We present a dynamical and spectral study of Ostrowski's map based on the use of transfer operators. The Ostroswki dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and interval fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operator, we show that the Ostroswki dynamical system admits an absolutely continuous invariant measure with piecewise holomorphic density and exponential mixing properties. We deduce a central limit theorem for associated random variables of an arithmetic nature and motivated by applications in inhomogeneous Diophantine approximation, we get Bowen--Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a certain bounded digit set.