The entropy of Nakada's α-continued fractions: analytical results

Abstract

We study the ergodic theory of a one-parameter family of interval maps Tα arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of Tα to be H¨older-continuous in the parameter α. Moreover, we prove a central limit theorem for possibly unbounded observables whose bounded variation grows moderately. This class of functions is large enough to cover the case of Birkhoff averages converging to the entropy.