Ruelle operators: Functions which are harmonic with respect to a transfer operator
Abstract
Let $ N \in \mathbb{N} $, $ N \geq 2 $, be given. Motivated by wavelet analysis, we consider a class of normal representations of the $ C^* $algebra $ \mathfrak{A}_{N} $ on two unitary generators $ U $, $ V $ subject to the relation \[ UVU^{1}=V^{N}. \] The representations are in onetoone correspondence with solutions $ h \in L^{1}(\mathbb{T}) $, $ h \geq 0 $, to $ R(h)=h $ where $ R $ is a certain transfer operator (positivitypreserving) which was studied previously by D. Ruelle. The representations of $ \mathfrak{A}_{N} $ may also be viewed as representations of a certain (discrete) $ N $adic $ ax+b $ group which was considered recently by J.B. Bost and A. Connes.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1998
 DOI:
 10.48550/arXiv.math/9805141
 arXiv:
 arXiv:math/9805141
 Bibcode:
 1998math......5141J
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Operator Algebras;
 46L60;
 47D25;
 42A16;
 43A65 (Primary);
 46L45;
 42A65;
 41A15 (Secondary)
 EPrint:
 56 pages, 4 figures (some with subfigures), AMSLaTeX v1.2r