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\def\equaldef{{\buildrel def \over =}}
\def\rr{\Bbb{R}}
\def\ttmath{\Bbb{T}}
\def\cc{\Bbb{C}}
\def\nn{\Bbb{N}}
\def\dz{\frac{\eps}{16}}
\def\cz{\frac{\eps}{8}}
\def\czt{\frac{\eps}{4}}
\def\om{\omega}
\def\Om{\Omega}
\def\vph{\varphi}
\def\cws{C(\Omega ,S)}
\def\cwl{C(\Omega ,L)}
\def\cgw{C(G\times \Omega)}
\def\flow{(\Omega ,\{T_t\}_{t\in \Bbb R},m)}
\def\skwflow{(G\times \Omega ,\{T^A_t\}_{t\in \Bbb R},m)}
\def\ltg{L^2(G,\nu)}
\def\ltgo{L^2(G\times \Omega ,\nu\times m)}
\def\av#1{\langle #1\rangle}
\def\cak{{\Cal K}}
\def\eps{\varepsilon}
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%
\topmatter
\title\nofrills Construction of ergodic cocycles that are \\
fundamental solutions to linear systems of special form
\endtitle

\footnotetext[1]{The research was supported by the NSF grant DMS-9972132}.
\footnotetext[2]{The research was supported by the NSF grant DMS-?????}.
\author Mahesh G. Nerurkar \footnotemark[1] and Hector J. Sussmann
\footnotemark[2]
\endauthor
\affil Departiment of Mathematics, Rutgers University, Camden, NJ 08102. \\
Departiment of Mathematics, Rutgers University, New Brunswick, NJ
08903.\endaffil
\leftheadtext{M. G. Nerurkar and H. J. Sussmann}
\rightheadtext{Ergodic Cocycles}
\email nmahesh\@.crab.rutgers.edu \\ sussmann\@hamilton.rutgers.edu
\endemail
\endtopmatter

\abstract

        Given a flow  $(\Om ,T,m)$, a closed convex subset $S\subset L$
of a Lie algebra $L$ and a continuous map $A:\Om \to S$ consider the
fundamental matrix solution $X_A$ of the (time-dependent family) of linear
system 
$$
x' = A(T_t\om )x,\quad x \in G,\  \om \in {\Om}\,;
$$
where $G$ is a connected Lie group with Lie algebra $L$. In this paper we
develop a new technique to show that the set of continuous maps $A:\Om \to
S$ for which cocycles $X_A$ is ergodic is residual provided $G$ is
compact and $S$ satisfies ``strong accessibility property''. We remark
that the dimension of $S$ can be much smaller than that of $L$ and thus
our result proves typicality of ergodicity even within very ``thin''
classes of cocycles. Our result covers a number of differential equations
arising in Mahthematical Physics, and in particular applies to the widely
studied example of the Rabi oscillator. As a consequence of our ergodicity
result, it follows that the spectrum of the quasi-energy operator
associated with the cocycle $X_A$ is purely continuous for a typical
$A$. In the case of the Rabi oscillator, this contrasts recent spectral
results obtained using the K.A.M. technique.

\endabstract
\endtopmatter
\document

\subhead \S 1. Introduction
\endsubhead

\medskip

	In this paper we develop a new technique to construct ergodic
cocycles that arise as the fundamental matrix solutions to linear
differential equations of a given form. These constructions are motivated
by questions regarding spectral and stability of forced quantum
oscillator systems (in particular the so called ``Rabi oscillator'' which 
models forced oscillations of a spin $1/2$ particle moving under an
external stationary stochastic field [BJL,DGP]). These questions lead to
problems about ergodic properties of certain skew-product flows (see 
[NJ]). In [NS1] we had established that the cocycle arising from a
generic Rabi oscillator is ``minimal''. In this paper we shall prove that
in fact it is ``ergodic''. This will help us establish that typically the
spectrum of the ``quasi-energy operator'' associated with such systems is
only continuous. This result contrasts a result of [BJL] where the authors
prove the existence of only discrete pure point spectrum using the KAM
technique.

	As in [NS1], our result will in fact be valid for a wide class of
time dependent linear differential systems of ``given specific form''
where the time dependence is ``recurrent'' (in particular quasi-periodic).
Such systems are given by specifying;
\roster
\item [1] a flow $(\Om ,T,m)$, where $\Om$ is a compact metric space
(with metric $d$) and $T = \{T_t\}_{r\in \Bbb R}$ is a one-parameter group
of homeomorphisms of $\Om$ that preserves a Borel probability measure $m$;
\item [2] a connected Lie group $G$ with Lie algebra $L$; and
\item [3] a subset $S$ of $L$.
\endroster
Given a continuous map $A\in C(\Om ,S)$-the space of continuous
$S$-valued maps on $\Om$, consider the family of linear differential
equations (parametrized by points $\om \in \Om$),
$$
x' = A(T_t\om )x,\quad x \in G,\  \om \in {\Om}\,. \tag 
1.1
$$  
Here we identify $L$ with the tangent space of $G$ at the identity. Then
the right hand side of \thetag{1.1} can be thought as the value at $x$ of
the right invariant vector field generated by $A({T_t}{\om})$. Let
$X_A(\om ,t)$ denote the fundamental matrix solution to \thetag{1.1}.
So for each $\omega\in \Omega$, the function $t\rightarrow 
X_A(\omega,t)$ is the solution of \thetag{1.1} such that $X_A(\om ,0) = 
e_G$- the identity of $G$. Then the map $X_A:\Om \times {\Bbb R} 
\rightarrow G$ is continuous and satisfies the following cocycle identity
$$
X_A(\om ,t+s) = X_A(T_t(\om ),s)X_A(\om ,t)
\qquad \text{ for all } \om \in \Om ,\ t,s \in {\Bbb R}.\tag 1.2
$$

	A proper analogue of the flow generated by a family of time 
dependent differential equations such as \thetag {1.1} is the $\bold
{skew}$-$\italic {product\ flow}$ define on $G\times \Om$ by 
$$
T^A_t(g,\om ) = (X_A(\om ,t)g,T_t(\om )).\tag 1.3
$$
Notice that the product measure $\nu \times m$ (where $\nu$ is a left 
Haar measure on $G$) is invariant under the skew-product flow. We shall
denote this skew-product system by $\skwflow$. When $G$ is compact, we 
shall assume that $\nu (G) = 1$.

	If $G$ is a matrix Lie group, say $G \subseteq GL(n,\Bbb C)$,
one defines a representation $V^A = \{V^A_t\}_{t\in {\Bbb R}}$ on 
$L^2(\Om ,{\Bbb C}^n,m)$ by setting
$$
V^A_tf(\om ) = X_A(\om ,t)^{-1} f(T_t(\om )). \tag 1.4
$$
If $G$ is a subgroup of the unitary group $U(n)$, this representation is 
unitary. In the Physics literature the infinitisimal generator of this 
representation is called the $\bold {quasi}$-$\bold {energy\ operator}$
and the stability properties of evolution of such systems are studied
through the spectral properties of this representation. Such a evolution
is regarded $\bold {stable}$ if the spectrum of the quasi-energy
operators is discrete pure point. In this paper we shall show that
(Theorem B) within ``very thin classes of linear differential systems of
special form'', those for which the quasi-energy operator has only purely
continuous spectrum are generic.

	The absence of point spectrum for a quasi-energy operator is very
closely related to a problem of ``lifting ergodicity''. We recall that a
flow $(X,\{T_t\}_{t\in \bold R},\lambda)$ is $\italic {ergodic}$ if the
$\lambda$-measure of any invariant Borel subset is either zero or one. The
ergodicity lifting problem involves constructing ergodic skew-product
flows $\skwflow$, where the map $A$ is constrained by the special form of 
the underlying family of differential equations. Under suitable
hypothesis on the ``nature of constraints'' we shall show (Theorem
A) that for a generic map $A\in \cws$, the skew-product flow is
ergodic. Then the spectral result (Theorem B) follows as a corollary.

	Now we describe the assumption imposed on constraint conditions 
which give the system their special form. The special class of systems is
described by specifying a fixed subset $S$ of the Lie algebra $L$ and then 
the special class results from the demand that the map $A$ be
$S$-valued. We illustrate this with the following example of a
Rabi-oscillator.

\medskip

\noindent $\bold {Example}$ Consider the system
$$
i\frac{d\psi}{dt} = \left( \matrix \lambda&f(t)&\\f(t)^*&-\lambda& 
\endmatrix \right)\psi, \qquad \psi \in {\Bbb C}^2. \tag 1.5
$$
This is the Schr\"odinger equation which describes the dynamics of a
``two level atom'' or a spin $1/2$ particle moving under external
magnetic field $f(t)$. The function $f: {\Bbb R} \rightarrow {\Bbb C}$
is a complex-valued potential, typically quasiperiodic in $t$, and
$\lambda \in {\Bbb R}$ is a fixed parameter. 

