Annealed and quenched representations of the Gauss-Rényi measure by "periodic points"

We consider independently identically distributed random compositions of the Gauss and Rényi 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-Rényi measure. We present both annealed (sample-averaged) and quenched (samplewise) results.