Open AccessShrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds
Author(s) -
Dubi Kelmer,
Hee Oh
Publication year - 2021
Publication title -
journal of modern dynamics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.668
H-Index - 25
eISSN - 1930-532X
pISSN - 1930-5311
DOI - 10.3934/jmd.2021014
Subject(s) - geodesic , mathematics , metric (unit) , logarithm , flow (mathematics) , cusp (singularity) , pure mathematics , hyperbolic manifold , combinatorics , mathematical analysis , geometry , hyperbolic function , operations management , economics
Let \begin{document}$ \mathscr{M} $\end{document} 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.