Random \(f\)-expansions (Q2778782)

The author studies the thermodynamic formalism for random piecewise expanding in average maps related to random \(f\)-expansions. The random \(f\)-expansion is a generalization of a continued fraction expansion of a number and can be written in the following form: \(x=f_\omega(a_1+f_{\sigma\omega}(a_2+f_{\sigma^2\omega}(a_3+\dots)\dots))\), where \(\sigma\) is an ergodic invertible transformation of a probability space, and \(f_\omega(\cdot)\) is a piecewise expanding one-dimensional map. The main results are the proof of the existence of Lebesgue absolutely continuous invariant measures for random transformations of this class and an estimate of the Hausdorff dimension of these measures. In particular, it is shown that the distribution of a continued fraction with independent digits whose distributions form a stationary sequence has the Hausdorff dimension strictly less than 1.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].