IPr -recurrence and nilsystems
نویسنده
چکیده
By a result due to Furstenberg, a homeomorphism T of a compact space is distal if and only if it possesses the property of IP-recurrence, meaning that for any x0 ∈ X, for any open neighborhood U of x0, and for any sequence (ni) in Z, the set RU (x0) = {n ∈ Z : T x0 ∈ U} has a non-trivial intersection with the set of finite sums {ni1 +ni2 + · · ·+nis : i1 < i2 < . . . < is, s ∈ N}. We show that translations on compact nilmanifolds (which are known to be distal) are characterized by a stronger property of IPr-recurrence, which asserts that for any x0 ∈ X and any neighborhood U of x0 there exists r ∈ N such that for any r-element sequence n1, . . . , nr in Z the set RU (x0) has a non-trivial intersection with the set {ni1 + ni2 + · · ·+ nis : i1 < i2 < . . . < is, s ≤ r}. We also show that the property of IPr-recurrence is equivalent to an ostensibly much stronger property of polynomial IPr-recurrence. (This should be juxtaposed with the fact that for general distal transformations the polynomial IP-recurrence is strictly stronger than the IP-recurrence.)
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