# Geometric law for numbers of returns until a hazard under $\phi$-mixing

## TL;DR

This paper investigates the distribution of return times in $\

## Contribution

It extends geometric return time distribution results to $\

## Key findings

- Returns approximately follow a geometric distribution when probabilities are small and similar.
- Results apply to geometric balls and Young towers under certain conditions.
- Provides insights into return times in open dynamical systems.

## Abstract

We consider a $\phi$-mixing shift $T$ on a sequence space $\Omega$ and study the number of returns $\{ T^k\omega\in U\}$ to a union $U$ of cylinders of length $n$ until the first return $\{ T^k\omega\in V\}$ to another union $V$ of cylinder sets of length $m$. It turns out that if probabilities of the sets $U$ and $V$ are small and of the same order then the above number of returns has approximately geometric distribution. Under appropriate conditions, we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymptotical behavior of numbers of returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".