	The class of cocycles $X_A$ arising from linear differential
systems of a specific form is described via a constraint on the values of
the map $A:\Omega \rightarrow L$. Namely, one considers only those maps
with values in a fixed compact convex subset $Q$ of the Lie algebra
$L$. We shall impose a certain Lie algebraic condition (the condition of
$\bold {strong\ accessibility}$) on the constraint set $Q$. With this
condition, the set $Q$ can still be very ``thin'' in $L$ in the sense
that $dim(G)$-the dimension of $G$-will in general be higher
than $dim(Q)$ which equals the number of free parameters that we are
allowed to vary to generate a cocycle with the desired properties. For
example, in the case of the Rabi oscillator the cocycle takes values in a
three-dimensional Lie group, but we only have two real degrees of freedom,
since the map $f$-which is our perturbation, takes values in a
two-dimensional space of complex numbers.

	To describe the condition ``of strong accessibility'', first, let
$L(Q)$ be the Lie subalgebra of $L$ generated by $Q$, and let $L_0(Q)$ be
the ideal of $L(Q)$ generated by the difference set $Q - Q = \{x - y\ |\
x\in Q,\ y\in Q\}$. 

\medskip

\proclaim {1.1 Definition} A subset $Q$ of the Lie algebra $L$ has the
$\bold {accessibility\ property}$ if $L(Q) = L$, and the $\bold {strong\
accessibility\ property}$ if $L_0(Q) = L$.
\endproclaim

\gnu {(1) Here we need to add some material on Accessibility Property and pointout
that the example of rabi-oscillator satisfies SAP}

	From now on, we choose an inner product $\langle \cdot ,\cdot 
\rangle$
on $L$ and let $||\cdot ||$ be the corresponding norm. 

\medskip

\proclaim {Theorem A} Let $\flow$ be a flow on a compact metric space
$\Omega$, where $m$ is a flow invariant Borel probability measure on   
$\Omega$. Let $G$ be a compact connected Lie group with Lie algebra $L$
and let $S$ be a subset of $L$. Let $\cws$ denote the metric
space of all continuous maps from $\Om$ to $S$ with the supremum
metric. Assume that \roster
\item [1] the flow $\flow$ is ergodic and aperiodic,
\item [2] $S$ is closed and convex,
\item [3] $S$ has the strong accessibility property.
\endroster
Then the set
$$
C_{erg}(\Omega ,S) = \{A\in C(\Omega ,S) : (G\times \Omega
,T^A,\nu\times m)\ \text {is ergodic}\}, \tag{1.6}
$$
is a residual subset of $C(\Omega ,S)$.
\endproclaim

\medskip

	Next, we state a consequence of Theorem A.  

\medskip

\proclaim {Theorem B} In addition to the hypothesis of Theorem A, suppose
that
\roster
\item [4] $G$ is a closed subgroup of the unitary group $U(n)$ and 
\item [5] $G$ does not fix any ray in the complex projective $n$-space 
$P({\Bbb C}^n)$.
\endroster
Let
$$
\align \Cal C_{cont}(& \Omega , S) \\
& = \{A\in C(\Omega ,S): \text{the representation}\  \{V^A_t\}_{t\in \bold 
R}\ \text{has only continuous spectrum}\}.\endalign
$$
Then $\Cal C_{cont}(\Omega ,S)$ is a residual subset of $C(\Omega ,S)$.
\endproclaim

\medskip

	First, we show that Theorem B can be deduced from Theorem A 
via the following lemma.

\medskip

\proclaim {1.2 Lemma} Suppose that the group $G$ satisfies conditions (4) 
and (5) of Theorem B and suppose that the skew-product flow $\skwflow$ is 
ergodic. Then the representation $V^A$ does not admit any point spectrum.
\endproclaim
\demo{Proof} Suppose $f\in L^2(\Omega ,{\Bbb C}^n,\mu)$ is an eigenvector  
of $V^A$. Since the representation is unitary, our supposition implies 
existence of $\lambda \in \Bbb R$ such that
$$
(V^A_tf)(\om ) = e^{i\lambda t}f(\om ),\quad \text{a.e.}\ \om \in 
\Om \ \text {and}\ \text {for all}\ t\in \Bbb R.
$$
Consider the map $\rho :U(n)\times \Om \rightarrow P({\Bbb C}^n)$, 
defined by
$$
\rho(g,\om) = \pi (g^{-1}f(\om )),
$$
where $\pi : {\Bbb C}^n\rightarrow P({\Bbb C}^n)$ is the 
canonical projection onto the projective space. Then,
$$\align \rho (T^A_t(g,\om )) & = \rho \big(X_A(\om ,t)g, T_t\om 
\big) \\
& = \pi \big(g^{-1}X_A(\om ,t)^{-1}f(T_t\om )\big)\\
& = \pi \big(g^{-1}f(\om )\big) \\
& = \rho (g,\om ). \endalign
$$
Thus $\rho$ is $T^A$ invariant and hence by the ergodicity assumption, it
is constant a.e. $(g,\om )$. Thus there is a non-zero vector $v\in {\Bbb
C}^n$ such that
$$
\pi (g^{-1}f(\om )) = \pi (v)\quad \text {a.e.}\ (g,\om ).
$$
In particular, this means that almost all $g\in G$, map the ray $\pi (v)$
into a fixed ray. This implies that the stabilizer of this fixed ray is a
closed subgroup with full Haar measure and hence must be all of $G$. This 
contradicts assumption (5) of Theorem B. \qid
\enddemo

\medskip

\subhead \S 2. PROOF OF THEOREM A
\endsubhead

\medskip

        To prove this result, first we introduce some notation. If
$f\in\ltg$, we use $\av{f}$ to denote the average of $f$ over $G$ with 
respect to $\nu$, so $\av{f}=\int_{G}f(g)\,d\nu (g)$. If $f$ belongs to
$\ltgo$ and $\om \in \Om$, we use $f^\omega$ to denote the function
$G\ni g\mapsto f(g,\om)\in\cc$.  Then $f^\om\in\ltg$ for almost all
$\om\in \Om$ and $\int_{\Om}\|f^\om\|_{L^2}^2dm(\om)
\le\|f\|_{L^2}^2$.

        For $f\in \ltgo$ set
$$
\hat f(g,\om ) = \av {f} \equaldef \int_{G}f(g,\om )d\nu (g).
$$
Then $\hat f$ is in fact the orthogonal projection of $f$ on the space of
square-integrable functions on $G\times \Om$ that are functions of
$\om$ only.

We use
$$
\align
{\Cal H}&=\ltgo\,,\\
{\Cal H}_0 &= \{f\in {\Cal H}:\int_{G \times \Om}f\,d(\nu \times m) =
0\}\,,\\
{\Cal H}_{0,av} &= \{f\in {\Cal H}:\hat f\equiv 0\}\,.
\endalign
$$
Given $A \in \cwl$ and $t\in \rr$ define a unitary operator $U_t^A$ on
$\Cal H$ by setting
$$
U_t^Af = f\circ T_t^A. \tag {2.1}
$$
Next, given a positive real number $M$, define the averaging operator 
$W_{M,A}$ on $\Cal H$ by setting
$$
W_{M,A}f = \frac {1}{M}\int_{0}^{M}U_t^Afdt .\tag {2.2}
$$
We will then prove the following.

\medskip

\proclaim{Theorem 2.1} Let $\flow$ be a flow on a compact metric space
$\Om$, where $m$ is a flow invariant Borel probability measure on
$\Om$. Let $G$ be a compact connected Lie group with Lie algebra $L$
and let $S$ be a subset of $L$. Assume that
\roster 
\item [1] the flow $\flow$ is ergodic and aperiodic,
\item [2] $S$ is closed and convex,
\item [3] $S$ has the strong accessibility property.
\endroster
Then the set
$$
C_{erg,av}(\Om ,S) = \{A\in \cws : \lim\limits_{M\to \infty}W_{M,A}f = 0
\ \ \text {for all}\ \ f\in {\Cal H}_{0,av}\},
$$
is a residual subset of $\cws$.
\endproclaim
  
\medskip

\noindent{\bf PROOF THAT THEOREM 2.1 IMPLIES THEOREM A.}\

\medskip

Assume that the conditions of Theorem A hold, so we can apply Theorem
2.1 as well. Let $A\in C_{erg,av}(\Omega ,S)$. If $f\in{\Cal H}_0$, then
$\tilde f\ \equaldef f-\hat f\in{\Cal H}_{0,av}$, so $W_{M,A}\tilde f\to 0$
as $M\to\infty$.  Moreover, since the flow $\flow$ is ergodic, the ergodic
average $W_Mh \equaldef \frac{1}{M}\int_{0}^M(h\circ T_t)dt$ converge to 
$0$, where we let $h(\om)\equaldef \langle f^\om\rangle$, because 
$\int_{\Om} h(\om)\,dm(\om)=0$. Observe that
$$
W_Mh = W_{M,A}\hat f.
$$
Therefore $W_{M,A}f = W_{M,A}\tilde f + W_{M,A}\hat f \to 0$ as
$M\to\infty$. Since this is true for all $f\in{\Cal H}_0$, we have shown
that $\skwflow$ is ergodic and subsequently $C_{erg}(Y,G)$ is
residual. \qid

\medskip

\noindent{\bf PROOF OF THEOREM 2.1.}

\medskip

        Given $f \in \Cal H_{0,av}$, $\eps > 0$ and $\bar n\in \nn$,
define a set $E(f,\eps,\bar n)$ as follows:
$$
E(f,\eps,\bar n) = \{A\in\cws : ||W_{n,A}f||_2<\eps\ \text{\rm for
some}\ n\in\nn\ \text{\rm such that}\ n>\bar n\}\,.\tag 2.3
$$

