The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative

Abstract

We study a class of dynamical systems in $L^2$ spaces of infinite products $X$. Fix a compact Hausdorff space $B$. Our setting encompasses such cases when the dynamics on $X = B^\mathbb {N}$ is determined by the one-sided shift in $X$, and by a given transition-operator $R$. Our results apply to any positive operator $R$ in $C(B)$ such that $R1 = 1$. From this we obtain induced measures $\Sigma$ on $X$, and we study spectral theory in the associated $L^2(X,\Sigma )$. For the second class of dynamics, we introduce a fixed endomorphism $r$ in the base space $B$, and specialize to the induced solenoid $\mathrm{Sol }(r)$. The solenoid $\mathrm{Sol }(r)$ is then naturally embedded in $X = B^\mathbb {N}$, and $r$ induces an automorphism in $\mathrm{Sol }(r)$. The induced systems will then live in $L^2(\mathrm{Sol }(r), \Sigma )$. The applications include wavelet analysis, both in the classical setting of $\mathbb {R}^n$, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism $r$ there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.

1 Introduction

The purpose of this paper is to offer a general framework for geometry and analysis of iteration systems. We offer a setting encompassing the kind of infinite product, or solenoid constructions arising in the study of iterated function systems (IFSs). Our aim is to give an operator theoretic construction of infinite product measures in a general setting that includes wavelet analysis of IFSs. To motivate this, recall, that to every affine function system $S$ with fixed scaling matrix, and a fixed set of translation points in $\mathbb {R}^n$, we may associate to $S$ a solenoid. By this we mean a measure space whose $L^2$ space includes $L^2(\mathbb {R}^n)$ in such a way that $\mathbb {R}^n$ embeds densely in the solenoid. (In the more familiar case of $n = 1$, we speak of a dense curve in an infinite-dimensional “torus”. The latter being a geometric model of the solenoid.

The need for this generality arose in our earlier investigations, for example in the building of wavelet systems on Cantor systems, of which the affine IFSs are special cases. In these cases (see Theorem 2.6 and Corollary 4.5 below) we found that one must pass to a suitable $L^2$ space of a solenoid. Indeed, we showed that such wavelet bases fail to exist in the usual receptor Hilbert space $L^2(\mathbb {R}^n)$ from wavelet theory.

For reference to earlier papers dealing with measures on infinite products, and their use in harmonic analysis and wavelet theory on fractals; see e.g., [DJ12, DJS12, DLS11, DJ11a, DS11, DJ11b, DHSW11, DJ10, DJP09, LN12, DL10, LW09].

The paper is organized as follows: Starting with a compact Hausdorff space $B$, and a positive operator $R$ in $C(B)$, we pass to a family of induced probability measures $\Sigma$ (depending on $R$) on the infinite product $\Omega = B^\mathbb {N}$. Among all probability measures on $\Omega$, we characterize those which are induced. In Sect. 2, we prove a number of theorems about $R$-induced measures on $\Omega$, and we include applications to random walks, and to fractal analysis. In Sect. 3, we then introduce an additional structure: a prescribed endomorphism $r$ in the base space $B$, and we study the corresponding solenoid $\mathrm{Sol }(r)$, contained in $\Omega$, and its harmonic analysis, including applications to generalized wavelets. The latter are studied in detail in Sect. 4 where we introduce wavelet-filters, in the form of certain functions $m$ on $B$.

2 Analysis of Infinite Products

Definition 2.1

Let $B$ be some compact Hausdorff space. $\mathcal {B}$ refes to a $\sigma$-algebra, usually generated by the open sets, so Borel. We will denote by $\mathcal {C}$ the cylinder sets, see below. We denote by $\mathcal {M}(B)$ the set of positive Borel masures on $B$, and by $\mathcal {M}_1(B)$ those that have $\mu (B)=1$. Let $V$ be some set.

$$B^V=\prod _V B=\text{ all } \text{ functions } \text{ from } V \text{ to } B.$$

For example $V=\mathbb {N}$ or $\mathbb {Z}$.

For $x\in B^V$ we denote by $\pi _v(x):=x_v$, $v\in V$. If $V=\mathbb {N}$, then we denote by

$$\pi _1^{-1}(x)=\{(x_1,x_2,\dots )\in B^\mathbb {N}: x_1=x\}.$$

Let $r:B\rightarrow B$ be some onto mapping, and $\mu$ a Borel probability measure on $B$, $\mu (B)=1$.

To begin with we do not introduce $\mu$ and $r$, but if $r$ is fixed and

$$\begin{aligned} 1\le \#r^{-1}(x)<\infty ,\quad \text{ for } \text{ all }\,\,x\in B, \end{aligned}$$

(2.1)

then we introduce two objects

• (1) $R=R_W$, the Ruelle operator;

• (2) $\mathrm{Sol }(r)$, the solenoid.

For (1), fix $W:B\rightarrow [0,\infty )$ such that

$$\sum _{r(y)=\,0x}W(y)=1,\text{ for } \text{ all }\,\,x\in B,$$

and set

$$\begin{aligned} (R_W\varphi )(x)=\sum _{r(y)=x}W(y)\varphi (y). \end{aligned}$$

(2.2)

For (2),

$$\begin{aligned} \mathrm{Sol }(r)=\left\{ x\in B^{\mathbb {N}} : r(x_{i+1})=x_i, i=1,2,\dots \right\} \end{aligned}$$

(2.3)

$$\begin{aligned} \sigma (x)_i=x_{i+1},\quad (x\in B^{\mathbb {N}}),\quad \quad \widehat{r}(x)=(r(x_1),x_1,x_2,\dots ). \end{aligned}$$

(2.4)

More generally, consider

$$\begin{aligned} R:C(B)\rightarrow C(B)\,\,\text{ or }\,\,R:M(B)\rightarrow M(B), \end{aligned}$$

(2.5)

where $M(B)$ is the set of all measurable functions on $B$.

Definition 2.2

We say that $R$ is positive iff

$$\begin{aligned} \varphi (x)\ge 0\,\,\text{ for } \text{ all }\,\,x\in B\,\,\text{ implies } (R\varphi )(x)\ge 0\text{, } \text{ for } \text{ all } x\in B. \end{aligned}$$

(2.6)

We will always assume $R1=1$ where $1$ indicates the constant function 1 on $B$. This is satisfied if $R=R_W$ in (2.2), but there are many other positive operators $R$ with these properties.

While what we call “the transfer operator” or a “Ruelle operator” has a host of distinct mathematical incarnations, each dictated by a particular family of applications, they are all examples of positive operators $R$ in the sense of our Definition 2.2. Our paper has two aims: One is to unify, and extend earlier studies; and the other is to prove a number of theorems on measures, dynamical systems, stochastic processes built from infinite products. Indeed there are many positive operators $R$ which might not fall in the class of operators studied as “transfer operators”. The earlier literature on transfer operators includes applications to physics [LR69], to the Selberg zeta function [FM12], to dynamical zeta functions [Rue02, Rue96, Nau12, MMS12]; to $C^*$-dynamical systems [Kwa12, ABL11]; to the study of Hausdorff dimension [Hen12]; to spectral theory [ABL12].

These applications are, in addition to the aforementioned, to analysis on fractals, and to generalized wavelets. For book treatments, we refer the reader to [Bal00], and [BJ02]. The literature on positive operators $R$, in the general sense, is much less extensive; but see [Arv86].

Definition 2.3

A subset $S$ of $B^{\mathbb {N}}$ is said to be shift-invariant iff $\sigma (S)\subset S$, where $\sigma$ is as in (2.4), $\sigma (x)_i=x_{i+1}$.

Remark 2.4

Every solenoid $\mathrm{Sol }(r)$ is shift-invariant.

Example 2.5

The solenoids introduced in connection with generalized wavelet constructions:

Let $r:B\rightarrow B$ as above and let $\mu$ be a strongly invariant measure, i.e.,

$$\int f\,d\mu =\int \frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}f(y)\,d\mu (x)$$

for all $f\in C(B)$.

A quadrature mirror filter (QMF) for $r$ is a function $m_0$ in $L^\infty (B,\mu )$ with the property that

$$\begin{aligned} \frac{1}{N}\sum _{r(w)=z}|m_0(w)|^2=1,\quad (z\in B) \end{aligned}$$

(2.7)

As shown by Dutkay and Jorgensen [DJ05, DJ07], every quadrature mirror filter (QMF) gives rise to a wavelet theory. Various extra conditions on the filter $m_0$ will produce wavelets in $L^2(\mathbb {R})$ [Dau92], on Cantor sets [DJ06, MP11], on Sierpinski gaskets [DMP08] and many others.

Theorem 2.6

[DJ05, DJ07] Let $m_0$ be a QMF for $r$. Then there exists a Hilbert space $\mathcal {H}$, a representation $\pi$ of $L^\infty (B)$ on $\mathcal {H}$, a unitary operator $U$ on $\mathcal {H}$ and a vector $\varphi$ in $\mathcal {H}$ such that

• (i) Covariance

  $$\begin{aligned} U\pi (f)U^*=\pi (f\circ r),\quad (f\in L^\infty (B)) \end{aligned}$$

  (2.8)

• (ii) Scaling equation

  $$\begin{aligned} U\varphi =\pi (m_0)\varphi \end{aligned}$$

  (2.9)

• (iii) Orthogonality

  $$\begin{aligned} \left\langle \pi (f)\varphi \, , \,\varphi \right\rangle =\int f\,d\mu ,\quad (f\in L^\infty (B)) \end{aligned}$$

  (2.10)

• (iv) Density

  $$\begin{aligned} \overline{\text{ span }}\left\{ U^{-n}\pi (f)\varphi : f\in L^\infty (B),n\ge 0\right\} =\mathcal {H}\end{aligned}$$

  (2.11)

The system $(\mathcal {H},U,\pi ,\varphi )$ in Theorem 2.6 is called the wavelet representation associated to the QMF $m_0$.

While, as we mentioned before, these representations can have incarnations on the real line, or on Cantor sets, they can be also represented using certain random-walk measures on the solenoid (see [DJ05, DJ07, Dut06]).

