Strong renewal theorems and Lyapunov spectra forα-Farey andα-Lüroth systems

Abstract

Abstract. In this paper we introduce and study the α-Farey map and its associated jump transformation, the α-Lüroth map, for an arbitrary countable partition α 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 α-sumlevel sets for the α-Lüroth 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 α-Farey map and the α-Lüroth 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 α.

Pointwise convergence of Birkhoff averages for global observables

“…Let us recall the definition of an α-Farey map, as introduced in [KMS,§1.4]. Start with a decreasing sequence (t k ) k∈Z + of real numbers such that t 1 = 1 and lim k→∞ t n = 0.…”

“…For later use, let us also recall the definition of the related α-Lüroth expansion (for more details we refer again to [KMS,§1.4]). Each partition α generates a series expansion of the numbers in the unit interval, in that we can associate to each x a sequence of positive integers ( i ) i≥1 for which…”

Stability and perturbations of countable Markov maps

“…Let α k be the partition where a n (α k ) = a n (α D ) for all n / ∈ {k, k+1}, and we modify the point t k+1 (α D ) in order to obtain the lengths a k (α k ) = 2 −k 2 and a k+1 = 2 −k + 2 −(k+1) − 2 −k 2 . Then the conjugacy map θ k between T k and T is exactly the map studied in [17], where in particular it was shown in [17,Lemma 2.3] that the Hölder exponent of θ k is given by κ(θ k ) = inf log a n (α D ) log a n (α k ) : n ∈ N .…”

Hausdorff dimension of the graph of the error-sum function of $\alpha$-Lüroth series

“…In [7], the authors studied some topological and ergodic theoretic properties of the α -Lüroth series and gave a complete description of its Lyapunov spectra in terms of the thermodynamical formalism. Meanwhile, in a related paper [8], Munday computed the Hausdorff dimension of some α -Good type sets.…”