\medskip

\proclaim {Lemma 2.2} Let $A\in \cws$, and let ${\Cal F}$ be a dense
subset of $\Cal H_{0,av}$. If $A\in E(f,\frac {1}{n},\overline n)$ for
all $f\in{\Cal F}$, $n\in\nn$, $\overline n\in\nn$, then $A\in
C_{erg,av}(\Om ,S)$.
\endproclaim
\demo{Proof} Fix an $f\in{\Cal F}$. By the $L^2$ ergodic theorem, the
sequence $W_{n,A}f$ converges in ${\Cal H}$ as $n\rightarrow \infty$
to some $f^*\in \Cal H$. Since $A\in E(f,\frac{1}{n},\overline n)$ for
all $n,\overline n\in\nn$, $f^*$ must be the zero function.  So
$W_{n,A}f\to 0$ for every $f\in{\Cal F}$. Since ${\Cal F}$ is dense in
$\Cal H_{0,av}$, and $\|W_{n,A}\|\le 1$ for all $n$, it follows that
$W_{n,A}f\rightarrow 0$ as $n\rightarrow\infty$ for all $f\in{\Cal
H}_{0,av}$, so $A\in C_{erg,av}(\Om ,S)$. \qid
\enddemo

\medskip

In view of Lemma 2.2, Theorem 2.1 will follow from the Baire
category theorem---together with the facts that (a) $C(\Om ,S)$ is a
complete metric space, because $G$ is a compact metric group, (b) $ 
\cgw \cap {\Cal H}_{0,av}$ is dense in ${\Cal H}_{0,av}$, and (c)
$\cgw$ is separable--- if we prove

\medskip

\proclaim {Lemma 2.3} Let $f\in \cgw \cap{\Cal H}_{0,av}$.
Then $E(f,\frac{1}{n},\overline n)$ is open and dense in $\cws$.
\endproclaim

\medskip

The openness of the sets $E(f,\eps,\bar n)$ is easily checked. We shall
establish their density in the next section. The proof of density
involves a series of constructions involving application of two general
results. In the remainder of this section we shall state these results. 
The first result is about approximating the integral of a function over a
compact group by the average of its translates and the second is a
version of Rokhlin's lemma (due to D. Lind [L]) for flows. We begin by
introducing the necessary notation.

\medskip

        Let $C(G)$ be the space of all continuous (complex valued) maps
on $G$ with the supremum metric. If $h\in C(G)$ and $g\in G$, we define
functions $\tau_gh$ by setting
$$
(\tau_gh)(x)=h(g^{-1}x),\qquad \text {for}\ x\in G.
$$
Clearly, each $\tau_g$ is an isometric linear maps from $C(G)$ onto
$C(G)$.

\medskip
        
\proclaim {Proposition 2.4} Let $G$ be a compact metric group. If
$\cak\subseteq C(G)$ is compact and $\eta>0$,  then there exists a
positive integer $P$, and members $\gamma_1,\ldots,\gamma_{P}$ of $G$,
such that
$$
\Big\|\av{h}-\frac {1}{P}\sum_{r=1}^{P}\tau_{\gamma_r^{-
1}}h\Big\|_{sup}\le\eta\,\,\,\text{\rm  for all}\,\,h\in\cak\,.
\tag{2.4}
$$
\endproclaim
\demo {Proof} Since $\cak$ is a compact subset of $C(G)$, it is bounded
and equicontinuous. Hence there exist (a) a constant $C$ such that
$|h(x)|\le C$ whenever $x\in G$ and $h\in\cak$, and (b) a neighborhood $W$ 
of $e_G$ such that $|h(x)-h(y)|<\frac{\eta}{2}$ whenever $xy^{-1}\in W$
and $h\in\cak$.  Partition $G$ into measurable subsets $G_1,\ldots,G_R$
such that each $G_r$ is contained in a left translate $\xi_rW$ of
$W$. Let $\tilde p_r=\nu (G_r)$.

If $h\in\cak$ and $x\in G$,  then
$$
\align
\av{h}&=\int_Gh(gx)\,d\nu(g)=\sum_{r=1}^R\int_{G_r}h(gx)\,d\nu(g)\\
&=\sum_{r=1}^R\int_{G_r}(h(gx)-h(\xi_rx))\,d\nu(g)+
\sum_{r=1}^R\tilde p_r(\tau_{\xi_r^{-1}}h)(x)\,.
\endalign
$$
If $g\in G_r$, then $(gx)(\xi_rx)^{-1}=g\xi_r^{-1}\in W$, so
$|h(gx)-h(\xi_rx)|\le\frac{\eta }{2}$.  Therefore
$$
\Big|\av{h}-\sum_{r=1}^R\tilde p_r\tau_{\xi_r^{-1}}h(x)\Big|
\le\frac{\eta }{2}
$$
for all $h\in\cak$, $x\in G$. Now, let $M$ be a positive integer, and  
write
$$
\tilde p_r=\frac {p_r}{M}+\hat p_r\,,
$$
where $0\leq \hat p _r<\frac{1}{M}$. Let $P= p_1+\ldots+ p_R$.
Then  
$1-\frac{R}{M}\le\frac{P}{M}\le 1$. Moreover,
$$
\Big|\sum_{r=1}^R\hat p_r\tau_{\xi_r^{-1}}h(x)\Big|\le\frac{CR}{M}\,.
$$
So
$$
\Big|\av{h}-\sum_{r=1}^R\frac{p_r}{M}\tau_{\xi_r^{-1}}h(x)\Big|\le
\frac{\eta}{2}+\frac{CR}{M}\,.
$$
Therefore
$$
\Big|\av{h}-\sum_{r=1}^R\frac{p_r}{P}\tau_{\xi_r^{-1}}h(x)\Big|\le     
\frac{\eta}{2}+\frac{2CR}{M}\,.
$$
Now choose $M$ such that $4CR<M\eta$. Then
$$
\Big\|\av{h}-\sum_{r=1}^R\frac{p_r}{P}\tau_{\xi_r^{-
1}}h\Big\|_{sup}\le\eta\,\,\,\text{\rm for all}\,\,h\in\cak\,.
$$
If we now define
$$
\align
\gamma_1&=\gamma_2=\cdots=\gamma_{p_1}=\xi_1\,,\\
\gamma_{p_1+1}&=\gamma_{p_1+2}=\cdots=\gamma_{p_1+p_2}=\xi_2\,,\\
&\cdots\\
\gamma_{p_1+p_2+\cdots+p_{R-1}+1}&=\gamma_{p_1+p_2+\cdot+p_{R-1}+2}
=\cdots=\gamma_{P}=\xi_R\,,\\
\endalign
$$
then
$$
\Big\|\av{h}-\frac {1}{P}\sum_{r=1}^{P}\tau_{\gamma_r^{-
1}}h\Big\|_{sup}\le\eta\,\,\,\text{\rm  for all}\,\,h\in\cak\,.
$$
\qid
\enddemo

\medskip

	Next, we recall a result of D. Lind (see [L]) which extends
the classical Rokhlin's lemma for aperiodic measure preserving
transformations to aperiodic, measure preserving $\Bbb R$ actions. First
we introduce some notation. Let $(X,\Cal B ,m)$ be a Lebesgue
space and let $(x,t)\to T_t(x):X\times \Bbb R \rightarrow X$ be a jointly
measurable, $m$ preserving flow (i.e. $\Bbb R$ action) on $X$. The flow
$(X,\Cal B,\{T_t\}_{t\in \Bbb R},m)$ is {\it aperiodic} if there exists a
null set $N\subset X$ such that if $x\notin N$ and $t\ne 0$, then $T_tx
\ne x$.

\medskip

\proclaim {Theorem 2.5} Let $(X,\Cal B,T,m)$ be
an aperiodic measure preserving flow on a Lebesgue space. Then, given $N \in
\Bbb N$ and $\eps > 0$ there exists a set $F\subset X$ such that 
\roster
\item [1] $F$ and $T_N(F) \equaldef \cup \{T_t(F) : 0 \leq t \leq N\}$ are 
measurable;
\item [2] $T_tF\cap T_sF = \emptyset$ for all $t\ne s$, $t,s \in [0,N]$;
\item [3] $m(T_N(F)) > 1 - \eps$;
\item [4] the map $\vph : F\times [0,N]\rightarrow T_N(F)$ defined by
$\vp (x,t) = T_t(x)$ is bijective, measurable and its inverse is
measurable (i.e. a measurable isomorphism). Furthermore there exists a
probability measure $\tilde m$ defined on (the measurable subsets) of $F$
such that $\vp^* (m|_{T_N(F)}) = \tilde m \times \frac {dl_N}{N}$, where 
$m|_{T_N(F)}$ is the normalized restriction of $m$ to the set $T_N(F)$ and 
$dl_N$ is the usual Lebesgue measure on $[0,N]$.
\endroster
\endproclaim

\medskip

	The set $F$ in the above theorem is a ``transversal'' to the
flow and $\tilde m$ is a ``transversal measure'' induced by $m$ on $F$.
Also notice that if the flow $(X,\Cal B ,\{T_t\}_{t\in \Bbb R},m)$ is
ergodic and aperiodic, then $\tilde m$ is non-atomic.

