Shrinking targets for discrete time flows on hyperbolic manifolds

Abstract

We prove dynamical Borel Canteli Lemmas for discrete time homogenous flows hitting a sequence of shrinking targets in a hyperbolic manifold. These results apply to both diagonalizable and unipotent flows, and any family of measurable shrinking targets. As a special case, we establish logarithm laws for the first hitting times to shrinking balls and shrinking cusp neighborhoods, refining and improving on perviously known results.

Shrinking targets and eventually always hitting points for interval maps

“…We note that the sufficient condition obtained by Kelmer [13] in the setting of discrete time homogeneous flows acting on a finite volume quotient of H n is slightly better than what we get in our setting. More precisely, [13, Theorem 2] states that in the mentioned setting…”

“…In this sense, being eventually always hitting for B m is a stronger property than hitting B m infinitely often. The term eventually always hitting was coined by Kelmer in [13] where this set was studied in the context of flows on hyperbolic manifolds. Kelmer proved necessary and sufficient conditions for the set of eventually always hitting points to be of full measure.…”

Shrinking target problems for flows on homogeneous spaces

“…Remark 7. In the other direction, it was shown in [Kel17] that for any ergodic one-parameter flow, for any monotone sequence, {B m } m∈N , of shrinking targets, if there is c < 1 such that the set {m : mµ(B m ) ≤ c} is unbounded then A ah is a null set. In particular, if we assume that µ(B m ) decays polynomially in the sense that µ(B m ) ≍ m −η for some fixed η, then Theorem 1.8, implies that A ah is a set of full measure when η < 1, and a null set when η > 1.…”

“…The results of [Kel17] were given for a more general setting of Z d -actions. For Z-actions we also have the following lemma stating that A ah is always either null or co-null.…”

“…as m → ∞ for any f ∈ L 2 (X ), where µ(f ) := X f dµ. In this section we adapt the method introduced in [GK17] and [Kel17], to prove two effective mean ergodic theorems using the explicit rate of decay of matrix coefficients obtained in the previous section. The arguments are slightly different for rank one groups and for groups with property (T ), so we will treat them separately.…”