Remark 2.7

In examples when the condition (2.1) is not satisfied, the modification of the family of relevant integral operators is as follows.

In the general case when $r:B\rightarrow B$ is given, but $\#r^{-1}(x)=\infty$, the modification of the operators $R$, extending those from Example 2.5, is as follows:

Consider

• (i) $W:B\rightarrow [0,\infty )$ Borel

• (ii) $p:B\times \mathcal {B}(B)\rightarrow [0,\infty )$ such that for all $x\in B$, $p(x,\cdot )\in \mathcal {M}(r^{-1}(x))$, so is a positive measure such that

  $$\int \limits _{r^{-1}(x)}W(y)p(x,dy)=1,\quad (x\in B).$$

Then set

$$(R\varphi )(x)=\int \limits _{r^{-1}(x)}\varphi (y)W(y)p(x,dy).$$

Example 2.8

$G=(V,E)$ infinite graph, $V$ are the vertices, $E$ are the edges.

$$\begin{aligned} i(e)=\text{ initial } \text{ vertex },\quad t(e)=\text{ terminal } \text{ vertex }. \end{aligned}$$

(2.12)

$$\begin{aligned} S^{(G)}=\text{ solenoid } \text{ of } G=\left\{ \tilde{e}\in E^{\mathbb {N}} : t(e_j)=i(e_{j+1})\,\,\text{ for } \text{ all }\,\,j\in \mathbb {N}\right\} . \end{aligned}$$

(2.13)

For example $V=\mathbb {Z}^2$ and the edges are given by $x\sim y$ iff $\Vert x-y\Vert =1$. For details and applications, see [JP10].

Definition 2.9

Now back to $\mathcal {C}$, the cylinder sets mentioned in Definition 2.1. $C\in \mathcal {C}$ are subsets of $B^{V}$ indexed by finite sytems $v_1,\dots , v_n$ , $O_1,\dots ,O_n$, $v_i\in V$, $O_i\subset B$ open subsets, $i=1,\dots ,n$, $n\in \mathbb {N}$.

$$\begin{aligned} C_{v_i,O_i}:=\left\{ \tilde{x}\in B^V : x_{v_i}\in O_i\,\,\text{ for } \text{ all }\,\,i=1,\dots ,n\right\} . \end{aligned}$$

(2.14)

Notation: $\mathcal {C}$ generates the topology and the $\sigma$-algebra of subsets in $B^V$ in the usual way, and $B^V$ is compact by Tychonoff’s theorem.

If $\varphi$ is a function on $B$, we denote by $M_\varphi$ the multiplication operator $M_\varphi f=\varphi f$, defined on functions $f$ on $B$. In the applications below, we will use $C(B)$, all continuous functions from $B$ to $\mathbb {R}$.

Lemma 2.10

Consider $B^{\mathbb {N}}$ and the algebra generated by cylinder functions of the form $f=\varphi _1\otimes \dots \otimes \varphi _n$, $\varphi _i\in C(B)$, $n\in \mathbb {N}$, $1\le i\le n$,

$$\begin{aligned} (\varphi _1\otimes \dots \otimes \varphi _n)(\tilde{x})=\varphi _1(x_1)\varphi _2(x_2)\dots \varphi _n(x_n),\quad (\tilde{x}\in B^{\mathbb {N}}), \end{aligned}$$

(2.15)

or

$$\begin{aligned} \varphi _1\otimes \dots \otimes \varphi _n=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n). \end{aligned}$$

(2.16)

Let $\mathcal {A}_\mathcal {C}$ be the algebra of all cylinder functions. Then $\mathcal {A}_\mathcal {C}$ is dense in $C(B^{\mathbb {N}})$.

Proof

Easy consequence of Stone-Weierstrass. $\square$

Theorem 2.11

Let $R$ be a positive operator as in (2.5), with $R1=1$. Then for each $x\in B$ there exists a unique Borel probability measure $\mathbb {P}_x$ on $B^{\mathbb {N}}$ such that

$$\begin{aligned} \int \limits _{B^{\mathbb {N}}}\varphi _1\otimes \dots \otimes \varphi _n\,d\mathbb {P}_x=\left( M_{\varphi _1}RM_{\varphi _2}\dots RM_{\varphi _n}1\right) (x),\quad (\varphi _i\in C(B), n\in \mathbb {N}). \end{aligned}$$

(2.17)

Proof

We only need to check that the right-hand side of (2.17) for $\varphi _1\otimes \dots \varphi _n$ equals the right-hand side of (2.17) for $\varphi _1\otimes \dots \varphi _n\otimes 1$; but this is immediate from (2.17) and the fact that $R1=1$. The existence and uniqueness of $\mathbb {P}_x$ the follows form the inductive method of Kolmogorov. $\square$

Corollary 2.12

Let $B$ and $R:C(B)\rightarrow C(B)$ be as in Theorem 2.11, and let $\mu \in \mathcal {M}_1(B)$ be given. Let $\Sigma =\Sigma ^{(\mu )}$ be the measure on $\Omega =B^{\mathbb {N}}$ given by

$$\begin{aligned} \int f\,d\Sigma :=\int \limits _B\int \limits _{\pi _1^{-1}(x)} f\,d\mathbb {P}_x\,d\mu (x). \end{aligned}$$

(2.18)

Then

• (i) $V_1:L^2(B,\mu )\rightarrow L^2(\Omega ,\Sigma )$ given by $V_1\varphi :=\varphi \circ \pi _1$ is isometric.

• (ii) For its adjoint operator $V_1^*$, we have $V_1^*:L^2(\Omega ,\Sigma )\rightarrow L^2(B,\mu )$ with

  $$\begin{aligned} (V_1^*f)(x)=\int \limits _{\pi _1^{-1}(x)}f\,d\mathbb {P}_x. \end{aligned}$$

  (2.19)

Proof

The assertion (i) is immediate from Theorem 2.11. To prove (ii) we must show that the following formula holds:

$$\begin{aligned} \int \limits _B\left( \int \limits _{\pi _1^{-1}(x)}f\,d\mathbb {P}_x\right) \psi (x)\,d\mu (x)=\int \limits _{\Omega }f\,\psi \circ \pi _1\,d\Sigma \end{aligned}$$

(2.20)

for all $f\in L^2(\Omega ,\Sigma )$ and all $\psi \in C(B)$.

Recall that $V_1^*$ is determined by

$$\begin{aligned} \left\langle V_1^*f\, , \,\psi \right\rangle _{L^2(\mu )}=\left\langle f\, , \,V_1\psi \right\rangle _{L^2(\Omega ,\Sigma )}. \end{aligned}$$

(2.21)

But by Lemma 2.10 (Stone-Weierstrass), to verify (2.20), we may restrict attention to the special case when $f$ has the form given in (2.16). Note that if $f=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n)$ then

$$f(\psi \circ \pi _1)=((\varphi _1\psi )\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n),$$

and so the right-hand side of (2.20) is equal to

$$=\int \limits _\Omega ((\varphi _1\psi )\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n)\,d\Sigma $$

$$=\int \limits _B\varphi _1(x)\psi (x)R(\varphi _2R(\dots \varphi _{n-1}R(\varphi _n))\dots )))(x)\,d\mu (x)=\int \limits _B\psi (x)\int f\,d\mathbb {P}_x\,d\mu (x)$$

which is the left-hand side of (2.20) and (ii) follows. $\square$

Remark 2.13

When $R:C(B)\rightarrow C(B)$ is a given positive operator, we induce measures on $\Omega =B^{\mathbb {N}}$ by the inductive procedure outlined in the proof of Theorem 2.11; but implicit in this construction is an extension of $\varphi \mapsto R(\varphi )$ from all $\varphi$ continuous to all Borel measurable functions. This extension uses the Riesz theorem in the usual way as follows: Fix $x\in B$ and then apply Riesz’ theorem to the positive linear functional $C(B)\ni \varphi \mapsto R(\varphi )(x)$. There is a unique regular Borel measure $\mu _x$ on $B$ such that

$$R(\varphi )(x)=\int \limits _B\varphi (y)\,d\mu _x(y),\quad (\varphi \in C(B)).$$

If $E\subset B$ is Borel, we define

$$\tilde{R}(E)(x)=\tilde{R}(\chi _E)(x):=\mu _x(E);$$

but we shall use this identification without overly burdening our notation with tildes.

Lemma 2.14

Let $B$ and $R$ be specified as above. Given $\mu \in \mathcal {M}_1(B)$, let $\Sigma =\Sigma ^{(\mu )}$ denote the corresponding measure on $\Omega$, i.e.,

$$\begin{aligned} \int \limits _\Omega f\,d\Sigma =\int \limits _B\int \limits _{\pi _1^{-1}(x)}f\,d\mathbb {P}_x^{(R)}\,d\mu (x) \end{aligned}$$

(2.22)

We shall consider $V_1:L^2(B,\mu )\rightarrow L^2(\Omega ,\Sigma )$ and its adjoint operator $V_1^*: L^2(\Omega ,\Sigma )\rightarrow L^2(B,\mu )$, where $V_1\varphi =\varphi \circ \pi$, for all $\varphi \in L^2(B,\mu )$. Note that the adjoint operator $V_1^*$ makes reference to the choice of $R$ at the very outset. The following two hold:

• (i) $R$ naturally extends to $L^2(B,\mu )$; and

• (ii)
  $$\begin{aligned} RV_1^*f=V_1^*(f\circ \sigma ),\quad (f\in L^2(\Omega ,\Sigma )) \end{aligned}$$

  (2.23)

Remark 2.15

Given $R$, we say that a function $\varphi \in B$ is harmonic iff $R\varphi =\varphi$. It follows that harmonic functions contain the range of $V_1^*$, applied to $\{f: f\circ \sigma =f\}$. For a stronger conclusion, see Corollary 2.21.