	In our set-up the underlying flow is actually a jointly continuous
flow on a compact metric space. Since the measure space obtained from the 
completion of the Borel $\sigma$-algebra with respect to a given non-atomic 
Borel measure on compact spaces is a Lebesgue space [R], we can apply
Lind's theorem in our setting. Furthermore, since the set $T_N(F)$ in
above theorem can be approximated from inside by compact sets and since
our flow is jointly continuous, without loss of generality in our
setting, we can assume that the transversal $F$ is compact.

\gnu {[2] We need to check whether [R] states the result we need in the 
form we desire}

\medskip

\subhead \S 3. Density of $E(f,\eps ,\bar n)$
\endsubhead

\medskip

        Now we turn to the proof of the density of $E(f,\eps ,\bar n)$.
For this purpose, we fix the following objects:
\roster
\item [1] $f\in \cgw\cap{\Cal H}_{0,av}$;
\item [2] $A_0\in \cws$;
\item [3] a positive number $\eps$;
\item [4] a positive number $\delta$ and
\item [5] a positive integer $\bar n$.
\endroster 
To prove the density of $E(f,\eps ,\bar n)$ we shall construct a map $A
\in \cws$ such that
\roster
\item [1] $||A_0 - A|| < \delta$ and
\item [2] $A \in E(f,\eps ,\bar n)$.
\endroster

First we sketch-in general terms-the idea underlying the construction of
map $A$. This map will be first defined on a ``strip'' or the Rokhlin
tower $T_N(E)$ which is identified with the set $E\times [0,N]$ (see the
notation in Theorem 2.5). Here $E$ is the ``base of the tower'' and $N$ is
suitably chosen large positive integer (height of the tower). The
existence of such a tower is guaranteed by D. Lind's theorm. Let $n$ be a
suitably chosen integer that is large but much smaller than $N$. The
aperiodicity of the flow allows us to identify the piece of the orbit
$\{T_t(\om ) : t\in [0,n]\}$ with the interval $[0,n]$ via the map
$t\rightarrow T_t(\om )$, for ``most $\om$'s'' in the strip. Let
$\gamma_{0,\om}(t) \equaldef  A_0(T_t(\om ))$. Then we want to construct
an $\eps$-perturbation $\gamma_{\om}$ of $\gamma_{0,\om}$ in the space
$C([0,n],S)$ of continuous $S$ valued functions so that the solution
$\Gamma_{\om}(t)$ of the Cauchy problem
$$
g_{\om}'(t) = \gamma_{\om}(t)g(t),\quad g_{\om}(0) = e_G,
$$
has the property that the ergodic (time) average on the time interval
$[0,n]$ of the given function $f\in \cgw\cap{\Cal H}_{0,av}$ along the
curve $(\Gamma_{\om}(t),T_t\om)$ is close to zero for ``most of $\om$''s
in the strip.  

	To solve this problem let us examine the Cauchy problem
$$
g' = \gamma_{\om}(t)g,
$$
where $\gamma_{\om}$ is a $S$-valued curve. This system can be regarded 
as a right invariant control system on the Lie group $G$ (see [NS1] for a 
detailed discussion on this point view). The desired perturbation is then 
thought as a ``control'' function that stears the original trajectory (the
one corresponding to $\gamma_{0,\om}$) to suitable ``targets''. The demand
that ``the time average of the given $f$ along the integral curves of this
system is close to zero for most the $\om$'s'' can be translated into a
``description of the targets''. The approximation result - Proposition
2.4 - plays a role in this description. The general strategy of stearing
trajectories to these targets involve ``Property $A^*$'' of the fiber group
$G$. We refer the reader to the discussion on page 316 of [NS1] for a
review of this strategy.

\gnu {[3] may be we need to descibe this strategy in a bit more detail
giving the definition of Property A*}

	The crucial point in implementing this strategy is the existence
of a neighbourhood $W$ of $e_G$ such that the ``time $1$ reachable set''
of the control process $g'(t) = \gamma_{\om}(t)g$, $g(0) = e_G$ contains
some right translate of $W$. It is crucial that this $W$ be independent of
$\om \in \Om$. In the following (Theorem 3.1) we state this crucial fact
precisely We begin by introducing the necessary notation.

	Given a closed interval $[a,b]$, let $C^{pc}([a,b],S)$ denote the
set of piecewise continuous maps $A:[a,b]\rightarrow S$ (i.e. continuous
$S$-valued maps with a finite number of jump discontinuities). If $0\in
[a,b]$, $A\in C^{pc}([a,b],S)$, let $\Gamma_A$ denote the unique
$G$-valued absolutly continuous solution curve of the initial value
problem
$$
\align g' & = A(t)g(t),\qquad t\in [a,b],\\
g(0) & = e_G. \endalign
$$
If $A\in C^{pc}([a,b],S)$ and $\delta > 0$, define the set $N_S(A,\delta
)$ as follows:
$$
N_S(A,\delta ) = \{B\in C^{pc}([a,b],S) : \sup_{t\in [a,b]}||A(t) -
B(t)|| < \delta\}.
$$
The following result is proved in [NS1, Proposition 2.1].

\medskip

\proclaim {Proposition 3.1} Let $G$ be a Lie group with Lie algebra $L$.
Let $S \subseteq L$ be as in Theorem 1.1. Let $\tau > 0$ and $\Cal F
\subseteq C([0,\tau],S)$ be a compact subset. Then given $\delta > 0$
there exist a neighbourhood $W$ of the identity of $G$, depending on
$\delta$ but independent of $A\in {\Cal F}$ such that the set
$\{\Gamma_B(\tau)\ :\ B\in N_S(A,\delta )\}$ contains some right
translate of $W$.
\endproclaim

\medskip

Now we begin with the construction of the desired map.

\medskip

\noindent {\it Step 1} : Let $\cak$ to be the closed convex hull of the
set of all functions given by
$$
\{\tau_g (f^\om) : \om \in \Om\}\subset C(G).
$$
Then $\cak$ is a compact convex subset of $C(G)$ that is also invariant
under $\tau_g$ for all $g\in G$. Furthermore $\av{h}=0$ for every
$h\in\cak$.

        Next, we fix a finite subset $\cak_0$ of $\cak$ such that every
$h\in\cak$ satisfies the inequality \def\refineqcz{3.4}
$$
\|h-h_0\|_{sup}<\dz \tag 3.1
$$
for some $h_0\in\cak_0$. We let $\hat\kappa$ be the cardinality of
$\cak_0$. We let
$$
\beta=\frac{\eps}{8\hat\kappa}\ .\tag3.2
$$
Next, let $\delta_1 > 0$ be such that
$$
\text {if}\ d(g_1,g_2) < \delta_1,\ \text {then}\ |H(g_1) - H(g_2)| <
\frac
{\beta}{2},\  g_1,g_2 \in G\ \  \text {and}    \tag 3.3
$$
for all $H \in \cak$.

\medskip   

\noindent {\it Step 2} : Now applying Proposition 2.4 to the above set $\cak$
and
with $\eta = \beta$,  we get a positive integer $P$ and $\gamma_1,\cdot
\cdot \cdot ,\gamma_P\in G$ such that
$$
\Big\|\frac {1}{P}\sum_{r=1}^P\tau_{\gamma_r^{-1}}h\Big\|_{sup}\le\beta
\,\,\,\text{\rm  for all}\,\,h\in\cak\,. \tag {3.4}
$$

\medskip
  
\noindent {\it Step 3} : Corresponding to the given $\delta$ pick a
neighbourhood $W$ of $e_G$ according to Proposition 3.1. (We apply
Proposition 3.1 by taking $\tau = 1$ and $\Cal F = \{(A_0)_\om : \om \in
\Om\} \subseteq C([0,1],S)$, where $(A_0)_\om (t) = A_0(T_t(\om ))$,
$t\in [0,1]$.) Hence for each $\om \in \Om$ there exists some $h(\om )\in
G$ satisfying  
$$
Wh(\om ) \subseteq \{\Gamma_B(1) : B \in N_S((A_0)_\om ,\delta/3)\}. \tag
{3.5}
$$

\noindent {\it Step 4} : Corresponding to $W$, pick $\kappa \in \Bbb N$
according to Property $A^*$.

\medskip

\noindent {\it Step 5} : Next, we choose $N = (\kappa + \mu )\lambda$
where $\mu$ and $\lambda$ are large positive integers, the precise
choice of which will be described later. Then using Theorem 2.5 pick a
subset $\hat E$ of $\Om$ with the following properties:
\roster
\item [1] $\hat E$ and $T_N(\hat E) \equaldef \cup \{T_t(\hat E) : t\in
[0,N]\}$ are Borel sets;
\item [2] $T_s(\hat E) \cap T_t(\hat E) = \emptyset$ for all $t\ne s$,
$t,s \in [0,N]$;
\item [3] $m(T_N(\hat E)) \ge 1-c_1$ where $c_1$ is a positive
constant whose choice will be described later.
\endroster
Let $\tilde m$ denote the induced measure on the transversal $\hat E$ as in
Theorem (2.5).

