Summary
LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,μ) and letf: X→ℝ be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={α∈ℝ:f−α is recurrent}. If\(\rho = \log \frac{{d\mu T}}{{d\mu }}\), then R(ρ)={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number α∈ℝ to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments.
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Schmidt, K. On recurrence. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 75–95 (1984). https://doi.org/10.1007/BF00535175
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DOI: https://doi.org/10.1007/BF00535175