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We consider a $\phi$-mixing shift $T$ on a sequence space $\Omega$ and studythe number of returns $\{ T^k\omega\in U\}$ to a union $U$ of cylinders oflength $n$ until the first return $\{ T^k\omega\in V\}$ to another union $V$ ofcylinder 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 hasapproximately geometric distribution. Under appropriate conditions, we extendthis result for some dynamical systems to geometric balls and Young towers withintegrable tails. This work is motivated by a number of papers on asymptoticalbehavior of numbers of returns to shrinking sets, as well as by the papers onopen systems studying their behavior until an exit through a "hole".

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