\medskip

\noindent {\it Step 6} : Using uniformly continuity of $f$ on $G\times
\Om$, choose
\roster
\item [1] a positive number $c_2$ such that
$|f(g,y)-f(g,z)|\le\cz$ whenever $g\in G$, $y,z\in \Om$, and $d(y,z)\le
c_2$;
\item [2] a neighborhood $V$ of the identity $e_G$ in $G$ such
that $|f(g,\om)-f(h,\om)|\le\cz$ whenever $g,h\in G$, $\om \in \Om$, and
$gh^{-1}\in V$.
\endroster

\medskip

\noindent {\it Step 7} : Let $c_3$ be a positive constant such that
$$
||A_0(T_t({\om}_1)) - A_0(T_t({\om}_2))|| < \frac {\delta}{3}\quad \text
{if}\ d({\om}_1,{\om}_2) < c_3\ \text {and}\ t\in [0,N].
$$

\medskip

\noindent {\it Step 8} : Using continuity of the map $T_t$ ($t\in
[0,N]$), pick a positive  $c_4$ such that
\roster
\item[1] $d(T_ty,T_tz)\le c_2$ whenever $d(y,z)\le c_4$ and $t\in [0,N]$,
\item[2] $c_4 < c_3$.
\endroster

\medskip

\noindent {\it Step 9} : We partition $\Om$ into finitely many
Borel-measurable sets $\Om_1,\ldots,\Om_s$ of diameter less than $c_3$,
and let $\hat E_i=\hat E\cap {\Om}_i$ for $i=1,\ldots,s$. Write
$I=\{1,\ldots,s\}$. Using the fact that $m$ is regular, for each $i\in
I$ pick compact sets $E_i$ such that $E_i\subseteq\hat E_i$ and $m(\hat
E_i\backslash E_i)<\frac{c_1}{sN}$. Note that $\{T_tE_i : i\in I,\ t\in
[0,N]\}$ is a family of mutually disjoint compact sets and 
$$ 
m\Biggl(\bigcup_{i\in I}\bigcup_{t\in [0,N-1]}T_t(E_i)\Biggr) \geq
m\Biggl(\bigcup_{t\in [0,N-1]}T_t(\hat E)\Biggr)(1 - c_1) \geq (1 -
c_1)^2 \ge 1-2c_1\,. \tag {3.6}
$$
We discard any $E_i$'s that have $\tilde m$-measure zero, so
$\tilde m(E_i)>0$, for all $i\in I$.

\medskip

\noindent {\it Step 10} : Let $J = [0,1,\cdot \cdot \cdot \cdot ,N-1]$.
We pick points ${\om}_i\in E_i$ for each $i\in I$, and then set
$\om_{ij}=T_j({\om}_i)$ for $j\in J$.

\medskip 
  
\noindent {\it Step 11} : Now we are going to define the desired map $A$.
For this purpose, we begin by dividing the integer interval $J =
[0,1,\cdot \cdot \cdot \cdot ,N-1]$ into $\lambda$ blocks 
$J_1,\ldots,J_\lambda$ of length $\kappa+\mu$, given by
$$
J_\ell=\{(\ell-1)(\kappa+\mu),(\ell-1)(\kappa+\mu)+1,\ldots,
\ell(\kappa+\mu)-1\}\,\,\,\text{\rm for}\,\,\,\ell\in L\,,
$$
where $L=\{1,2,\ldots,\lambda\}$. We then split each block $J_\ell$ into
two blocks $J_\ell^1$, $J_\ell^2$, of lengths $\kappa$ and $\mu$, by
letting
$$
\align
J_\ell^1&=\{(\ell-1)(\kappa+\mu),(\ell-1)(\kappa+\mu)+1,\ldots,
\ell\kappa+(\ell-1)\mu-1\}\,,\\
J_\ell^2&=\{\ell\kappa+(\ell-1)\mu,\ell\kappa+(\ell-1)\mu+1,
\ldots,\ell(\kappa+\mu)-1\}\,.
\endalign
$$
For each $i\in \{1,\cdot \cdot \cdot ,s\}$ and $j\in J_\ell^1$, where
$\ell \in \{1,\cdot \cdot \cdot , \lambda\}$, consider the map $A_{ij}\in
\Cal F$ defined by
$$
A_{i,j}(t) = A_0(T_t({\om}_{ij})),\qquad t\in [0,1].
$$
Letting $h_{i,j} = h({\om}_{ij})$, \thetag {3.5} yields
$$
Wh_{ij} \subseteq \{\Gamma_B(1) : B\in N_S(A_{ij},\delta /3)\}. \tag {3.7}
$$
For each $(i,j)\in I\times J$ a choice of $g_{ij}\in G$ is made. This
choice will be described later in detail. Then, using \thetag {3.7},
select $\hat A_{ij}\in N_S(A_{ij},\delta )$, (for each $(i,j)\in I\times
J$) such that
$$
g_{ij} = \Gamma_{\hat A_{ij}}(1). \tag {3.8}
$$
Next, for each $\om \in \cup_{i=1}^sE_i$ define $\tilde A_\om :
\Bbb R\rightarrow S$ by setting
$$
\align \tilde A_\om (t) & = A_{ij}(t-j) \quad \text {if}\ j\leq t\leq
j+1,\ j\in J_{\ell}^1,\ l\in \{1,\cdot \cdot \cdot ,\lambda\}\ \text
{and}\ \om \in E_i, \\
& = A_0(T_t\om ) \quad \text {otherwise}\,.\endalign
$$
Note that the map $\tilde A_{\om}$ remains the same as $\om$ varies over
$E_i$, (for each $1\leq i\leq s$) and hence we shall also write $\tilde
A_{\om} = \tilde A_i$ if $\om \in E_i$. Clearly for each $\om \in
\cup_{i=1}^sE_i$, $\tilde A_{\om} \in C^{ps}(\Bbb R,S)$ and 
$$
||\tilde A_{\om}(t) - A_0(T_t\om )|| < \frac {\delta}{3}\quad \om\in 
\cup_{i=1}^{s}E_i\ \text {and}\ t\in [0,N]\,. \tag {3.9}
$$

\medskip

\noindent {\it Step 12} : Now we construct a continuous approximation to
the map $\tilde A_{\om}$. A general approximation procedure involves defining the
following family of maps (parametrized by a positive real parameter $\nu$).
$$
\tilde A_\om^\nu (t) = \nu \int_{t}^{t+\frac {1}{\nu}}\tilde
A_{\om}(s)ds,\qquad \om \in \cup_{i=1}^sE_i\,. \tag {3.10}
$$
We claim that (see [NS1] for full details) by considering large enough values of
$\nu$, we can find a $\bar \nu$ such that
$$
||\tilde A_\om^{\bar \nu}(t) - A_0(T_t\om )|| < \delta, \quad \text {for all}\
t\in [0,N],\ \om \in \cup_{i=1}^sE_i\,, \quad \text {and}\tag {3.11}
$$
$$
\Gamma_{\tilde A_{\om}}(t)\Gamma_{\tilde A_{\om}^{\bar \nu}}(t)^{-1}\in
V^*,\quad \text {for all}\ t\in [0,N],\ \om \in \cup_{i=1}^sE_i\,, \tag
{3.12}
$$
where $V^*$ is a neighbourhood of $e_G$ such that $V^*\cdot V^* \subseteq
V$ and $gV^*g^{-1} \subseteq V^*$ for all $g\in G$.

\gnu {(4) need to insert proof of this claim ?}

\medskip

\noindent {\it Step 13} : Set $A_\om^* = \tilde A_\om^{\bar \nu}$. Then
the map $A^*(\om ,t)\rightarrow A_\om^*(t)$ is continuous on
$\cup_{i=1}^sE_i\times [0,N]$. (In fact $A_\om^* \equiv \tilde A_i^{\bar
\nu}$ if $\om \in E_i$). We may regard $A^*$ as a (continuous) map defined
on the compact subset $\cup \{T_t(\om) : \om\in E_i,\ 1 \leq i \leq s,\
t\in [0,N]\}$; and then extend it continuously to a map $A:\Om \rightarrow L$
satisfying
\roster
\item [1] $A(\om)\in S$ for all $\om \in \Om$ and
\item [2] $||A(\om) - A_0(\om)|| < \delta$ for all $\om \in \Om$.
\endroster
We refer to page 320 of [NS1] for the details regarding the construction
of this extension.

\medskip

\noindent {\it Step 14} : Now we are now going to estimate the norm
$\|W_{n,A}f\|_{L_2}$ for large $n$. We will choose $n=(\kappa+\mu)\rho$ 
such that
$$
\frac {2}{n}\|f\|_{sup} < \frac {\eps}{8},
$$
where $\rho$ is a large positive integer whose precise choice will be 
described later, but it is such that $\frac{\lambda}{\rho}$ is still a
large positive integer. In fact the choice of $\mu$, $\rho$ and then
$\lambda$ (and $N$) will be made in this order. 

