**Memoirs of the American Mathematical Society**

2001;
60 pp;
Softcover

MSC: Primary 46; 42; 43;
Secondary 41

Print ISBN: 978-0-8218-2688-1

Product Code: MEMO/152/720

List Price: $49.00

AMS Member Price: $29.40

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-0313-3
Product Code: MEMO/152/720.E**

List Price: $49.00

AMS Member Price: $29.40

MAA Member Price: $44.10

# Ruelle Operators: Functions which Are Harmonic with Respect to a Transfer Operator

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*Palle E. T. Jorgensen*

Let \(N\in\mathbb{N}\), \(N\geq2\), be given. Motivated by wavelet analysis, we consider a class of normal representations of the \(C^{\ast}\)-algebra \(\mathfrak{A}_{N}\) on two unitary generators \(U\), \(V\) subject to the relation \(UVU^{-1}=V^{N}\). The representations are in one-to-one correspondence with solutions \(h\in L^{1}\left(\mathbb{T}\right)\), \(h\geq0\), to \(R\left(h\right)=h\) where \(R\) is a certain transfer operator (positivity-preserving) 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.

#### Readership

Graduate students and research mathematicians interested in functional analysis.

#### Table of Contents

# Table of Contents

## Ruelle Operators: Functions which Are Harmonic with Respect to a Transfer Operator

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. A discrete ax + b group 716 free
- Chapter 3. Proof of Theorem 2.4 1322
- Chapter 4. Wavelet filters 2332
- Chapter 5. Cocycle equivalence of filter functions 3342
- Chapter 6. The transfer operator of Keane 3847
- Chapter 7. A representation theorem for R-harmonic functions 4756
- Chapter 8. Signed solutions to R(f) = f 5362
- Bibliography 5867