Proofs of ergodicity of piecewise Möbius interval maps using planar extensions (Q6595853)

This paper is a contribution to a systematic approach to the study of dynamical properties of families of maps generalizing the Gauss map associated to the classical continued fraction. The class of maps considered here are piecewise Möbius interval maps, and the broad approach is to extend the maps to planar maps and deduce dynamical properties from these extensions. The work is grouped around two main themes. The first of these builds on earlier work by the same authors on an infinite family of maps for each of a countable number of Fuchsian triangle groups, see [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 20, No. 3, 951--1008 (2020; Zbl 1475.37009)]. A feature of many of these maps is that they are nonexpansive but eventually expansive. \N\NTheorem 1 gives general criteria that may be used to show eventual expansivity along with a definition weaker than Markov which in conjunction with conditions on the planar extension imply ergodicity with respect to an invariant measure absolutely continuous with respect to Lebesgue measure. \N\NTheorem 2 exploits the `quilting' techniques introduced by \textit{C. Kraaikamp} et al. [J. Math. Soc. Japan 62, No. 2, 649--671 (2010; Zbl 1209.11078)] to show a form of stability for dynamical properties of `nearby' maps. \N\NBoth of these broad ideas are shown to readily recover known results on the Nakada \(\alpha\)-continued fractions as well as for the interval maps derived from an infinite family of noncommensurable Fuchsian groups.