# Shrinking targets for discrete time flows on hyperbolic manifolds

## TL;DR

This paper establishes new dynamical Borel Cantelli lemmas for discrete flows on hyperbolic manifolds, providing refined logarithm laws for hitting times to shrinking targets, applicable to various flows and target families.

## Contribution

It introduces general Borel Cantelli lemmas for discrete hyperbolic flows with shrinking targets, improving existing results on hitting times and logarithm laws.

## Key findings

- Proves Borel Cantelli lemmas for discrete flows on hyperbolic manifolds.
- Establishes refined logarithm laws for hitting times to shrinking targets.
- Applicable to both diagonalizable and unipotent flows.

## 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.