Albert Fisher (University of Sao Paulo)
A flow crossection for Moeckel's theorem on continued fractions
Abstract: We construct a cross-section to the principal congruence modular flow which is represented as a skew product transformation over the natural extension of the Gauss map. This leads to a new proof of Moeckel's theorem on rational approximants. For an irrational number $x$ in the unit interval with continued fraction expansion $[n_0 n_1...]$, let $p_k/q_k= $[n_0 n_1..n_k]$ $ be the rational approximants for $x$. Writing these in lowest terms, they can be of three types: $\frac{O}{E}$, $\frac{E}{O}$, or $\frac{O}{O}$ where $O$ stands for odd and $E$ for even. Moeckel's theorem states that the frequency of each of these exists almost surely. What is unusual in the proof is that this does not follow directly from the ergodic theorem applied to an observable on the Gauss map (the shift on continued fractions): one must first enlarge the space. Moeckel's approach makes use of the geodesic flow on a three-fold cover of the modular surface, together with a geometric argument for counting the time that geodesics spend in cusps. Ergodicity of the flow is automatic (via the Hopf argument) but the counting is somewhat involved. Later Jager and Liardet found a second purely ergodic theoretic proof, constructing a skew product over the Gauss map. There the counting is direct, but the proof of ergodicity is more difficult. Our proof unifies the two earlier arguments, inheriting these strong points of each. |
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