Cross sections for geodesic flows and α-continued fractions

Abstract

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the α-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced α-variants.

20 pages, 2 figures

References in corpus (1)

• Natural extensions and entropy of $\alpha$-continued fractions

Cited by in corpus (4)

• Coding of geodesics on some modular surfaces and applications to odd and even continued fractions
• Distribution of approximants and geodesic flows
• $\alpha$-Expansions with odd partial quotients
• $\mathbf{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometry