A limit theorem for the Perron-Frobenius operator of transformations on [0,1] with indifferent fixed points
Let \(T\) be a transformation on \([0, 1]\) satisfying the author's conditions [in Isr. J. Math. 46, 67-96 (1983; Zbl 0528.28011)]. Here, it is shown that the Perron-Frobenius operator \(P\) associated to \(T\) satisfies \[ {1\over a_n} \sum^{n- 1}_{k= 0} P^k f\to h \int f \] for all integrable \(f\), where \(h\) denotes the invariant density and where the convergence is uniform away from parabolic points. In particular, this result implies the Darling-Kac property [see \textit{J. Aaronson}, \textit{M. Urbański} and the reviewer, Trans. Am. Math. Soc. 337, No. 2, 495-548 (1993; Zbl 0789.28010)].
scientific article; zbMATH DE number 820094