It is shown that if, for a piecewise $C^2$ mapping of the unit interval into itself where the absolute value of the derivative is greater than 1, an invariant measure is weak-mixing, then a central limit theorem holds for a class of real Holder functions.
"A Central Limit Theorem for Piecewise Monotonic Mappings of the Unit Interval." Ann. Probab. 7 (3) 500 - 514, June, 1979. https://doi.org/10.1214/aop/1176995050