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\def\rr{\Bbb{R}}
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\def\cc{\Bbb{C}}
\def\nn{\Bbb{N}}
\def\dz{\frac{\eps}{16}}
\def\cz{\frac{\eps}{8}}
\def\czt{\frac{\eps}{4}}
\def\sg{\sigma}
\def\bsg{\bar \sigma}
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\def\bnu{\bar \nu}
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\def\ltbg{L^2(\bar G,\bar \nu)}
\def\ltbgy{L^2(\bar G\times Y,\bar \nu\times m)}
\def\av#1{\langle #1\rangle}
\def\cak{{\Cal K}}
\def\eps{\varepsilon}
\def\vp{\varphi}
\def\bvp{\bar \varphi}
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%
\topmatter
\title\nofrills Lifting Ergodicity in $(G,\sigma )$-Extensions
\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{Affine Extensions}
\email nmahesh\@.crab.rutgers.edu \\ sussmann\@hamilton.rutgers.edu
\endemail
\endtopmatter

\abstract 

	Given a dynamical system $(Y,S,m)$ and a pair $(G,\sigma )$ 
consisting of a compact group $G$ and a continuous group automorphism 
$\sigma$ of $G$, we consider the twisted skew-product transformation on
$G\times Y$ given by
$$
S_\vp (g,y) = (\sg [(\vp (y)g],Sy),
$$
where $\vp : Y\rightarrow G$ is a continuous map. If $(Y,S,m)$ is
ergodic and aperiodic, we develop a new technique to show that for a
large class of groups $G$, the set of $\vp$'s for which the map $S_\vp$
is ergodic (with respect to the product measure $\nu\times m$, where
$\nu$ is the normalized Haar measure on $G$) is residual in the space of
continuous maps from $Y$ to $G$. The class of groups for which the
result holds contains the class of all connected abelian as well as the
class of all connected Lie groups. For the class of non-abelian 
fiber groups, this result is the only one of its kind.

\noindent KEY WORDS: Affine extensions, Dynamical systems, ergodicity, 
cocycles.
\endabstract

\document

\head \S 1. Introduction 
\endhead

	By a {\it dynamical system} we mean a triple
$(Y,S,m)$ where $Y$ is a compact metric space, $m$ is
a Borel probability measure on $Y$, and $S$ is an $m$-preserving
homeomorphism of $Y$. Let $G$ be a compact metric topological group and
let $\sg$ be a continuous group automorphism of $G$. Letter $d$
will denote metric on both, $Y$ and $G$.

	Given a continuous map $\vp :Y\rightarrow G$, define a
$\sg$-$\italic {skew}$-$\italic {product}$ transformation $S_\vp
:G\times Y\rightarrow Y\times G$ by the rule
$$
S_\vp (g,y) = (\sg [\vp (y)g],Sy)\qquad (g,y)\in G\times Y.
$$
Let $\nu$ denote the Haar measure on $G$, normalized so that 
$\nu (G)=1$. Then $\nu\times m$ is invariant under $S_\vp$. The 
dynamical system $(G\times Y,S_\vp ,\nu \times m)$ is called a 
$(G,\sg )$-extension or an $\italic {affine\ extension}$ of $(Y,S,m)$.
The more commonly studied `group extension' is a special case when $\sg =
I$-the identity automorphism of $G$.

	Such skew-product extensions have been extensively studied in 
ergodic theory and topological dynamics. However, inspite of the vast 
literature on the subject, several basic questions remain unanswered, 
particularly when the fiber group $G$ is $\bold {non-abelian}$. One such 
question is : Is the set
$$
\Cal C_{erg} \equiv \{\varphi \in \cyg : (G\times Y,S_\varphi 
,\nu \times m)\ \text {is ergodic}\}
$$
non-empty? and if so, how `big' is it? In this paper we shall show that
for large class of compact groups $G$, this set residual in $\cyg$-the
metric space of continuous maps from $Y$ to $G$ with the supremum
metric. The only other generic ergodicity lifting result for $(G,\sg )$
extensions was proved in [KNe] when $G$ is an -n-torus and the
genericity was in the-`not so natural'-class of `closures of
coboundaries'.

