Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

“…The shrinking target problems for sets with complicated geometry is discussed in [69,70,72,98,99,100,135,140].…”

Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces

“…Then, by (12) there exist g ∈ G such that x = gΓ, 1 ≤ i ≤ q, k ∈ K, and u ∈ U , such that (16) g = ka −R ug i .…”

“…In [16], Kelmer and Oh showed a strengthening of the above, considering excursion to individual cusps and obtaining a limit for the shrinking target problem of the geodesic flow. Note also that the result stated in [16] is for x ∈ T 1 (G/Γ), but since the distance function there is assumed to be K-invariant, where H n = K\G, and the set C 0 is K-invariant as well (see §2.2), we can deduce the form above. It follows from (3) that the limit on the left hand side of ( 2) is zero for almost every point x ∈ G/Γ (with respect to the invariant volume measure) in this case.…”

“…Sullivan's logarithm law for geodesics when Γ is geometrically finite with δ Γ > (n−1)/2 was shown in [16,38] (and is a strengthening of Sullivan's logarithm law for non-compact lattices ( [41, §9])), and implies that for almost all x ∈ G/Γ, (3) lim sup…”

Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

“…Theorem 1.1 and 9.1 are known to have many immediate applications in number theory and geometry. To name a few, see [26] for counting closed geodesics and [22] for shrinking target problems.…”