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

Published online by Cambridge University Press:  02 September 2011

MARC KESSEBÖHMER ,
SARA MUNDAY  and
BERND O. STRATMANN
MARC KESSEBÖHMER
Affiliation:
Fachbereich 3 – Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
SARA MUNDAY
Affiliation:
Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, Scotland (email: sam20@maths.st-and.ac.uk)
BERND O. STRATMANN
Affiliation:
Fachbereich 3 – Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
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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 linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical 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-sum-level 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 989 - 1017
Copyright
Copyright © Cambridge University Press 2011

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