An Indicator Function Limit Theorem in Dynamical Systems

Abstract

We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n{)}_{n\in \mathbb{N}}\subset {\mathbb{R}}_{+}$ such that $a_n\nearrow \infty$ and $\frac{a_n}{n}\to 0$ as $n\to \infty$, there exists a set $A\in\A$ having the property that the sequence of the distributions of $(\frac{1}{a_{n}}S_{n}(\ind_A-μ(A)))_{n\in\N}$ is dense in the space of all probability measures on $\mathbb{R}$.