\medskip

	Now we begin estimating $\|W_{n,A}f\|_{L_2}$. We let
$$
C = \bigcup_{t\in [0,N-n-1]}\,\bigcup_{i=1}^s T_t(E_i)\,.
$$
Since
$$
m\Big(\bigcup_{t\in [N-n-1,N]}\,\bigcup_{i=1}^s T_t(E_i)\Big)\le
\tilde m(\cup_{i=1}^sE)\cdot \frac {n+1}{N} \le \frac {n+1}{N} = \frac
{\rho}{\lambda} + \frac {1}{N} < \frac {\rho + 1}{\lambda}\,,
$$
the set $C$ satisfies
$$
m(C) \geq 1 - c_1 - \frac {\rho + 1}{\lambda}\,.
$$
Outside $G\times C$, the function $W_{n,A}f$ is bounded pointwise by
$\|f\|_{sup}$, so \def\refestione{3.13}
$$
\int_{G\times(\Om \backslash C)}|W_{n,A}f(g,\om
)|^2d\nu(g)\times m(\om ) \le\|f\|_{sup}^2\,\Big(c_1+\frac
{\rho + 1}{\lambda}\Big)\,.\tag 3.13
$$
Now let $(g,\om )\in G\times C$. Then $\om$ is of the form
$\om = T_{j+t'}\tilde \om$ for some $\tilde \om\in E_i$ where $i\in
\{1,\cdot
\cdot \cdot ,s\}$, $j\in J\equiv \{0,\cdot \cdot \cdot ,N-1\}$ such that 
$j \leq N - n -1$ and $t'\in
[0,1]$ are uniquely determined by $\om$. Note that
$$
\align W_{n,A}f(g,\om ) & = \frac {1}{n}\int_{0}^{n}f(X_A(\om
,s)g,T_s\om)ds \\
& = \frac {1}{n}\int_{0}^{n}f(X_A(\om ,s)g,T_{s+j+t'}\tilde
\om )ds \\
& \overset {\frac {\eps}{8}}\to \approx \frac {1}{n}\int_{0}^{n}f(X_A(\om 
,s)g,T_{s+j+t'}{\om}_i)ds,\quad \text {by (1) of Step 6 and Step
8}\,;\endalign
$$ 
where we write $a \overset {\eps}\to \approx b$ to mean $|a - b| < \eps$.
Thus, writing $w_{ij} \equaldef T_j{\om}_i$ we get
$$
W_{n,A}f(g,\om ) \overset {\frac {\eps}{8}}\to \approx   \frac
{1}{n}\int_{0}^{n}f(X_A(\om ,s)g,T_{s+t'}{\om}_{ij})ds = \frac
{1}{n}\int_{t'}^{n+t'}f(X_A(\om ,s'-t')g,T_{s'}{\om}_{ij})ds'\,.
$$
The choice of $n$ in Step 14 yields $\frac {2t'}{n}\|f\|_{sup} \leq \frac
{2}{n}\|f\|_{sup} < \frac
{\eps}{8}$ and hence
$$
\frac {1}{n}\int_{t'}^{n+t'}f(X_A(\om
,s'-t')g,T_{s'}{\om}_{ij})ds'\  \overset {\frac {\eps}{8}}\to \approx
\ \frac {1}{n}\int_{0}^{n}f(X_A(\om ,s'-t')g,T_{s'}{\om}_{ij})ds'\,.
$$
Thus, using cocycle identity we obtain
$$
\align W_{n,A}f(g,\om ) & \ \overset {\frac {\eps}{4}}\to \approx \ \frac
{1}{n}\int_{0}^{n}f(X_A(\om ,s'-t')g,T_{s'}{\om}_{ij})ds'\\
& \ \overset {\frac {\eps}{4}}\to \approx \ \frac 
{1}{n}\int_{0}^{n}f(X_A(T_j\tilde \om ,s)X_A(T_j\tilde \om 
,t')^{-1}g,T_{s}{\om}_{ij})ds\,.
\endalign
$$
Now we claim that for any $\tilde \om \in E_i$ and $j\in J$ such that
$j\leq N-n-1$,
$$
X_A(T_j\tilde \om,s) = X_A(\om_{ij},s)\quad \text {for all}\ s\in [0,n]\,.
$$
To prove this claim first notice that
$$
A(T_t\tilde \om) = A_{\tilde \om}^*(t) = \tilde A_{\tilde \om}^{\bar
\nu}(t) = \tilde A_{\om_i}^{\bar \nu}(t) =  A_{\om_i}^*(t) =
A(T_t\om_i)\,,
$$
for all $t\in [0,N]$. Thus, $X_A(\tilde \om ,t) = X_A(\om_i,t)$ for all
$t\in [0,N]$. Now the claim follows by using the cocycle identity as
follows;
$$
\align X_A(T_j\tilde \om ,s) & = X_A(\tilde \om ,s+j)X_A(\tilde \om
,j)^{-1}\\
& =  X_A(\om_i ,s+j)X_A(\om_i ,j)^{-1}\qquad (\text {since}\ s+j\in
[0,N])\\
& = X_A(\om_{ij},s)\,. \endalign
$$
Thus,
$$
\align W_{n,A}f(g,\om ) & \overset {\frac {\eps}{4}}\to \approx \frac 
{1}{n}\int_{0}^{n}f(X_A({\om}_{ij},s)X_A({\om}_{ij},t')^{-
1}g,T_{s}{\om}_{ij})ds. \tag {3.14} \endalign
$$

