Local limit theorems for expanding maps

We prove local central limit theorems for partial sums of the form \,Sₙ=∑ⱼ₌₀ⁿ⁻¹fⱼ∘ Tⱼ₋₁∘⋯∘ T₁∘ T₀ where fⱼ are uniformly H\"older functions and Tⱼ are expanding maps. Using a symbolic representation a similar result follows for maps Tⱼ in a small C¹ neighborhood of an Axiom A map and H\"older continuous functions fⱼ. All of our results are already new when all maps are the same Tⱼ=T but observables (fⱼ) 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.