbMATH DE number 6967039
scientific article