On the central limit theorem for dynamical systems

Abstract:

Given an aperiodic dynamical system $(X,T,\mu )$ then there is an $f \in {L^2}(\mu )$ with $\smallint fd\mu = 0$ satisfying the Central Limit Theorem, i.e. if ${S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}$ and ${\sigma _m} = {\left \| {{S_m}f} \right \|_2}$ then \[ \mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} < u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .\] The analogous result also holds for flows.