Proof

of Lemma 2.14 Using the Stone-Weierstrass theorem, applied to $C(\Omega )$, we note that it is enough for us to check the validity of formula (2.23) on the algebra $\mathcal {A}^{(cyl)}$ spanned by all cylinder functions

$$\begin{aligned} f=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n) \end{aligned}$$

(2.24)

$n\in \mathbb {N}$, $\varphi _i\in C(B)$. But note that if $f$ is as in (2.24) then

$$\begin{aligned} f\circ \sigma =(\varphi _1\circ \pi _2)(\varphi _2\circ \pi _3)\dots (\varphi _n\circ \pi _{n+1}) \end{aligned}$$

(2.25)

Using then (2.19) in Corollary 2.12 above, we conclude that

$$(V_1^*(f\circ \sigma ))(x)=R(\varphi _1R(\varphi _2(R\dots \varphi _{n-1}R(\varphi _n))\dots ))(x)=(RV_1^*f)(x);$$

The extension from the cylinder functions $\mathcal {A}^{(cyl)}$ to all of $L^2(\Omega ,\Sigma )$ now follows from the usual application of Stone-Weierstrass; recall that $C(\Omega )$ is dense in $L^2(\Omega ,\Sigma )$ relative to the $L^2$-norm; and we have the desired conclusion. $\square$

2.1 What Measures on $B^\mathbb {N}$ have a Transfer Operator?

Below we characterize, among all Borel probability measures $\Sigma$ on $B^\mathbb {N}$, precisely those which arise from a pair $\mu$ and $R$ with a transfer operator $R$ and $\mu$ a measure on $B$. The characterization is general and involves only the one-sided shift $\sigma$ on $B^\mathbb {N}$.

Lemma 2.16

Let $\Sigma \in \mathcal {M}_1(B^\mathbb {N})$ and set $\mu :=\Sigma \circ \pi _1^{-1}\in \mathcal {M}(B)$; then for $\mu$-almost all $x\in B$ there is a field $\mathbb {P}_x\in \mathcal {M}(\pi _1^{-1}(x))$ such that

$$\begin{aligned} d\Sigma =\int \limits _B\,d\mathbb {P}_x\,d\mu (x) \end{aligned}$$

(2.26)

and the following hold

• (i) The operator $V_1:L^2(B,\mu )\rightarrow L^2(B^\mathbb {N},\Sigma )$ given by

  $$\begin{aligned} V_1\varphi =\varphi \circ \pi _1 \end{aligned}$$

  (2.27)

  is isometric.

• (ii) Its adjoint operator $V_1^*:L^2(B^\mathbb {N},\Sigma )\rightarrow L^2(B,\mu )$ satisfies

  $$\begin{aligned} (V_1^*f)(x)=\int _{\pi _1^{-1}(x)}f\,d\mathbb {P}_x=:\mathbb {E}_x(f),\quad (x\in B). \end{aligned}$$

  (2.28)

Proof

(i) For $\varphi \in C(B)$, we have

$$\begin{aligned} \Vert V_1\varphi \Vert _{L^2(\Sigma )}^2=&\,\int _{B^\mathbb {N}}|\varphi \circ \pi _1|^2\,d\Sigma =\int _{B^\mathbb {N}}|\varphi |^2\circ \pi _1\,d\Sigma \\ =&\,\int _B|\varphi |^2\,d(\Sigma \circ \pi _1^{-1})=\int _B|\varphi |^2\,d\mu . \end{aligned}$$

(ii) For $\varphi \in C(B)$ and $f\in L^2(B^\mathbb {N},\Sigma )$, we have

$$\begin{aligned} \int _{B^\mathbb {N}}(V_1\varphi )f\,d\Sigma =\int _B\varphi (x)(V_1^*f)(x)\,d\mu (x), \end{aligned}$$

(2.29)

where $V_1^*f\in L^2(B,\mu )$. Hence

$$\begin{aligned} \int _{B^\mathbb {N}}(\varphi \circ \pi _1)f\,d\Sigma =\int _B\varphi (x)(V_1^*f)(x)\,d\mu (x) \end{aligned}$$

(2.30)

and $(V_1^*f)(x)$ is well defined for $\mu$-almost all $x\in B$. Moreover, the mapping

$$\begin{aligned} C(B^\mathbb {N})\ni f\mapsto (V_1^*f)(x) \end{aligned}$$

(2.31)

is positive; i.e., $f\ge 0$ implies $(V_1^*f)(x)\ge 0$. This follows from (2.30). For if $E\subset B$, $\mu (E)>0$, and $V_1^*f<0$ on $E$ then there exists $\varphi \in C(B)$, $\varphi >0$ such that $\int _B\varphi (x)(V_1^*f)(x)\,d\mu (x)<0$, which contradicts (2.30). Now the conclusion in (2.28) follows from an application of Riesz’ theorem to (2.31). $\square$

Proposition 2.17

Let $\Sigma \in \mathcal {M}(B^\mathbb {N})$, then $(\mathbb {P}_x)_{x\in B}$ from Lemma 2.16 has the form (2.17) in Theorem 2.11 if and only if there is a positive operator $R$ such that $R1=1$ and

$$\begin{aligned} \mathbb {E}_x(f\circ \sigma )=(R(\mathbb {E}\bullet f))(x) \end{aligned}$$

(2.32)

holds for all $x\in B$ and for all $f\in L^2(B^\mathbb {N},\Sigma )$, where in (2.32) we use the notation

$$\begin{aligned} \mathbb {E}_x(\dots )=\int \limits _{\pi _1^{-1}(x)}\dots \,d\mathbb {P}_x=\mathbb {E}^{(\Sigma )}(\dots \,|\,\pi _1=x) \end{aligned}$$

(2.33)

for the field of conditional expectations, and  $\mathbb {E}\bullet f$ denotes the map $x\mapsto \mathbb {E}_xf$.

Proof

The implication (2.17) $\Rightarrow$ (2.32) is already established. It is Lemma 2.14(ii). Now assume some positive operator $R$ exists such that (2.32) holds. We will then prove that $\Sigma$ is the measure determined in Theorem 2.11 from $R$ and $\mu =\Sigma \circ \pi _1^{-1}$. It is enough to verify (2.17) on all finite tensors

$$\begin{aligned} f=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n) \end{aligned}$$

(2.34)

as in (2.16); and we now establish (2.17) by induction, using the assumed (2.32).

The case $n=1$ is

$$\mathbb {E}_x(\varphi \circ \pi _1)=\varphi (x),\quad (\varphi \in C(B),x\in B);$$

and this follows from Lemma 2.16.

For $n=2$, we compute as follows

$$\begin{aligned} \mathbb {E}_x(\varphi _1\circ \pi _1\,\varphi _2\circ \pi _2)=\varphi _1(x)(R\varphi _2)(x). \end{aligned}$$

(2.35)

To do this, we shall prove the following fact, obtained from assumption (2.32):

For $\psi \in C(B)$ and $f\in L^2(B^\mathbb {N},\Sigma )$ we have

$$\begin{aligned} \mathbb {E}_x((\psi \circ \pi _1)f)=\psi (x)\mathbb {E}_x(f). \end{aligned}$$

(2.36)

Using (2.28) in Lemma 2.16(ii), note that (2.36) is equivalent to

$$\int \limits _{B^\mathbb {N}}(\varphi \circ \pi _1)(\psi \circ \pi _1)f\,d\Sigma =\int \limits _B\varphi \psi V_1^*f\,d\mu ,$$

which in turn follows from $V_1(\varphi \psi )=(V_1\varphi )(V_1\psi )$ since $\varphi \mapsto \varphi \circ \pi _1$ is multiplicative.

Returning to (2.35), we then get

$$\begin{aligned} \mathbb {E}_x((\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2))=&\,\varphi _1(x)\mathbb {E}_x(\varphi _2\circ \pi _2)=\varphi _1(x)\mathbb {E}_x(\varphi _2\circ \pi _1\circ \sigma )\\ =&\,\varphi _1(x)R(\mathbb {E}\bullet (\varphi _2\circ \pi _1))(x)=\varphi _1(x)(R\varphi _2)(x). \end{aligned}$$

We shall now be using $\pi _i\circ \sigma =\pi _{i+1}$.

Assume that

$$\begin{aligned} \mathbb {E}_x(f)=\varphi _1(x)R(\varphi _2 R(\dots R(\varphi _n)\dots ))(x) \end{aligned}$$

(2.37)

holds when $f$ in (2.34) has length $n-1$; then we show it must hold if it has length $n$. We set

$$\mathbb {E}_x(f)=\mathbb {E}_x((\varphi _1\circ \pi _1)(g\circ \sigma )),$$

where $g$ is a tensor of length $n-1$. Hence the induction hypothesis yields

$$\mathbb {E}_x(f)=\varphi _1(x)\mathbb {E}_x(g\circ \sigma )=\varphi _1(x)R(\mathbb {E}\bullet (g))$$

which is the right-hand side of (2.37). $\square$

2.2 Subalgebras in $L^\infty (\Omega ,\Sigma )$ and a Conditional Expectation

Let $B$, $R:C(B)\rightarrow C(B)$, $\mu \in \mathcal {M}_1(B)$ and $\Sigma =\Sigma ^{(\mu )}$ be as specified. The only assumptions on $R$ are that

• (i) it is linear;

• (ii) it is positive and

• (iii) $R1=1$.

We will be using Theorem 2.11 and Corollaries 2.12 and 3.14 referring to the measures

$$\begin{aligned} \{\mathbb {P}_x^{(R)} : x\in B\}\,\,\text{ on }\,\,\pi _1^{-1}(x),\quad (x\in B). \end{aligned}$$

(2.38)

The theorem below is about the operators $\{V_n : n\in \mathbb {N}\}$, $V_n:L^2(B,\mu )\rightarrow L^2(\Omega ,\Sigma )$ given by

$$V_n\varphi =\varphi \circ \pi _n,\quad (\varphi \in C(B),n\in \mathbb {N}).$$

Since $V_1: L^2(B,\mu )\rightarrow L^2(\Omega ,\Sigma )$ is isometric, it follows that

$$\begin{aligned} Q_1:=V_1V_1^* \end{aligned}$$

(2.39)

is a projection in each of the Hilbert spaces $L^2(\Omega ,\Sigma ^{(\mu )})$.

