# Local limit theorems in ergodic theory

# Local limit theorems in ergodic theory

We use Stone's version of a local limit theorem from 1969: Let $(X,{\cal F},T,m)$ be a measure preserving dynamical system. A measurable function $f:X\to \mathbb R$ satisfies a local limit theorem, if there are constants $A_n$ and $B_n\to\infty$ such that $$ B_nm( f+f\circ T+...+f\circ T^{n-1} \in x_n+I) \to g(x)|I|,$$ where $(x_n-A_n)/B_n \to x$ and where $g$ is the density of some stable distribution. An analogous definition applies in the lattice case.For Gibbs-Markov dynamical systems (including certain Markov shifts), such results can be established when the function is in the domain of attraction of a stable distribution. It also generalizes to non-Markov situations for certain maps of the interval, including beta-transformations. Applications to conservativity of dynamical systems and to Poincaré exponents are briefly discussed.