scholarly journals Symmetric Birkhoff sums in infinite ergodic theory

2016 ◽  
Vol 37 (8) ◽  
pp. 2394-2416 ◽  
Author(s):  
JON AARONSON ◽  
ZEMER KOSLOFF ◽  
BENJAMIN WEISS
Keyword(s):  
Ergodic Theory ◽  
Recurrent Events ◽  
Integrable Functions ◽  
Infinite Ergodic Theory

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity. Also, we consider the latter phenomenon and characterize it among transformations admitting generalized recurrent events.

Infinite ergodic theory for Kleinian groups

2005 ◽  
Vol 25 (4) ◽  
pp. 1305-1323 ◽  
Author(s):  
MANUEL STADLBAUER ◽  
BERND O. STRATMANN
Keyword(s):  
Ergodic Theory ◽  
Kleinian Groups ◽  
Infinite Ergodic Theory

scholarly journals Infinite ergodic theory meets Boltzmann statistics

2020 ◽  
Vol 138 ◽  
pp. 109890 ◽  
Author(s):  
Erez Aghion ◽  
David A. Kessler ◽  
Eli Barkai
Keyword(s):  
Ergodic Theory ◽  
Infinite Ergodic Theory ◽  
Boltzmann Statistics

scholarly journals LIMIT LAWS FOR DISTORTED CRITICAL RETURN TIME PROCESSES IN INFINITE ERGODIC THEORY

2007 ◽  
Vol 07 (01) ◽  
pp. 103-121 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
MEHDI SLASSI
Keyword(s):  
Asymptotic Behaviour ◽  
Uniform Distribution ◽  
Ergodic Theory ◽  
Ergodic Measure ◽  
Return Time ◽  
Limit Laws ◽  
Measure Spaces ◽  
Distribution Laws ◽  
Infinite Ergodic Theory ◽  
Measure Preserving Transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling–Kac condition. We identify two critical cases for which we prove uniform distribution laws.

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.

scholarly journals Infinite Ergodic Theory and Non-extensive Entropies

2011 ◽  
Vol 41 (4-6) ◽  
pp. 297-303 ◽  
Author(s):  
Luis M. Gaggero-Sager ◽  
E. R. Pujals ◽  
O. Sotolongo-Costa
Keyword(s):  
Ergodic Theory ◽  
Infinite Ergodic Theory
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