Geometric law for numbers of returns until a hazard under ϕ-mixing

We consider a ϕ-mixing shift T on a sequence space Ω and study the number of returns {T^kω ∈ U} to a union U of cylinders of length n until the first return {T^kω ∈ 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”.