Theorem 2.18

With $B,R,\mu , \Sigma =\Sigma ^{(\mu )}$ and $V_n$ specified as above, we have the following formulas:

• (i) $V_1^*V_{n+1}=R^n$ on $L^2(B,\mu )$, $n=0,1,2\dots$;

• (ii) $Q_1:=V_1V_1^*$ is a conditional expectation onto

  $$\mathcal {A}_1:=\{\varphi \circ \pi _1 :\varphi \in L^\infty (B,\mu )\}$$
  $$\begin{aligned} Q_1((\varphi \circ \pi _1)f)=(\varphi \circ \pi _1)Q_1(f)\,\, \text{ for } \text{ all } \varphi \in L^\infty (B,\mu )\text{, } f\in L^\infty (\Omega ,\Sigma )\text{. } \end{aligned}$$

  (2.40)

• (iii) $Q_1(\varphi \circ \pi _{n+1})=(R^n\varphi )\circ \pi _1$   for all   $\varphi \in C(B)$, $n=0,1,2,\dots$.

Proof

(i) As a special case of Theorem 2.11, we see that

$$\begin{aligned} \int \limits _{\pi _1^{-1}(x)}(\varphi \circ \pi _{n+1})\,d\mathbb {P}_x^{(R)}=(R^n\varphi )(x) \end{aligned}$$

(2.41)

holds for all $\varphi \in C(B)$. We further see that (2.41) extends to both $L^\infty (B,\mu )$ and to $L^2(B,\mu )$. Hence

$$\begin{aligned} (V_1^*V_{n+1}\varphi )(x)=\int \limits _{\pi _1^{-1}(x)}(\varphi \circ \pi _{n+1})\,d\mathbb {P}_x^{(\mathbb {R})}=(R^n\varphi )(x),\quad (x\in B). \end{aligned}$$

(2.42)

(ii) By Lemma 2.10, we see that to verify (2.40), it is enough to check it for cylinder functions $f$, i.e.,

$$\begin{aligned} f=(\psi _1\circ \pi _1)(\psi _2\circ \pi _2)\dots (\psi _n\circ \pi _n), \end{aligned}$$

(2.43)

$n\in \mathbb {N}$, $\psi _i\in C(B)$. But if $f$ is as in (2.43), then

$$\begin{aligned} (\varphi \circ \pi _1)f=((\varphi \psi _1)\circ \pi _1)(\psi _2\circ \pi _2)\dots (\psi _n\circ \pi _n), \end{aligned}$$

(2.44)

and the desired formula (2.40) is immediate.

(iii) Given (i), we may apply $V_1$ to both sides in (2.42), and the desired formula (iii) follows. $\square$

It is important to stress that one obtains a closed-form expression for $V_1^*$ where the operator $V_1:\varphi \mapsto \varphi \circ \pi _1$ is introduced in Corollary 2.12. Indeed $V_1^*$ is a conditional expectation:

$$\begin{aligned} (V_1^*f)(x)=\mathbb {E}^{(\Sigma )}(f\,|\,\pi _1=x)=\mathbb {E}_x^{(\Sigma )}(f),\quad (x\in B,f\in L^2(\Omega ,\Sigma )) \end{aligned}$$

(2.45)

By contrast, the situation for $V_n^*$, $n>1$ is more subtle.

Proposition 2.19

Let $B$, $R:C(B)\rightarrow C(B)$ and $\Sigma =\Sigma ^{(\mu )}\in \mathcal {M}_1(\Omega )$ be as above, i.e., $\mu =\Sigma \circ \pi _1^{-1}$. Let $R^*$ be the adjoint of the operator $R$ when considered as a bounded operator in $L^2(B,\mu )$. For $V_2^*$ we have

$$\begin{aligned} V_2^*((\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n))=R^*(\varphi _1)\varphi _2R(\varphi _3\dots R(\varphi _n)\dots ). \end{aligned}$$

(2.46)

Proof

Let $\psi$ be the function given on the right hand side in (2.46). The operator $V_2:\varphi \mapsto \varphi \circ \pi _2$ maps from $L^2(B,\mu )$ into $L^2(\Omega ,\Sigma )$. The assertion in (2.46) follows if we check that, for all $\xi \in C(B)$, we have the following identity:

$$\begin{aligned} \int \limits _\Omega (\varphi _1\circ \pi _1)((\xi \varphi _2)\circ \pi _2)(\varphi _3\circ \pi _3)\dots (\varphi _n\circ \pi _n)\,d\Sigma =\int \limits _B\xi \psi \,d\mu . \end{aligned}$$

(2.47)

But we may compare the left-hand side in (2.47) with the use of Theorem 2.11:

$$=\int \limits _B\varphi _1R((\xi \varphi _2)R(\varphi _3R(\dots R(\varphi _n)\dots )))\,d\mu =\int \limits _B(R^*\varphi _1)\xi \varphi _2R(\varphi _3R(\dots R(\varphi _n)\dots ))\,d\mu ,$$

which is the desired conclusion (2.46).

Recall that, by Theorem 2.18, we have $R=V_1^*V_2$ , and so $R^*=V_2^*V_1$. $\square$

The next result is an extension of Lemma 2.14(ii). Note that (2.23) is the assertion that $V_1^*$ intertwines the two operations, $R$ and $f\mapsto f\circ \sigma$. The next result shows that, by contrast, $V_2^*$ acts as a multiplier.

Corollary 2.20

Let $B,R,\Sigma$ and $\mu =\Sigma \circ \pi _1^{-1}$ be as in Proposition 2.19, and set $\rho :=R^*1\in L^2(B,\mu )$; then

$$(V_2^*(f\circ \sigma ))(x)=\rho (x)\mathbb {E}_x(f)=\rho (x)(V_1^*f)(x),\quad (x\in B, f\in L^2(\Omega ,\Sigma )).$$

Proof

This is immediate from Proposition 2.19, see (2.46). Recall that the span of the tensors is dense in $L^2(\Omega ,\Sigma )$ and that if $f=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n)$, then $f\circ \sigma =(\varphi _1\circ \pi _2)(\varphi _2\circ \pi _3)\dots (\varphi _n\circ \pi _{n+1})$. $\square$

In Proposition 4.6 we calculate the multiplier $\rho$ for the special case of the wavelet representation from Example 2.5.

Corollary 2.21

Let $B$ and $R$ be as in Theorem 2.18, and let $\mu \in \mathcal {M}_1(B)$ be given. The induced measure on $\Omega =B^\mathbb {N}$ is denoted $\Sigma ^{(\mu )}$ and specified as in (2.18). We then have the following equivalence:

• (i) $h\in L^2(B,\mu )$ and $Rh=h$, i.e., $h$ is harmonic; and

• (ii) There exists $f\in L^2(\Omega ,\Sigma ^{(\mu )})$ such that $f=f\circ \sigma$ and

  $$\begin{aligned} h(x)=\int \limits _{\pi _1^{-1}(x)} f\,d\mathbb {P}_x^{(R)}. \end{aligned}$$

  (2.48)

Proof

The implication (ii)$\Rightarrow$(i) follows from Lemma 2.14 and Remark 2.15. For (i)$\Rightarrow$(ii), let $h$ be given and assume it satisfies (i). An application of (iii) from Theorem 2.18 now yields

$$Q_1(h\circ \pi _{n+1})=R^n(h)\circ \pi _1=h\circ \pi _1=V_1h.$$

Using (2.39), we get $V_1(h-V_1^*(h\circ \pi _{n+1}))=0$ for all $n=0,1,2,\dots$; and therefore

$$\begin{aligned} h=V_1^*(h\circ \pi _{n+1}),\quad (n\in \mathbb {N}). \end{aligned}$$

(2.49)

Recalling

$$\begin{aligned} V_1^*(h\circ \pi _{n+1})(x)=\int \limits _{\pi _1^{-1}(x)}h\circ \pi _{n+1}\,d\mathbb {P}_x^{(R)} \end{aligned}$$

(2.50)

and using Theorem 2.11, we conclude that $\{h\circ \pi _{n+1}\}_{n\in \mathbb {N}}$ is a bounded $L^2(\Omega ,\Sigma ^{(\mu )})$-martingale.

By Doob’s theorem, there is a $f\in L^2(\Omega ,\Sigma ^{(\mu )})$ such that

$$\lim _{n\rightarrow \infty }\Vert f-h\circ \pi _{n+1}\Vert _{L^2(\Sigma ^{(\mu )})}=0.$$

Since $\pi _{n+1}\circ \sigma =\pi _n$, it follows that $f\circ \sigma =f$. Taking the limit in (2.49) and using that the operator norm of $V_1^*$ is one, we get that $h=V_1^*f$ and therefore the desired formula (2.48) holds. $\square$

2.3 A Stochastic Process Indexed by $\mathbb {N}$

Remark 2.22

In the literature one has a number of theorems dealing with the existence of measures $\mu$ satisfying the various conditions; and if $\mu \circ R=\mu$ is satsified, then the measure is called a Ruelle equilibrium measure.

Theorem 2.23

Let $B$ be compact Hausdorff and $R:C(B)\rightarrow C(B)$ positive, $R1=1$. Let $\mu \in \mathcal {M}_1(B)$ such that $\mu (B)=1$, $\mu \circ R=\mu$. Set

$$X_n(\varphi )=\varphi \circ \pi _n,\quad (\varphi \in C(B), n\in \mathbb {N})$$

and

$$\begin{aligned} \int f\,d\Sigma =\int \limits _B\int \limits _{\pi _1^{-1}(x)} f\,d\mathbb {P}_x^{(R)}\,d\mu (x) \end{aligned}$$

(2.51)

Then

$$\mathbb {E}(\dots )=\int \dots \,d\Sigma $$

satisfies

$$\begin{aligned} \mathbb {E}(X_n(\varphi )X_{n+k}(\psi ))=\int \limits _B\varphi (x)(R^k\psi )(x)\,d\mu ,\quad (n,k\in \mathbb {N},\varphi ,\psi \in C(B)) \end{aligned}$$

(2.52)

i.e., $R^k$ is the transfer operator governing distances $k$. Asymptotic properties as $k$ goes to infinity govern long-range order.