	Now fix $i\in I = \{1,\cdot \cdot \cdot ,s\}$ and $j\in J$ such
that $j\leq N - n - 1$. Set
$$
\align W_{n,A;i,j,t'}f(g) & \equaldef
\int_{0}^{n}f(X_A({\om}_{ij},s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{ij})ds
\\
& = \sum\limits_{k=0}^{n-1}\int_{k}^{k+1} 
f(X_A({\om}_{ij},s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{ij})ds \\
& = \sum_{k=0}^{n-1}\int_{0}^{1} 
f(X_A({\om}_{ij},s'+k)X_A({\om}_{ij},t')^{-1}g,T_{s'}{\om}_{i(j+k)})ds'. 
\endalign
$$
As $k$ varies from $0$ to $n-1$, the index $j+k$ takes values in the
integer interval
$$
\Gamma_n(j) = \{j,j+1,\cdot \cdot \cdot ,j+n-1\},
$$
which is a subset of $J$ because $j \leq N-n-1$. We can then write
$\Gamma_n(j) = \Gamma_n^c(j)\cup \Gamma_n^b(j)$, where the ``central
part'' $\Gamma_n^c(j)$ is a disjoint union of intervals $J_\ell$ and the
``bad part'' $\Gamma_n^b(j)$ has at most $2(\kappa + \mu)$ members. Let
$\Delta_n(j)$ be the set of indices $\ell$ such that $J_\ell \subset
\Gamma_n(j)$. Then
$$
W_{n,A;i,j,t'}f(g) = W_{n,A;i,j,t'}^{c}f(g) + W_{n,A;i,j,t'}^bf(g)\,,\tag
3.15
$$
where $W_{n,A;i,j,t'}^{c}f(g)$, $W_{n,A;i,j,t'}^bf(g)$ are respectively
the contributions to the right-hand side of \thetag {3.14} of the terms
for which $k+j\in\Gamma_n^{c}(j)$ and those for which 
$k+j\in\Gamma_n^b(j)$. Thus, in particular,
$$
\align W_{n,A;i,j,t'}^{c}f(g) & = \sum_{k+j\in\Gamma_n^c(j)}\int_{0}^{1}
f(X_A({\om}_{ij},k+s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(j+k)})ds \\
& = \sum_{\ell\in\Delta_n(j)}\,\,\sum_{k+j\in J_{\ell}}\int_{0}^{1} 
f(X_A({\om}_{ij},k+s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(j+k)})ds,
\endalign
$$
and
$$
W_{n,A;i,j,t'}^{b}f(g) = \sum_{k+j\in\Gamma_n^b(j)}\int_{0}^{1} 
f(X_A({\om}_{ij},k+s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(j+k)})ds.
$$
Since $\Gamma_n^b(j)$ has at most $2(\kappa + \mu)$ indices, we get the
bound
$$
\frac {1}{n}\Big|W_{n,A;i,j,t'}^bf(g)\Big|\le
\frac {2(\kappa+\mu)\|f\|_{sup}}{n} = \frac {2\|f\|_{sup}}{\rho}\,.\tag
{3.16}
$$
Now let us analyze the contribution to $W_{n,A;i,j,t'}^{c}f(g)$ by terms
such as following.
$$
\align W_{n,A;i,j,t';\ell}^{c}f(g) & \equaldef \sum_{k+j\in
J_{\ell}}\int_{0}^{1} 
f(X_A({\om}_{ij},k+s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(j+k)})ds \\
& = \sum_{u\in J_{\ell}}\int_{0}^{1} 
f(X_A({\om}_{ij},u-j+s)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{iu})ds\,.
\endalign
$$
Let $J_\ell\!=\!\{a_\ell,a_\ell\!+\!1,\ldots,a_\ell\!+\!\kappa\!+\!\mu\!-
\!1\}$. Rewriting the summation with $u\!=\!a_\ell\!+\!v$ and using the
cocycle identity we get
$$
\align W_{n,A;i,j,t';\ell}^{c} & f(g) = \sum_{v=0}^{\kappa + \mu -
1}\int_{0}^{1}f(X_A({\om}_{ij},a_\ell+v-j+s)
X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(a_\ell+v)})ds
\\
& = \sum_{v=0}^{\kappa + \mu - 1}\int_{0}^{1} 
f(X_A(T_{a_\ell - j}{\om}_{ij},v+s)X_A({\om}_{ij},a_\ell 
-j)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(a_\ell+v)})ds \\
& =  \sum_{v=0}^{\kappa + \mu - 1}\int_{0}^{1} 
f(X_A({\om}_{ia_\ell},v+s)X_A({\om}_{ij},a_\ell -
j)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(a_\ell+v)})ds\,. \endalign
$$
Now we split the above sum once more, by separating out the first
$\kappa$ terms from the remaining $\mu$ ones. We get
$$
W_{n,A;i,j;t';\ell}^{c}f(g) = W_{n,A;i,j,t';\ell}^{c,-}f(g) +
W_{n,A;i,j,t';\ell}^{c,+}f(g)\,,
$$
where 
$$
W_{n,A;i,j,t';\ell}^{c,-}f(g)=  
\sum_{v=0}^{\kappa-1}\int_{0}^{1}f(X_A({\om}_{ia_\ell},v+s)
X_A({\om}_{ij},a_\ell-j)X_A({\om}_{ij},t')^{-1}g,T_{s}{\om}_{i(a_\ell+v)})ds
$$
and
$$
\align W_{n,A;i,j,t';\ell}^{c,+} & f(g) \\ 
& =\sum_{q=0}^{\mu-1}\int_{0}^{1}f(X_A({\om}_{ia_\ell},\kappa +
q+s)X_A({\om}_{ij},a_\ell-j)X_A({\om}_{ij},t')^{-1}g,
T_{s}{\om}_{i(a_\ell+\kappa + q)})ds\,. \endalign
$$
Note that each function $W_{n,A;i,j,t';\ell}^{c,-}f$ is pointwise bounded
by $\kappa\|f\|_{sup}$. Since the number of members of $\Delta_n(j)$ is at
most $\rho$, we have the estimate
$$
\frac{1}{n}\Big|\sum_{\ell\in\Delta_n(j)}W_{n;A,i,j,t';\ell}^{c,-}
f(g)\Big|\le\frac{\rho\kappa\|f\|_{sup}}{n}=
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}\,. \tag 3.17
$$   
Next, we estimate $W_{n,A;i,j,t';\ell}^{c,+}f(g)$. Recall that
$$
\align W_{n,A;i,j,t';\ell}^{c,+} & f(g) \\
& =\sum_{q=0}^{\mu-1}\int_{0}^{1}f(X_A({\om}_{ia_\ell},\kappa +
q+s)X_A({\om}_{ij},a_\ell-j)X_A({\om}_{ij},t')^{-1}g,
T_{s}{\om}_{i(a_\ell+\kappa + q)})ds\,, \endalign
$$
First we compute the following expression using the cocycle identity,
$$
\align X_A({\om}_{ia_\ell},\kappa + & 
q+s) X_A({\om}_{ij},a_\ell-j)X_A({\om}_{ij},t')^{-1} \\
& = X_A({\om}_{i(a_\ell + 
\kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)X_A({\om}_{ij},a_\ell -
j)X_A({\om}_{ij},t')^{-1} \\
& = X_A({\om}_{i(a_\ell +
\kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)X_A({\om}_{ij},a_\ell -
j)X_A({\om}_{ij},t')^{-1} \\
& = X_A({\om}_{i(a_\ell + \kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)
X_A({\om}_i,a_\ell)X_A({\om}_i,j)^{-1}X_A({\om}_{ij},t')^{-1} \\
& = X_A({\om}_{i(a_\ell + 
\kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)
X_A({\om}_i,a_\ell)\big(X_A({\om}_{ij},t')X_A({\om}_i,j)\big)^{-1}\\
& = X_A({\om}_{i(a_\ell + \kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)
X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}\,.\endalign
$$
Thus,
$$
\align & W_{n,A; i,j,t';\ell}^{c,+}f(g)\\
&  = \sum_{q=0}^{\mu-1}\int_{0}^{1}
f(X_A({\om}_{i(a_\ell + \kappa)},q+s)X_A({\om}_{ia_\ell},\kappa)
X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}g,T_{s}{\om}_{i(a_\ell+\kappa
+ q)})ds\,. \endalign
$$
We now turn to the crucial estimate, namely, the bound for the functions
$W_{n,A;i,j,t';\ell}^{c,+}f$. It is at this point that we will choose the
$g_{ij}$. For this purpose, we will want to assign to each index $i\in I$
and each $\ell\in L$ an integer $r(i,\ell )\in\{1,\ldots,P\}$. We let
$F_{i,\ell}$ and $F^*_{i,\ell}$ be the functions on $G$ defined by
$$
F_{i,\ell}(g)=\frac{1}{\mu}\sum_{q=0}^{\mu-1}\int_{0}^{1}
f(X_A({\om}_{i(a_\ell + \kappa)},q+s)g,T_{s}{\om}_{i(a_\ell+\kappa +
q)})ds\,\quad \text {and}
$$
$$
F^*_{i,\ell}(g)=\frac{1}{\mu}\sum_{q=0}^{\mu-1}\int_{0}^{1}
f(X_{A_0}({\om}_{i(a_\ell + \kappa)},q+s)g,T_{s}{\om}_{i(a_\ell+\kappa +
q)})ds\,\quad \text {and}
$$
Notice that the function $F^*_{i,\ell}$ does not depend at all on the
choice of $g_{ij}$. Now \thetag {3.12}, Step 6, along with the choice of
$V^*$ and the definition of $A$ implies that
$$
||F^*_{i,\ell} - F_{i,\ell}||_{sup} < \cz \,.
$$
Next, observe that
$$
W_{n,A;i,j,t';\ell}^{c,+}f(g)=\mu F_{i,\ell}
\big(X_A({\om}_{ia_\ell},\kappa)X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}g
\big).
$$
Note that for each $q\in \{0,1,\cdot \cdot \cdot ,\mu -1\}$ the map 
$$
g\rightarrow \int_{0}^{1}f(X_{A_0}({\om}_{i(a_\ell + 
\kappa)},q+s)g,T_{s}{\om}_{i(a_\ell+\kappa + q)})ds
$$
belongs to $\Cal K$-the closed convex hull of the set
$\{\tau_g(f^{\om})\}$. Subsequently
each $F^*_{i,\ell}$ also belongs to $\Cal K$. Thus, or each $i,\ell$ we
can choose  $H_{i,\ell}\in\cak_0$ such that
$$
\|F^*_{i,\ell}-H_{i,\ell}\|_{sup}\le \cz.
$$
Thus,
$$
\|F_{i,\ell}-H_{i,\ell}\|_{sup}\le \czt. \tag {3.18}
$$

	For each $i\in I,H\in\cak_0$, let $A(i,H)$ be the set of those
indices $\ell\in L$ such that $H_{i,\ell}=H$.  Enumerate the members
of $A(i,H)$, from left to right, so 
$$
A(i,H)=\{\alpha_{i,H,1},\ldots,\alpha_{i,H,\Lambda(i,H)}\}\,,
$$
with $\alpha_{i,H,1}<\alpha_{i,H,2}<\ldots<\alpha_{i,H,\Lambda(i,H)}$.
Then for each $i\in I$, $\ell\in L$, there exists a unique 
$k=k(i,\ell)\in\{1,\ldots,\Lambda(i,H_{i,\ell})\}$ such that 
$\ell=\alpha_{i,H_{i,\ell},k}$.  We then define $r(i,\ell)$ to be the
remainder of $k(i,\ell)$ modulo $P$, so $k(i,\ell)=\hat k(i,\ell)P +
r(i,\ell)$, where $\hat k(i,\ell)$ and $r(i,\ell)$ are non-negative
integers and $0\le r(i,\ell)<P$. 

	We now define the $g_{ij}$ for $i\in I$, $j\in J$. As explained
above, we let $g_{ij}=e_G$ if $j\in\cup_{\ell}J_{\ell}^2$.  
We then define the family $\{g_{ij}\}_{i\in I,j\in J_{\ell}^1}$
for each $j\in J_{\ell}^1$, inductively with respect to $\ell$. 

