Large deviations for denominators of continued fractions

Abstract

We give exponential upper bounds on the probability with which the denominator of the nth convergent in the regular continued fraction expansion stays away from the mean . The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation. We also establish the large deviation principle (LDP) for denominators. As corollaries, we derive the LDPs for denominators of periodic continued fractions and continued fraction preimages. Proofs of the main results rely on the thermodynamic formalism for finite topological Markov shifts.