Proof

From the definition of $\mathbb {P}_x^{(R)}$ we have

$$\begin{aligned} \mathbb {P}_x^{(R)}(X_n(\varphi ))=R^{n-1}(\varphi )(x),\quad \varphi \in C(B) \end{aligned}$$

(2.53)

Now let $n,k,\varphi ,\psi$ as in the statement in (2.52). Let $\Sigma$ be the measure on $B^{\mathbb {N}}$ in (2.51). Then

$$\begin{aligned} \mathbb {E}(X_n(\varphi )X_{n+k}(\psi ))=&\,\int \limits _{B^{\mathbb {N}}}(\varphi \circ \pi _n)(\psi \circ \pi _{n+k})\,d\Sigma \\ =&\,\int \limits _B R^{n-1}(\varphi R^k(\psi ))(x)\,d\mu (x)\\=&\,\int \limits \varphi (x) R^k(\psi )(x)\,d\mu (x) \end{aligned}$$

which is the desired conclusion. $\square$

Definition 2.24

We say that $\{X_k(\varphi )\}$ is independent at $\infty$ if

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathbb {E}(X_n(\varphi )X_{n+k}(\psi ))=\left( \int \varphi \,d\mu \right) \left( \int \psi \,d\mu \right) ,\quad (\varphi ,\psi \in C(B),n\in \mathbb {N}). \end{aligned}$$

(2.54)

Corollary 2.25

Suppose for all $\varphi$ in $C(B)$ we have

$$\lim _{k\rightarrow \infty } R^k(\varphi )=\left( \int \varphi \,d\mu \right) 1,$$

then (2.54) is satisfied.

Proof

We proved

$$\mathbb {E}(X_n(\varphi )X_{n+k}(\psi ))=\int \varphi R^k\psi \,d\mu .$$

Now take the limit as $k\rightarrow \infty$, the desired conclusion (2.54) follows. $\square$

The next result answers the question: what is the distribution of the random variable $X_n(\varphi )$?

Corollary 2.26

Assume $\mu \circ R=\mu$. The distribution of $X_n(\varphi )$ is $\mu (\{x\in B: \varphi (x)\le t\})$ for all $n$.

Proof

Take $\varphi$ real valued for simplicity.

For $t\in \mathbb {R}$,

$$\begin{aligned} \Sigma (\{\tilde{x}\in B^\mathbb {N}: \varphi \circ \pi _n(\tilde{x})\le t\})=&\,\int \limits _B\int \limits _{\pi _1^{-1}(x)}\chi _{\{\varphi \le t\}}\circ \pi _n\,d\mathbb {P}_x^{(R)}\,d\mu (x)\\ =&\,\int \limits _B R^{n-1}\chi _{\{\varphi \le t\}}\,d\mu =\int \limits _B\chi _{\{\varphi \le t\}}\,d\mu . \end{aligned}$$

In particular, it follows that all the random variables $X_n(\varphi )$ have the same distribution. $\square$

2.4 Application to Random Walks

Corollary 2.27

Let $(r,W)$ be as in Definition 2.1, and let $R_W$ be the Ruelle operator in (2.2), $\mathbb {P}_x^{(W)}$—the random walk measure with transition probability specified as follows

$$\begin{aligned} \mathrm{* }{Prob}(x\rightarrow y)=\left\{ \begin{array}{ll} W(y),&{}\,\text{ if }\,r(y)=x\\ 0,&{}\text{ otherwise }\end{array}\right. \end{aligned}$$

(2.55)

Then $\mathbb {P}_x$ from Theorem 2.11 is equal to $\mathbb {P}_x^{(W)}$.

Proof

We apply Theorem 2.11 to $R=R_W$ in (2.2) and we compute the right-hand side in (2.15) with induction

$$(M_{\varphi _1}R_W\dots R_W M_{\varphi _{n+1}})(x)=\varphi _1(x)\sum _{y_1}\dots \sum _{y_n}W(y_1)W(y_2)\dots W(y_n)\varphi _2(y_1)\dots \varphi _{n+1}(y_n),$$

where $r(y_{i+1})=y_i$, $1\le i<n$, $r^{(n)}(y_n)=x$. Further,

$$=\sum \dots \sum \mathrm{* }{Prob}(x\rightarrow y_1)\mathrm{* }{Prob}(y_1\rightarrow y_2\, |, y_1)\dots \mathrm{* }{Prob}(y_{n-1}\rightarrow y_n\,|\,y_{n-1})\varphi (y_1)\dots \varphi (y_n)$$

$$\begin{aligned} =\int d\mathbb {P}_x^{(W\text{-transition } R_W\text{-measure })}\varphi _1\otimes \dots \otimes \varphi _n \end{aligned}$$

(2.56)

$\square$

Remark 2.28

The assertion in (2.56) applies to any random walk measure, for example, the one in Example 2.8.

Let $G=(V,E)$ be as in Example 2.8, with $E$ un-directed edges. Let $c:E\rightarrow [0,\infty )$ be such that

$$\begin{aligned} c_{(xy)}=c_{(yx)}\,\,\text{ for } \text{ all }\,\,(xy)\in E,\quad c_{(xy)}\ne 0\,\,\text{ if }\,\,(xy)\not \in E. \end{aligned}$$

(2.57)

A function as in (2.57) is called conductance.

Set $p=p^c$, where

$$\begin{aligned} p_{xy}=\frac{c_{xy}}{\sum _{z, z\sim x}c_{xz}}=\frac{c_{xy}}{c(x)}, \end{aligned}$$

(2.58)

where

$$c(x)=\sum _{z, z\sim x}c_{xz},\text{ and }\,\, z\sim x\,\,\text{ means }\,\,(zx)\in E.$$

Then there is a unique $\mathbb {P}_x^{(c)}$ such that

$$\int \varphi _1\otimes \dots \otimes \varphi _n\,d\mathbb {P}_x^{(c)}=\sum _{y_1}\dots \sum _{y_n}p_{xy_1}p_{y_1y_2}\dots p_{y_{n-1}y_n}\varphi _1(y_1)\dots \varphi _n(y_n),$$

where the sums are over all $y_1,y_2,\dots ,y_n$ such that $(y_iy_{i+1})\in E$.

Note that $\mathbb {P}_x^{(W)}$ is supported on the solenoid, and $\mathbb {P}_x^{(c)}$ is supported on $S^{(G)}$ (see (2.13)).

Remark 2.29

The last application is useful in the setting of harmonic functions on graphs $G=(V,E)$ with prescribed conductance function $c$ as in (2.57). Set

$$\begin{aligned} (\Delta \varphi )(x)=\sum _{y\in V, y\sim x}c_{xy}(\varphi (x)-\varphi (y)) \end{aligned}$$

(2.59)

the graph Laplacian with conductance $c$.

A function $\varphi$ on $V$ satisfies $\Delta \varphi \equiv 0$, iff

$$\begin{aligned} \varphi (x)=\sum _{y\in V,y\sim x}p_{xy}^{(c)}\varphi (y), \end{aligned}$$

(2.60)

where $p_{xy}^{(c)}=\frac{c_{xy}}{c(x)}$ as in (2.58).

Application. Use $\mathbb {P}_x$ to get harmonic functions. The study of classes of harmonic functions is of interest for infinite networks (see Remarks 2.28 and 2.29), and in Corollary 2.21 is is shown that the harmonic functions $h$ are precisely those that arise from applying $\mathbb {E}_x$ to functions $f$, $f\circ \sigma =f$ on $B^{\mathbb {N}}$, i.e.,

$$h(x)=\int f(x\dots )\,d\mathbb {P}_x,$$

and conversely a martingale limit constructs $f$ from $h$. For more details on this construction, see Corollary 2.21.

2.5 An Application to Integral Operators

Let $K:B\times B\rightarrow [0,\infty )$ be a continuous function and let $\mu$ be a probability measure on $B$ such that

$$\begin{aligned} \int \limits _B K(x,y)\,d\mu (y)=1\,\,\text{ for } \text{ all }\,\,x\in B. \end{aligned}$$

(2.61)

Define

$$R_Kf(x)=\int \limits _BK(x,y)f(y)\,d\mu (y),\quad (x\in B,f\in C(B)).$$

Then $R=R_K$ defines a positive operator as in Definition 2.2, $R_K1=1$ and then $\mathbb {P}_x$ in Theorem 2.11 satisfies

$$\begin{aligned}&\int _{B^{\mathbb {N}}}\varphi _1\otimes \dots \otimes \varphi _{n+1}\,d\mathbb {P}_x=\\&\varphi _1(x)\int \dots \int K(x,y_1)K(y_1,y_2)\dots K(y_{n-1},y_n)\varphi _2(y_1)\dots \varphi _{n+1}(y_n)\,d\mu (y_1)\dots \,d\mu (y_n) \end{aligned}$$

We get a measure $\Sigma$ on $B^{\mathbb {N}}$ as follows

$$\begin{aligned} \int f\,d\Sigma =\int f\,d\mathbb {P}_x\,d\mu (x) \end{aligned}$$

(2.62)

since the right-hand side in (2.62) is independent of $x$.

3 Positive Operators and Endomorphisms

3.1 Preliminaries About $r:B\rightarrow B$

Given an endomorphism $r$, we form the solenoid $\mathrm{Sol }(r)\subset B^{\mathbb {N}}$. Below we will study $\widehat{r}:\mathrm{Sol }(r)\rightarrow \mathrm{Sol }(r)$,

$$\widehat{r}(x_1x_2\dots )=(r(x_1)x_1x_2\dots )$$

and $\widehat{r}\in \mathrm{* }{Aut}(\mathrm{Sol }(r))$.

Given a positive operator $R:C(B)\rightarrow C(B)$, $R1=1$ we then form the measure $\mathbb {P}_x^{(R)}$ in the usual way. We will prove the following property $\mathbb {P}_x^{(R)}\circ \widehat{r}^{-1}=\mathbb {P}_x^{(R)}$ on the solenoid but not on $B^{\mathbb {N}}$.

We will impose the conditon (3.10)

$$R((\varphi \circ r)\psi )=\varphi R\psi $$

as the only axiom. It may or may not be satisfied for some examples of positive operators $R$. But it does hold in the following two examples:

$$(R\varphi )(x)=\sum _{r(y)=x}W(y)\varphi (y)\,\,\text{ and }$$

$$(R\varphi )(x)=\frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}|m(y)|^2\varphi (y),$$

where the functions $W$ and $m$ are given subject to the usual conditions.

