Abstract
Let π:(T,X)→(T,Z) be an extension of flows with phase group T. A point x in X is π-distal if x is proximal to at most itself in π−1πx∩Tx‾ under (T,X). We study the π-distal points using combinatorial methods. We present characterizations of π-distal points using product IP/Cw/C-recurrence, dynamics syndetic sets, distal sets, IP⁎-sets, and C⁎-sets in T. Moreover, we give the dynamics realization of IP-set of any discrete group by IP-recurrent point and the dynamics realization of C-set of Zd by C-recurrent point. The IP⁎-recurrence of π-distal points introduced here is useful for simplifying the proof of Furstenberg's structure theorem.
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