The random continued fraction transformation

Abstract

We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.2010 Mathematics Subject Classification. Primary: 37C40, 11K50.

Linear response for random dynamical systems

“…The random system consists of choosing randomly the Gauss and the Rényi map, with respective probabilities p and 1 − p, with p ∈ [0, 1]. We model this example with ℓ = 2, I 1 = I 2 = {0}, T 1,0 = G, T 2,0 = R, π 1 = p, π 2 = 1 − p which determines P. This random dynamical system is used to study random continued fractions [23].…”

“…We apply our results to iid compositions of uniformly expanding circle maps, to iid compositions of the Gauss-Rényi maps and to iid compositions of Pomeau-Manneville maps. The latter family models intermittent transition to turbulence and is of central interest for both mathematicians [15,17,19,21,26,27,28,33,35] and physicists [31], while the former family provides fundamental links between ergodic theory and number theoretic questions [13,14,23]. Indeed, for the Gauss-Rényi maps we use our results to approximate the invariant density governing the statistics of random continued fractions by the well known invariant density of the Gauss map, 1 log 2 1 1+x , and its linear response with respect to a Bernoulli distribution (see subsection 5.3 for more details; in particular (31)).…”

Random N-continued fraction expansions

“…In the following, we use a similar technique from [10] to show that µ p is in fact equivalent with the Lebesgue measure.…”

“…Defining the random dynamical system as a skew product allows one to use results from ergodic theory in order to gain information about the asymptotic behaviour of the expansions. This is done in [10] for expansions like (1), where N ∈ {−1, 1}. In [8] more invariant measures for random β-expansions are obtained by constructing an isomorphism between the skew product for the random β-expansion and the digit sequences it induces.…”

“…In this paper we will prove the existence of an invariant measure for the random transformation generating 2-continued fraction expansions, so expansions of the form (1) where N = 2. We will use the approach of [10] to show that an accelerated version of our system has an invariant measure of the form m × µ, where m is a Bernoulli measure and µ is equivalent with the Lebesgue measure. Using standard techniques, we lift the obtained invariant measure for the accelerated system to an invariant measure ρ for the original random system.…”