For reference to earlier papers dealing with measures on infinite products, random walk, and stochastic processes; see e.g., [JP11, JP10, AJ12].

Example 3.1

Classical wavelet theory on the real line. Let $N=2$, $B=\mathbb {T}=\{z\in \mathbb {C}: |z|=1\}\simeq \mathbb {R}/\mathbb {Z}\simeq (-\frac{1}{2},\frac{1}{2}]$ via $z=e^{2\pi i\theta }$, $\theta \in \mathbb {R}/\mathbb {Z}$; $\mu =d\theta$; $L^2(B,\mu )=L^2((-\frac{1}{2},\frac{1}{2}],d\theta )$, $r:B\rightarrow B$,

$$\begin{aligned} r(z)=z^2,\text{ or } \text{ equivalently }r(\theta \mod \mathbb {Z})=2\theta \mod \mathbb {Z}. \end{aligned}$$

(3.1)

Let

$$\begin{aligned} m_0(\theta )=\sum _{n\in \mathbb {Z}}h_ne^{2\pi i n\theta },\text{ or } \text{ equivalently }\,\,m_0(z)=\sum _{n\in \mathbb {Z}}h_nz^n, \end{aligned}$$

(3.2)

where we assume

$$\sum _{n\in \mathbb {Z}}h_n=\sqrt{2},\quad \sum _{n\in \mathbb {Z}}|h_n|^2<\infty .$$

Lemma 3.2

With $m_0$ as in (3.2), the condition (2.7) is equivalent to

$$\begin{aligned} \sum _{k\in \mathbb {Z}}h_k\overline{h}_{k-2n}=\frac{1}{2}\delta _{n,0}. \end{aligned}$$

(3.3)

Proposition 3.3

([BJ02, DJ05]) Suppose that $m_0$ is as above, and that there is a solution $\varphi \in L^2(\mathbb {R})$ satisfying

$$\begin{aligned} \frac{1}{\sqrt{2}}\varphi \left( \frac{x}{2}\right) =\sum _{n\in \mathbb {Z}}h_n\varphi (x-n), \end{aligned}$$

(3.4)

and

$$\begin{aligned} \text{ The } \text{ translates } \varphi (\cdot -n) \text{ are } \text{ orthogonal } \text{ in } L^2(\mathbb {R})\text{, } n\in \mathbb {Z}. \end{aligned}$$

(3.5)

Set $W:L^2(\mathbb {T})\rightarrow L^2(\mathbb {R})$,

$$\begin{aligned} (W\xi )(x)=\sum _{n\in \mathbb {Z}}\widehat{\xi }(n)\varphi (x-n)=:\pi (\xi )\varphi , \end{aligned}$$

(3.6)

where $\xi \in L^2(\mathbb {T})$, and $\widehat{\xi }(n)=\int _{\mathbb {T}}\overline{e}_n\xi \,d\mu$;

$$\begin{aligned} (S_0\xi )(z)=m_0(z)\xi (z^2),\quad (z\in \mathbb {T}); \end{aligned}$$

(3.7)

and

$$\begin{aligned} (Uf)(x)=\frac{1}{\sqrt{2}}f\left( \frac{x}{2}\right) ,\quad f\in L^2(\mathbb {R}). \end{aligned}$$

(3.8)

• (i) Then $S_0$ is isometric, and $(L^2(\mathbb {R}),\varphi ,\pi ,U)$ is a wavelet representation.

• (ii) The dilation $W:L^2(\mathbb {T})\rightarrow L^2(\mathbb {R})$ then takes the following form: $W$ is isometric and it intertwines $S_0$ and the unitary operator $U$, i.e., we have

  $$\begin{aligned} (WS_0\xi )(x)=(UW\xi )(x)=\frac{1}{\sqrt{2}}(W\xi )\left( \frac{x}{2}\right) ,\quad (\xi \in L^2(\mathbb {T}), x\in \mathbb {R}). \end{aligned}$$

  (3.9)

Remark 3.4

With $m_0$ as specified in Proposition 3.3, we conclude that the wavelet representation can be realized on $L^2(\mathbb {R})$. On the other hand, we will see in Corollary 4.5 that it can be also realized on the solenoid. The two representations have to be isomorphic. The identifications can be done via the usual embedding of $\mathbb {R}$ into the solenoid $x\mapsto (e^{2\pi i x},e^{2\pi ix/2},e^{2\pi i x/2^2},\dots )$. The measure $\Sigma$ in this case is supported on the image of $\mathbb {R}$ under this embedding. For details, see [Dut06].

Axioms. $B$ compact Hausdorff space, $R:C(B)\rightarrow C(B)$ positive linear operator such that $R1=1$, $r:B\rightarrow B$ onto, continuous.

Assume

$$\begin{aligned} R((\varphi \circ r)\psi )=\varphi R(\psi ),\quad (\varphi ,\psi \in C(B)) \end{aligned}$$

(3.10)

Note that (3.10) is the only property that we assume on the operator $R$.

Lemma 3.5

On the solenoid

$$\mathrm{* }{Sol}(r)=\{(x_1,x_2,\dots )\in B^\mathbb {N}: r(x_{i+1})=x_i\},$$

$$\pi _i\circ \widehat{r}=r\circ \pi _i.$$

Proof

For $\tilde{x}=(x_1,x_2,\dots )$,

$$\begin{aligned} \widehat{r}(\tilde{x})=(r(x_1),x_1,x_2,\dots ). \end{aligned}$$

(3.11)

$$\pi _i\circ \widehat{r}(\tilde{x})=x_{i-1}=r(x_i)=r\circ \pi _i(\tilde{x}).$$

$\square$

Remark 3.6

Our initial setup for a given endomorphism $r$ in our present setup is deliberately left open to a variety of possibilities. Indeed, the literature on solenoid analysis is vast, but divides naturally into cases when $r: B\rightarrow B$ has only one contractivity degree; as opposed to a mix of non-linear contractive directions. The first case is common in wavelet analysis, such as those studied in [DJ06, DJ07, DJ10, DJ12]. Examples of the second class, often called “hyperbolic” systems, includes the Smale-Williams attractor, with the endomorphism $r$ there prescribed to preserve a foliation by meridional disks; see e.g., [Kuz10, KP07, KP07, Rue04]. Or the study of complex dynamics and Julia sets; see e.g., [BCMN04] .

Lemma 3.7

Let $r:B\rightarrow B$ be given and let $\widehat{r}\in \mathrm{* }{Aut}(\mathrm{Sol }(r))$ be the induced automorphism on the solenoid. Then

$$\widehat{r}(\pi _1^{-1}(x))=\pi _1^{-1}(r(x))\cap \pi _2^{-1}(x),\quad (x\in B).$$

Proof

Use the definition of $\widehat{r}$ in (3.11). $\square$

Definition 3.8

Given $\mu$ and $R$, they generate the probability measure $\Sigma =\Sigma ^{(\mu )}$ on $B^\mathbb {N}$. We assume $R1=1$ and $\mu (B)=1$. Define

$$\begin{aligned} \mathbb {E}(f)=\int \limits _{B^\mathbb {N}}f\,d\Sigma \end{aligned}$$

(3.12)

$$\begin{aligned} \mathbb {E}_x(f):=\mathbb {E}(f\, |\, \pi _1=x)=\int \limits _{\pi _1^{-1}(x)}f\,d\Sigma \end{aligned}$$

(3.13)

$$\begin{aligned} \mathbb {E}_{x_1,x_2}(f):=\mathbb {E}(f \,|\, \pi _1=x_1,\pi _2=x_2)=\int \limits _{\pi _1^{-1}(x_1)\cap \pi _2^{-1}(x_2)}f\,d\Sigma . \end{aligned}$$

(3.14)

for all $x_1,x_2\in B$. As before we take

$$\begin{aligned} \mathbb {E}(f)=\int \limits _B\int \limits _{\pi ^{-1}(x)}f\,d\mathbb {P}_x\,d\mu (x) \end{aligned}$$

(3.15)

and we then get

$$\begin{aligned} \mathbb {E}_x(f)=\int \limits _{\pi ^{-1}(x)}f\,d\mathbb {P}_x \end{aligned}$$

(3.16)

Lemma 3.9

Let $B$, $R$ and $r$ be given as above.

$$\begin{aligned} R((\varphi \circ r)\psi )=\varphi R(\psi ) \end{aligned}$$

(3.17)

Then the following two are equivalent for some measure $\mu$ on $B$:

• (i) $\mu \circ R=\mu$

• (ii) $\int (\varphi \circ r)\psi \,d\mu =\int \varphi R\psi \,d\mu$.

Proof

(i)$\Rightarrow$(ii). Assume (i) and (3.17). Then

$$\int \varphi \circ r\cdot \psi \,d\mu =\int R((\varphi \circ r)\psi )\,d\mu =\int \varphi R\psi \,d\mu $$

which is condition (ii).

(ii)$\Rightarrow$(i). Assume (ii). Then set $\varphi =1$ in (ii) and we get $\int \psi \,d\mu =\int R\psi \,d\mu$ which is the desired property (i). $\square$

Lemma 3.10

Assume the basic axiom (3.17). For $f\in L^1(\Sigma )$, we denote by $\mathbb {E}\bullet (f)$, the function $x\mapsto \mathbb {E}_x(f)$, $x\in B$. Then

$$\begin{aligned} \mathbb {E}_x(f\circ \sigma )=R(\mathbb {E}\bullet (f))(x) \end{aligned}$$

(3.18)

Also,

$$\begin{aligned} \mathbb {E}_x(f\circ \widehat{r})=\mathbb {E}_{r(x),x}(f) \end{aligned}$$

(3.19)

for all $x\in B$, $f\in L^1(\Sigma )$, or equivalently

$$\begin{aligned} \mathbb {E}(f\circ \widehat{r}\,|\, \pi _1=x)=\mathbb {E}(f\,|\,\pi _1=r(x),\pi _2=x); \end{aligned}$$

(3.20)

see the notations in Definition 3.8.

