Strong renewal theorems and Lyapunov spectra for $α$-Farey and $α$-Lüroth systems

Abstract: In this paper we introduce and study the $\alpha$-Farey map and its associated jump transformation, the $\alpha$-L\"uroth map, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $\alpha$-sum-level sets for the $\alpha$-L\"uroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $\alpha$-Farey map and the $\alpha$-L\"uroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition $\alpha$.