On the local limit theorem in dynamical systems

In 1987, Burton and Denker proved the remarkable result that in every aperiodic dynamical system (including irrational rotations for example) there is a square integrable, zero mean function such that its corresponding time series satisfies a CLT. Subsequently, Volny showed that one can find a function which satisfies the strong (almost sure) invariance principle. All these constructions resulted in a non-lattice distribution.

In a joint work with Dalibor Volny we show that there exists an integer valued cocycle which satisfies the local limit theorem.