A generalized local limit theorem for Lasota-Yorke transformations (Q1813922)

Let \(T\) be a transformation of the interval [0,1] to itself. Under some restrictions on \(T\) the author proves a local limiting distribution theorem about the sums \(S_ n f = f(x) + f(Tx) +\cdots +f(T^{n-1}x)\), where \(f(x)\) is any function of bounded variation. The conditions, which are rather complicated, ensure that for \(x\) considered as a random variable with the distribution the invariant measure of \(T,x,Tx,T^ 2 x,\dots\) behave essentially as independent random variables. The paper concludes with examples. Among else it turns out that the regular continued fraction expansion is contained in the main result of the paper; namely, the transformation \(T(x) = (1/x), (.)\) being the fractional part, satisfies the conditions of the main theorem.