Local limit theorems for expanding maps

We prove local central limit theorems for partial sums of the form \newline $S_n={\sum }_{j=0}^{n-1}{f}_j\circ T_{j-1}\circ \cdots \circ T_1\circ T_0$ where $f_j$ are uniformly Hölder functions and $T_j$ are expanding maps. Using a symbolic representation a similar result follows for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$. All of our results are already new when all maps are the same $T_j=T$ but observables $(f_j)$ are different. The current paper compliments [43] where Berry–Esseen theorems are obtained. An important step in the proof is developing an appropriate reduction theory in the sequential case.