Large deviation principle for the backward continued fraction expansion

Abstract

We investigate stochastic properties of the backward continued fraction expansion of irrational numbers in (0,1). For the mean process associated with a real-valued observable which depends only on the first digit of the expansion, we establish the large deviation principle. For any such observable which is non-negative, we completely determine the set of minimizers of the rate function in terms of a growth rate of the observable. Our method of proof employs the thermodynamic formalism for topological Markov shifts, and a multifractal analysis of pointwise Lyapunov exponents for the Rényi map generating the backward continued fraction expansion.