Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures

摘要

In infinite ergodic theory, two distributional limit theorems are well-known.One is characterized by the Mittag-Leffler distribution for time averages of$L^1(m)$ functions, i.e., integrable functions with respect to an infiniteinvariant measure. The other is characterized by the generalized arc-sinedistribution for time averages of non-$L^1(m)$ functions. Here, we provideanother distributional behavior of time averages of non-$L^1(m)$ functions inone-dimensional intermittent maps where each has an indifferent fixed point andan infinite invariant measure. Observation functions considered here arenon-$L^1(m)$ functions which vanish at the indifferent fixed point. We callthis class of observation functions weak non-$L^1(m)$ function. Our main resultrepresents a first step toward a third distributional limit theorem, i.e., adistributional limit theorem for this class of observables, in infinite ergodictheory. To prove our proposition, we propose a stochastic process induced by arenewal process to mimic a Birkoff sum of a weak non-$L^1(m)$ function in theone-dimensional intermittent maps.