Counterexamples in ergodic theory and number theory

1979 ◽  
Vol 245 (3) ◽  
pp. 185-197 ◽  
Author(s):  
Andres del Junco ◽  
Joseph Rosenblatt
Keyword(s):  
Number Theory ◽  
Ergodic Theory

Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory

2012 ◽  
Vol 9 (4) ◽  
pp. 2985-3059
Author(s):  
Vitaly Bergelson ◽  
Nikos Frantzikinakis ◽  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Combinatorial Number Theory

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

2011 ◽  
Vol 32 (3) ◽  
pp. 989-1017 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
SARA MUNDAY ◽  
BERND O. STRATMANN
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Level Sets ◽  
Gauss Map ◽  
Unit Interval ◽  
Elementary Number Theory ◽  
The Core ◽  
Lyapunov Spectra ◽  
Infinite Ergodic Theory ◽  
Farey Map

AbstractIn this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof 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 calledα-sum-level sets for theα-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 theα-Farey map and theα-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 partitionα.

The remarkable effectiveness of ergodic theory in number theory

2009 ◽  
Vol 17 (1) ◽  
Author(s):  
Alexander Arbieto ◽  
Carlos Matheus ◽  
Carlos Moreira
Keyword(s):  
Number Theory ◽  
Ergodic Theory
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