Proof

Equation (3.18) is proved in (2.32). For (3.19), we use the Stone-Weierstrass approximation as before. If

$$f=(\varphi _1\circ \pi _1)(\varphi _2\circ \pi _2)\dots (\varphi _n\circ \pi _n),$$

then

$$f\circ \widehat{r}=(\varphi _1\circ r\circ \pi _1)(\varphi _2\circ r\circ \pi _2)\dots (\varphi _n\circ r\circ \pi _n),$$

and so

$$\begin{aligned} \mathbb {E}_x(f\circ \widehat{r})=&\,\varphi _1(r(x))R(\varphi _2\circ rR(\varphi _3\circ r\dots R(\varphi _n\circ r))\dots )(x)\\ =&\,\varphi _1(r(x))\varphi _2(x)R(\varphi _3R(\dots \varphi _{n-1}R(\varphi _n))\dots )(x)=\mathbb {E}_{r(x),x}(f), \end{aligned}$$

or equivalentlly (3.20). $\square$

Proposition 3.11

Let $B$ and $R:C(B)\rightarrow C(B)$ be as stated in Theorem 2.11. For every $\mu \in \mathcal {M}_1(B)$ we denote the induced measure on $\Omega =B^{\mathbb {N}}$ by $\Sigma ^{(\mu )}$. If some $r:B\rightarrow B$ satisfies

$$\begin{aligned} R((\varphi \circ r)\psi )=\varphi R(\psi ),\quad (\varphi ,\psi \in C(B)) \end{aligned}$$

(3.21)

then every one of the induced measures $\Sigma ^{(\mu )}$ has its support contained in the solenoid $\mathrm{* }{Sol}(r)$.

Proof

Using Lemma 2.14, it is enough to prove that each of the measures $\mathbb {P}_x^{(R)}$ with $x$ fixed (from Corollary 2.12) has its support equal to

$$\begin{aligned} \pi _1^{-1}(x)\cap \mathrm{* }{Sol}(r) \end{aligned}$$

(3.22)

For every $n$, consider all infinite words indexed by $y\in r^{-n}(x)$ and specified on the beginning length-$n$ segments as follows $\Omega _n(r,x): (x,r^{n-1}(y),\dots ,r(y),y,\text{ free } \text{ infinite } \text{ tail })$ and note that

$$\begin{aligned} \pi _1^{-1}(x)\cap \mathrm{* }{Sol}(r)=\bigcap _n\Omega _n(r,x) \end{aligned}$$

(3.23)

For $n=1$, we have

$$\mathbb {P}_x^{(R)}(\Omega _1(r,x))=R(\chi _{\{x\}}\circ r)(x)=\chi _{\{x\}}(x)R(1)=1,$$

where we used assumption (3.21) in the last step in the computation.

The remaining reasoning in the proof is an induction. Indeed, one checks that

$$\begin{aligned} \mathbb {P}_x^{(R)}(\Omega _n(r,x))=&\,R((\chi _{\{x\}}\circ r)R((\chi _{\{x\}}\circ r^2)R(\dots (\chi _{\{x\}}\circ r^{n-1})R(\chi _{\{x\}}\circ r^n)\dots )))(x)\\ =&\,R((\chi _{\{x\}}\circ r)R(\dots R(\chi _{\{x\}}\circ r^{n-1})\dots ))(x). \end{aligned}$$

Hence the assertion for $n-1$ implies the next step $n$. By induction, we get

$$\mathbb {P}_x^{(R)}(\Omega _n(r,x))=1,\quad (n\in \mathbb {N},x\in B).$$

Using (3.23), we get

$$\mathbb {P}_x^{(R)}(\pi _1^{-1}(x)\cap \mathrm{* }{Sol}(r))=\lim _{n\rightarrow \infty }\mathbb {P}_x^{(R)}(\Omega _n(r,x))=1.$$

As a consequence, the measure $\mathbb {P}_x^{(R)}$ assigns value 1 to the indicator function of $\pi _1^{-1}(x)\cap \mathrm{* }{Sol}(r)$. But

$$\mathrm{* }{Sol}(r)=\bigcup _{x\in B}\pi _1^{-1}(x)\cap \mathrm{* }{Sol}(r).$$

So if $\mu (B)=1$, it follows from (2.51) that

$$\Sigma ^{(\mu )}(\mathrm{* }{Sol}(r))=\int \limits _{B^\mathbb {N}}\chi _{\mathrm{* }{Sol}(r)}\,d\Sigma ^{(\mu )}=1;$$

and as a result that

$$\Sigma ^{(\mu )}(B^\mathbb {N}\!\setminus \!\mathrm{* }{Sol})=0$$

which is the desired conclusion. $\square$

Corollary 3.12

Let $B,r,\mu ,R$ be as above and assume (3.17). Then $\Sigma$ is supported on $\mathrm{Sol }(r)$ and $\widehat{r}$ is invertible on $\mathrm{Sol }(r)$ with $\widehat{r}^{-1}=\sigma$. The measure $\Sigma$ is invariant (for $\widehat{r}$) if and only if

$$\begin{aligned} \mu \circ R=\mu . \end{aligned}$$

(3.24)

Proof

It is enough to prove that

$$\begin{aligned} \int \limits _{\mathrm{Sol }(r)}f\circ \sigma \,d\Sigma =\int \limits _{\mathrm{Sol }(r)}f\,d\Sigma \end{aligned}$$

(3.25)

holds for all $f\in L^1(\Sigma )$ if and only if (3.24) holds. But, by (3.18) we have

$$ \int \limits _{\mathrm{Sol }(r)}f\circ \sigma \,d\Sigma =\int \limits _B\mathbb {E}_x(f\circ \sigma )\,d\mu (x)=\int \limits _BR(\mathbb {E}\bullet (f))(x)\,d\mu (x), $$

and

$$ \int \limits _{\mathrm{Sol }(r)}f\,d\Sigma =\int \limits _B\mathbb {E}_x(f)\,d\mu (x). $$

But the functions $x\mapsto \mathbb {E}_x(f)$ are dense in $L^1(B,\mu )$ as $f$ varies in $L^1(\Sigma )$ (consider for example $f=g \circ \pi _1$ for $g\in C(B)$). Thus the equivalence of (3.24) and (3.25) is immediate from this. $\square$

Corollary 3.13

Let $R$ be a positive operator in $C(B)$ satsifying the axioms above, $R1=1$, $R((\varphi \circ r)\psi )=\varphi R(\psi )$ for all $\varphi ,\psi \in C(B)$. Let $\mu$ be a Borel measure on $B$, and set $\Sigma =\Sigma ^{(\mu )}$

$$\int \limits _{\mathrm{Sol }(r)}f\,d\Sigma :=\int \limits _B\int \limits _{\pi _1^{-1}(x)} f\,d\mathbb {P}_x^{(R)}\,d\mu (x);$$

then

$$\begin{aligned} \mathcal {U}^{(R)}f:=f\circ \widehat{r} \end{aligned}$$

(3.26)

defines a unitary operator on $L^2(\mathrm{Sol }(r),\Sigma )$ if and only if $\mu =\mu \circ R$.

Proof

Since $\widehat{r}$ is invertible in $\mathrm{Sol }(r)$, we conclude that $\mathcal {U}^{(R)}$ maps onto $L^2(\mathrm{Sol }(r),\Sigma )$. Recall $\widehat{r}^{-1}=\sigma$

$$\begin{aligned} \sigma (x_1,x_2,x_3,\dots )=(x_2,x_3,x_4,\dots ) \end{aligned}$$

(3.27)

Since, by Proposition 3.11, the measure $\Sigma$ is supported on $\mathrm{Sol }(r)$, the result follows from Corollary 3.12. $\square$

Corollary 3.14

Let $B$ and $R:C(B)\rightarrow C(B)$ be as in Corollary 2.12. Let $\mu \in \mathcal {M}_1(B)$ and consider $\Sigma =\Sigma ^{(\mu )}$. Let $r:B\rightarrow B$ be an endomorphism.

• (i) For the operators $V_1:L^2(B,\mu )\rightarrow L^2(\mathrm{Sol }(r),\Sigma )$ and  $\mathcal {U}^{(R)}: f\mapsto f\circ \widehat{r}$ acting in $L^2(\mathrm{Sol }(r),\Sigma )$ we have the following covariance relation: $V_1$ is isometric and

  $$(V_1^*\mathcal {U}^{(R)}V_1)(\varphi )=\varphi \circ r,\quad (\varphi \in C(B)).$$

• (ii) Assume that $\mu =\mu \circ R$ and

  $$\begin{aligned} R((\varphi \circ r)\psi )=\varphi R(\psi ),\quad (\varphi ,\psi \in C(B)) \end{aligned}$$

  (3.28)

  holds. For functions $F$ on $\Omega$, say $F\in L^\infty (\mathrm{Sol }(r))$, let $M_F$ be the multiplication operator defined by $F$. Then $\mathcal {U}^{(R)}$ is unitary and the following covariance relation holds:

  $$(\mathcal {U}^{(R)})^*M_F\mathcal {U}^{(R)}=M_{F\circ \sigma }.$$

Proof

(i) We will make use of the formula (2.19) for $V_1^*$, see Corollary 2.12(ii). We now compute

$$(V_1^*\mathcal {U}^{(R)}V_1\varphi )(x)=\mathbb {E}_x\mathcal {U}^{(R)}V_1\varphi =\mathbb {E}_x(\varphi \circ \pi _1\circ \widehat{r})=\mathbb {E}_x(\varphi \circ r\circ \pi _1)=(\varphi \circ r)(x),$$

which is the conclusion in (i). (ii) We proved in Corollary 3.13 that $\mathcal {U}^{(R)}$ is unitary. The covariance relation follows from a simple computation. $\square$

For reference to earlier papers dealing with measures on infinite products, and shift-invariant systems; see e.g., [CGHU12, CH94].

3.2 Compact Groups

As a special case of our construction, we mention the compact groups; this will include the case of wavelet theory.