	Fix an $\ell$, and assume that the $g_{ij}$ have already been
chosen for $j\in J_{\ell'}^1$ for all $\ell'$ such that $\ell'<\ell$,
but not yet for $j\!\in\!J_\ell^1$.   Then the $g_{ij}$ are in fact
determined for all $j$ such that $j<a_{\ell}$. For any $i\in I$, we have
$$
\align
X_A( & {\om}_{i,a_\ell}, \kappa) X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}
\\
& = X_A({\om}_{i,a_\ell+\kappa-1},1)X_A({\om}_{i,a_\ell+\kappa-2},1)\cdot 
\cdot \cdot
X_A({\om}_{i,a_\ell},1)X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}\\
& =  g_{i,a_\ell+\kappa-1}\cdot z_{\kappa-1}\cdot g_{i,a_\ell+\kappa-
2}\cdot z_{\kappa-2}\cdot \cdot \cdot \cdot g_{i,a_\ell}z_0\,,\tag {3.19}
\endalign
$$
where $z_k = X_{A_0}({\om}_{i,a_\ell+k},1)$ for $0<k<\kappa$, and $z_0 = 
X_{A_0}({\om}_{i,a_\ell},1)X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-1}$. Note 
that $z_0$ is determined, because the $g_{ij}$ have already been selected 
for $j<a_\ell$. Now, our choice of $\kappa$ in Step 4 guarantees that 
$Wz_{\kappa-1}\cdot z_{\kappa-2}\cdot \cdot \cdot \cdot W z_0= G$.
Thus we can choose $g_{i0},g_{i1},\cdot \cdot \cdot ,g_{i(\kappa -1)}$ such 
that
$$
X_A({\om}_{i,a_\ell},\kappa)X_A({\om}_i,a_\ell)X_A({\om}_i,j+t')^{-
1} = \gamma_{r(i,\ell )}\,. \tag 3.20
$$
Then, for any $i,j$, 
$$
\align \sum_{\ell\in\Delta_n(j)}W_{n,A;i,j,t';\ell}^{c,+}f(g)& = \mu
\sum_{\ell\in\Delta_n(j)}F_{i,\ell}\big(
\gamma_{r(i,\ell)X_A({\om}_i,j+t')^{-1}}g\big)\\
& = \mu\sum_{H\in\cak_0}\,\,\sum_{\ell\in A(i,H)\cap\Delta_n(j)}
\tau_{\gamma_{r(i,\ell)}^{-1}}F_{i,\ell}\big(X_A({\om}_i,j+t')^{-1}g\big).
\endalign
$$
Let
$$
{\Cal W}_{n,A;i,j}(g) = \mu\sum_{H\in\cak_0}\sum_{\ell\in
A(i,H)\cap\Delta_n(j)} \tau_{\gamma_{r(i,\ell )}^{-1}}H(g).
$$   
Since $\|H-F_{i,\ell}\|_{sup}\le\czt$ whenever $\ell\in A(i,H)$, we have
the bound
$$
\Big|{\Cal W}_{n,A;i,j}(X_A({\om}_i,j+t')^{-1}g)-
\sum_{\ell\in\Delta_n(j)}W_{n,A;i,j,t';\ell}^{c,+}f(g)\Big|
\le\frac{\mu\rho\eps}{4}\,,
$$
using the fact that $\Delta_n(j)$ has at most $\rho$ members. Observing
that $\frac {\mu \rho}{n} \leq 1$, we get
$$
\frac{1}{n}\Big|{\Cal W}_{n,A;i,j}(X_A({\om}_i,j+t')^{-1}g)-
\sum_{\ell\in\Delta_n(j)}W_{n,A;i,j,t';\ell}^{c,+}f(g)\Big| \le\czt\,.
\tag {3.21}
$$ 

        We now turn to the  task of estimating ${\Cal W}_{n,A;i,j}$. We
write
$$
{\Cal W}_{n,A;i,j}(g) = \mu\sum_{H\in\cak_0}{\Cal W}_{n,A;i,j;H}(g),
$$
where
$$
{\Cal W}_{n,A;i,j;H}(g) = \sum_{\ell\in A(i,H)\cap\Delta_n(j)}
\tau_{\gamma_{r(i,\ell)}^{-1}}H(g). \tag {3.22}
$$
Now suppose that $A(i,H)\cap\Delta_n(j)$ has $q_{i,j,H}P+p_{i,j,H}$
members, where $q_{i,j,H}$ and $p_{i,j,H}$ are integers such that
$q_{i,j,H}\ge 0$ and $0\le p_{i,j,H}<P$. Then the sum of the first
$q_{i,j,H}P$ terms of \thetag {3.22} is equal to
$q_{i,j,H}\sum_{r=1}^{P}\tau_{\gamma_r^{-1}}H(g)$,
whose absolute value is bounded by $P\beta q_{i,j,H}$, because $\av{H}=0$.
The sum of the remaining $p_{i,j,H}$ terms is bounded by
$p_{i,j,H}\|f\|_{sup}$. Thus
$$
\Big|{\Cal W}_{n,A;i,j;H}(g)\Big|\le P\beta
q_{i,j,H}+p_{i,j,H}\|f\|_{sup}\,.
$$
Each number $q_{i,j,H}$ is bounded by $\frac{\rho}{P}$, and
$p_{i,j,H}\le P$. So
$$
\Big|{\Cal W}_{n,A;i,j;H}(g) \Big|\le \rho\beta + P\|f\|_{sup}\,.
$$
Therefore
$$
\|{\Cal W}_{n,A;i,j}\|_{sup}\le \mu\hat\kappa\rho\beta + \mu\hat\kappa
P\|f\|_{sup}\,.\tag {3.23}
$$
If we combine \thetag {3.21} and \thetag {3.23}, we find   
$$
\frac{1}{n}\Big\|\sum_{\ell\in\Delta_n(j)}W_{n,A;i,j,t';\ell}^{c,+}f(g)
\Big\|_{sup}\le\czt+\frac{\mu\hat\kappa\rho\beta}{n}+
\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,, \tag 3.24
$$
We now use \thetag {3.17} and get
$$
\frac{1}{n}\Big|W_{n,A;i,j,t'}^{c}\Big|_{sup}
=\frac{1}{n}\Big|\sum_{\ell\in\Delta_n(j)}
W_{n,A;i,j,t';\ell}^{c}\Big|_{sup}
\le\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\czt+
\frac{\mu\hat\kappa\rho\beta}{n}+\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,.
$$
Then \thetag {3.15} and \thetag {3.16} implies
$$
|W_{n,A,i,j,t'}(g)|\le\frac{2\|f\|_{sup}}{\rho}+
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\czt +
\frac{\mu\hat\kappa\rho\beta}{n}+\frac{\mu\hat\kappa P}{n}
\|f\|_{sup}\,. \tag 3.25
$$
Finally, we use \thetag {3.14} and get the pointwise estimate
$$
|W_{n,A}f(g,y)|\le\czt+\frac{2\|f\|_{sup}}{\rho}+
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\czt +
\frac{\mu\hat\kappa\rho\beta}{n}+
\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,, \tag 3.26
$$
valid  whenever $\bar g\in G$ and $y\in C$. Now note that depending on
the initial given data (i.e. $A_0,f,\eps,\delta$ and $\bar n$ we have
so far chosen $\kappa ,\, \hat \kappa$ and $P$. Next, we choose $\rho$ and
$\mu$ such that
$$
n= (\kappa+\mu)\rho>\bar n\,,\quad \frac
{\|f\|_{sup}}{\rho} \le \dz \,,\quad
\frac {\kappa \|f\|_{sup}}{\mu} \le \dz\,,\quad\text{\rm and}\ \
\frac {P\hat \kappa \|f\|_{sup}}{\rho} \le \dz\,.
$$
Then \thetag {3.26} and \thetag {3.2} implies that
$$
|W_{n,A}f(\bar g,y)|\le\frac{7\eps}{8}\quad\text{\rm
whenever}\ \ g\in G\ \text{\rm and}\ y\in C\,.\tag 3.27
$$
This, together with \thetag {3.13}, gives
$$
\int_{G\times Y}|W_{n,A}f|^2
\le\frac{49\eps^2}{64}+\Big(2c_1+\frac{\rho}{\lambda}\Big)
\|f\|_{sup}^2\,.
$$
Finally we choose $\lambda$ and $c_1$ so that
$(2c_1+\frac{\rho}{\lambda})\|f\|_{sup}^2\!\le\!\frac{14\eps^2}{64}$,
and conclude that $\|W_{n,A}f\|_{L^2}\!<\!\eps$.

        Since $n>\bar n$, we have proved that $\psi\in E(f,\eps,\bar n)$.
By construction, $||A - A_0|| < \delta$ and $A_0\in E(f,\eps,\bar
n)$, concluding the proof of Lemma 2.3 and subsequently the proof of
Theorem 2.1 as well.\qid

\medskip

\head REFERENCES
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\enddocument
\end

\gnu {(3) We summarize various the order in which choices of various
constants and parameters are made in this proof.

Given data (1) $A_0\in C(\Om ,S)$, (2) $f\in \Cal _H{0,av}\cap C(G\times
\Om )$, (3) $\epsilon > 0$, (4) $\delta > 0$, (5) $\bar n\in \Bbb N$.

Level (1) :

(i) $\hat kappa = \hat \kappa (f)$
(ii)  $W = W(f,\delta)$, $kappa = \kappa (W)$   
(iii) $c_2 = c_2 (f,\epsilon)$  
(iv) $V = V(f,\epsilon )$
(v) $c_1 = c_1 (f,\epsilon)$

Level (2)

(i) $\beta = \beta (\hat kappa ,\epsilon), $P = P(\beta )$

Level (3)

(i) $\mu = \mu (f,\kappa ,\epsilon)$
(ii) $\rho = \rho (f,P,\kappa ,\hat \kappa , \epsilon ,\mu ,\bar n)  
(iii) n = (\kappa + \mu)\rho�

Level (4)

(i) $\lambda = \lambda (\f,\rho ,\epsilon ,c_1)$
(ii) N = (\kappa + \mu )\lambda$

Level (5)

(i) $c_3 = c_3 (\A_0 ,\delta ,N)$
(ii) $\bar \nu = \bar \nu (A_0 , \delta ,N)

Level (6)

$c_4 = c_4(c_2,c_3,N)$

Remark $\bar \nu$ does not depend on how the perturbation is defined.}



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