A generalization of a lemma of Kac

Arevalo C.D.M. · Archiv der Mathematik (2023) · DOI: 10.1007/s00013-023-01849-y

Resumen

Denote by τ(k,.) : A→ N the k-th return function for a set A with positive measure and a transformation T: Ω → Ω that preserves μ . Kac’s lemma asserts that ∫ Aτ(1 ,.) = μ(Ω) - μ(E∗) . Where E∗ is the set of points that never enter A. We generalize this formula for arbitrary time of return: ∫Aτ(k,.)=k(μ(Ω)-μ(E∗)). We also prove that the distributions of τ(1 ,.) and τ(k,.) - τ(k- 1 ,.) are equal for arbitrary k.

Instituciones: Universidad Nacional de Ingeniería