Annealed and quenched representations of the Gauss-R\'enyi measure by "periodic points"

Abstract

We consider independently identically distributed random compositions of the Gauss and R\'enyi maps that generate random continued fractions. Using methods of ergodic theory, thermodynamic formalism and large deviations, we show that weighted cycles of this random dynamical system equidistribute with respect to the Gauss-R\'enyi measure. We present both annealed (sample-averaged) and quenched (samplewise) results.