The fractional Poisson process and other limit point processes for rare events in infinite ergodic theory

Abstract

We study the process of suitably normalized successive return times to rare events in the setting of infinite-measure preserving dynamical systems. Specifically, we consider small neighborhoods of points whose measure tends to zero. We obtain two types of results. First, we conduct a detailed study of a class of interval maps with a neutral fixed point and we fully characterize the limit processes for all points, highlighting a trichotomy and the emergence of the fractional (possibly compound) Poisson process. This is the first time that these processes have been explicitly identified in this context. Second, we prove an abstract result that offers an explanation for the emergence of the fractional Poisson process, as the unique fixed point of a functional equation, drawing a parallel with the well-established behavior of the Poisson process in finite-measure preserving dynamical systems.