Proposition 3.15

Assume $B$ is a compact group with normalized Haar measure $\mu$. Let $r:B\rightarrow B$ be a homomorphism $r(xy)=r(x)r(y)$ for all $x,y\in B$, and assume for $N\in \mathbb {N}$, $N>1$

$$\#r^{-1}(x)=N,\quad (x\in B).$$

Set

$$\begin{aligned} (R\varphi )(x)=\frac{1}{N}\sum _{r(y)=x}\varphi (y),\quad (\varphi \in C(B)); \end{aligned}$$

(3.29)

then

• (i) $\mathrm{Sol }(r)$ is a compact subgroup of $B^\mathbb {N}$.

• (ii) The induced measure $\Sigma =\Sigma ^{(\mu ,R)}$ is the Haar measure on the group $\mathrm{Sol }(r)$.

Proof

(i) If $\tilde{x}=(x_1,x_2,\dots ), \tilde{y}=(y_1,y_2,\dots )\in \mathrm{Sol }(r)$ then $r(x_{i+1}y_{i+1})=r(x_{i+1})r(y_{i+1})=x_iy_i$, so $\tilde{x}\tilde{y}\in \mathrm{Sol }(r)$.

(ii) From (3.29) we see that the measures $\mathbb {P}_x$ in the decomposition

$$\begin{aligned} d\Sigma =\int \limits _B\mathbb {P}_x\,d\mu (x) \end{aligned}$$

(3.30)

$\mathbb {P}_x\in \mathcal {M}(\pi _1^{-1}(x))$, $x\in B$, are random-walk measures with uniform distributions on the points in $r^{-1}(x)$, for all $x\in B$. If $\mathbb {E}$ denotes the $\Sigma$-expectation and $\mathbb {E}_x$ the $\mathbb {P}_x$-expectation, we have

$$\mathbb {E}_x(f)=\mathbb {E}(f\,|\,\pi _1=x),\quad (x\in B).$$

For points $\tilde{y}=(y_1,y_2,\dots )\in \mathrm{Sol }(r)$, denote by $f(\cdot \tilde{y})$ the translated function on $\mathrm{Sol }(r)$. Then

$$\begin{aligned} \mathbb {E}(f(\cdot \tilde{y})\,|\,\pi _1=x)=\mathbb {E}(f\,|\,\pi _1=xy_1) \end{aligned}$$

(3.31)

where we use the terminology in Proposition 2.17.

Now, combining (3.30) and (3.31), we arrive at the formula:

$$\begin{aligned} \int \limits _{\mathrm{Sol }(r)}f(\cdot \tilde{y})\,d\Sigma {=}\mathbb {E}(f(\cdot \tilde{y}))=&\,\int \limits _B\mathbb {E}(f(\cdot \tilde{y})\,|\,\pi _1{=}x)\,d\mu (x)\,{=}\int \limits _B\mathbb {E}(f\,|\,\pi _1{=}xy_1)\,d\mu (x)\\ =&\,\int \limits _B\mathbb {E}(f\,|\,\pi _1=x)\,d\mu (x)=\int \limits _{\mathrm{Sol }(r)}f\,d\Sigma . \end{aligned}$$

$\square$

Remark 3.16

In wavelet theory, one often takes $B=\mathbb {R}^n/\mathbb {Z}^n$, and a fixed $n\times n$ matrix $A$ over $\mathbb {Z}$ such that the eigenvalues $\lambda$ satisfy $|\lambda |>1$. For $r:B\rightarrow B$, then take

$$r(x\!\!\!\!\mod \mathbb {Z}^n)=Ax\!\!\!\!\mod \mathbb {Z}^n,\quad (x\in \mathbb {R}^n)$$

and it is immediate that $r$ satisfies the multiplicative property in Proposition 3.15.

There are important examples when $r:B\rightarrow B$ does not satisfy this property.

Example 3.17

(Non-group case: the Smale-Williams attractor) Take $B=\mathbb {T}\times \mathbb D$, where $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ and $\mathbb D=\{z\in \mathbb {C}: |z|\le 1\}$ the disk. For $(t,z)\in \mathbb {T}\times \mathbb D$, set

$$r(t,z)=(2t\!\!\!\!\mod \mathbb {Z},\frac{1}{4} z+\frac{1}{2} e^{2\pi i t}).$$

Then $\mathrm{Sol }(r)$ is the Smale-Williams attractor, see [KP07], a hyperbolic strange attractor.

4 Isometries

Below we study condition on functions $m:B\rightarrow \mathbb {C}$ which gurantees that $L^2(\mu )\ni f\mapsto m\cdot f\circ r\in L^2(\mu )$ defines an isometry in $L^2(B,\mu )$; and we will study the unitary dilations $L^2(\mathrm{Sol }(r),\Sigma )\ni \tilde{f}\mapsto \tilde{f}\circ \widehat{r}\in L^2(\mathrm{Sol }(r),\Sigma )$.

Setting. $B$ fixed compact Hausdorff space. We introduce

• (i) $r:B\rightarrow B$ measurable, onto such that

  $$\begin{aligned} 1\le \#r^{-1}(x)<\infty ,\quad (x\in B) \end{aligned}$$

  (4.1)

• (ii) $\mu$ Borel measure on $B$, $\mu (B)=1$.

• (iii) $m:B\rightarrow \mathbb {C}$ a fixed function on $B$.

Question: Given two of them what are the conditions that the third should satisfy such that

$$\begin{aligned} L^2(\mu )\ni f\mapsto m\cdot f\circ r\in L^2(\mu ) \end{aligned}$$

(4.2)

is an isometry.

Definition 4.1

Transformations of measures. Given $\nu$ measure on $B$, $\nu \in \mathcal {M}_1(B)$ and $r:B\rightarrow B$, set $\nu \circ r^{-1}\in \mathcal {M}_1(B)$. For $A\in \mathcal {B}(B)$ a Borel set, $(\nu \circ r^{-1})(A):=\nu (r^{-1}(A))$ where $r^{-1}(A):=\{x\in B : r(x)\in A\}$.

Fact: $\nu \circ r^{-1}$ is determined uniquely by the condition

$$\begin{aligned} \int \limits _B\varphi \circ r\,d\nu =\int \varphi \,d(\nu \circ r^{-1}),\quad (\varphi \in C(B)) \end{aligned}$$

(4.3)

Lemma 4.2

Fix $r,\mu ,m$; then (4.2) is satisfied iff

$$\begin{aligned} (|m|^2\,d\mu )\circ r^{-1}=\mu \end{aligned}$$

(4.4)

Definition 4.3

Fix $r$, then we say that $\mu$ is strongly invariant iff

$$\begin{aligned} \int \varphi (x)\,d\mu (x)=\int \frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}\varphi (y)\,d\mu (x) \end{aligned}$$

(4.5)

Lemma 4.4

Given $r$ and assume $\mu$ is strongly invariant, then the isometry property (4.2) holds iff the corresponding positive operator

$$(R\varphi )(x):=\frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}|m(y)|^2\varphi (y)$$

satisfies $R1=1$.

Proof

Substitute (4.5) into (4.4). Note that then the equation

$$\int \limits _B\varphi (r(x))|m(x)|^2\,d\mu (x)=\int \limits _B\varphi (x)\frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}|m(y)|^2\,\mu (x),\quad (\varphi \in C(B))$$

holds; so $f\mapsto m\cdot f\circ r$ is isometric in $L^2(\mu )$ iff

$$\frac{1}{\#r^{-1}(x)}\sum _{r(y)=x}|m(y)|^2=1$$

$\mu$-a.e. $x\in B$. $\square$

The next corollary appears in [DJ07, Theorem 5.5].

Corollary 4.5

Let $B$ and $r$ be as specified in Sect. 3, let $\mu$ be strongly invariant and let the function $m$ be quadrature mirror filter, as in Example 2.5. Assume in addition that $m$ is non-singular, i.e.

$$\mu (\{x: m(x)=0\})=0.$$

Then, with $R$ as in Lemma 4.4, we get a wavelet representation as in Theorem 2.6 to $L^2(\mathrm{* }{Sol}(r),\Sigma ^{(\mu )})$ with

• (i) $\mathcal {H}=L^2(\mathrm{Sol }(r),\Sigma ^{(\mu )})$;

• (ii) $\mathcal {U}f=(m\circ \pi _1)(f\circ \widehat{r})$, for all $f\in L^2(\mathrm{Sol }(r),\Sigma ^{(\mu )})$;

• (iii) $\pi (g)f=(g\circ \pi _1)f$, for all $g\in L^\infty (B)$, $f\in L^2(\mathrm{Sol }(r),\Sigma ^{(\mu )})$;

• (iv) $\varphi =1$.

Proof

The details are contained in [DJ07, Theorem 5.5] and require just some simple computations. We only have to check that our measure $\Sigma ^{(\mu )}$ coincides with the one defined in [DJ07]. For this, we use [DJ07, Theorem 5.3] and we have to check that

$$\int \varphi \circ \pi _n\,d\Sigma ^{(\mu )}=\int R^n(\varphi )\,d\mu ,\quad (\varphi \in C(B)).$$

But this follows immediately from the definition of $\Sigma ^{(\mu )}$ in (2.17). $\square$

Proposition 4.6

Let $B,r,\mu$ and $m_0$ as in Example 2.5, i.e., $m_0$ is a QMF and the measure $\mu$ on $B$ is assumed strongly invariant with respect to $r$, and let $R$ as in Lemma 4.4. Then the function $\rho =R^*1$ in Corollary 2.20 is $\rho =|m_0|^2$.

Proof

The result follows if we verify the formula for $R^*$; we have

$$\begin{aligned} (R^*\psi )(x)=|m_0(x)|^2\psi (r(x)). \end{aligned}$$

(4.6)

The derivation of (4.6) may be obtained as a consequence of strong invariance as follows: for all $\varphi ,\psi \in C(B)$, we have:

$$\int \limits _B|m_0|^2(\psi \circ r)\varphi \,d\mu =\int \limits _B\psi (x)\frac{1}{N}\sum _{r(y)=x}|m_0(y)|^2\varphi (y)\,d\mu (x)=\int \limits _B\psi (x)(R\varphi )(x)\,d\mu (x);$$

and the assertion (4.6) follows. $\square$