	We shall begin by describing the conditions we need to impose
on the pair $(G,\sigma )$.

\medskip

\proclaim {1.1 Definition} Consider a pair $(G,\sigma )$ consisting of
a compact metric group $G$ and its continuous automorphism $\sigma$.
\roster
\item [1] The pair $(G,\sigma )$ has {\it Property A} if given
neighbourhood $W$ of identity $e_G$ there exist a positive integer
$\kappa$ such that, if $W_0,\ldots,W_{\kappa - 1}$ are any $\kappa$ right
translates of $W$,
then
$$
\prod\limits_{i=0}^{\kappa - 1}\sg^{i}\big(W_i\big) \equaldef \sg
\big(W_{\kappa -1}\big)\cdot \sg^2\big(W_{\kappa - 2}\big)\cdot \cdot 
\cdot \cdot \sg^{\kappa}\big(W_0\big) = G.
$$
\item [2] The pair $(G,\sg )$ has {\it Property B} if given $\vp \in
\cyg$ and $\eps > 0$ there exists a $\delta > 0$ such that if $F$ is a 
finite subset of $Y$ and $\psi :F\rightarrow G$ is a function that
satisfies $d(\vp (y),\psi (y)) < \delta$ for all $y \in F$, then
$\psi$ can be extended to a continuous map on $Y$ such that $d(\vp 
(y),\psi (y)) < \eps$ for all $y \in Y$.
\endroster
Following [KN3], we shall call the pair $(G,\sigma )$ {\it admissible}
if it satisfies both, Property A and B.
\endproclaim

\medskip

	We refer to [KN1,3] for the proof of the following.

\medskip

\proclaim {1.2 Proposition} If $G$ is either
\roster
\item [1] a compact, metric connected abelian group or
\item [2] a compact, connected Lie group or
\item [3] a compact connected metric group with finite center,
\endroster
then $(G,\sg )$ is admissible for any continuous automorphism $\sigma$. 
\endproclaim

	Before stating the main theorem, we recall that $(Y,S,m)$ {\it
aperiodic} if $m(\{y\in Y:S^ky=y\ \text{\rm for some}\ k\in\nn\}=0$ and
{\it ergodic} if every $S$-invariant Borel subset $E$ of $Y$ has $m$
measure either zero or one.

\medskip

\proclaim {Theorem 1.3} Let $(Y,S,m)$ be a dynamical system and 
$(G,\sg )$ be a pair where $G$ is a compact group and $\sg$ its continuous
automorphism. Suppose that
\roster
\item [1] $(Y,S,m)$ is a ergodic and aperiodic, and
\item [2] $(G,\sg )$ is admissible.
\endroster
Then the set 
$$
C_{erg}(Y,G) = \{\varphi\in \cyg : (G\times Y,S_{\vp},\nu_G\times m)
\ \text {is ergodic}\}, \tag{1.1}
$$
is a residual subset of $\cyg$.
\endproclaim

\medskip

\subhead \S 2. Reduction of the Proof
\endsubhead

\medskip

	First we describe a crucial result (Proposition 2.2, which is
Corollary 2.4 in [KN2]) that reduces the problem of lifting ergodicity
from the case of a general $\sg$ to the case when $\sg$ is
equicontinuous. ( The automorphism $\sg$ is {\it equicontinuous} if the
family of map $\{{\sg}^i : i \in \nn \}$ is an equicontinuous family of
maps from $G$ to $G$.) First we begin with some notation.

\medskip

\proclaim {2.1 Definition} Consider a given pair $(G,\sg)$ with $G$
compact.
\roster
\item [1] Let $\hat G$ denote the `dual object' i.e. the set of
equivalence classes of finite dimensional irreducible representations
of $G$. A typical element of $\hat G$ will be denoted by $[\gamma ]$
where $\gamma$ denotes a representative of the class $[\gamma ]$.
\item [2] Let $\hat \sg :\hat G\rightarrow \hat G$ be the map induced
by $\sg$, i.e. $\hat \sg [\gamma ] = [\gamma \circ \sg ]$.
\item [3] Given a $[\gamma ] \in \hat G$, let $\Bbb Z_{[\gamma ]}$
denote the {\it stabilizer} subgroup of $[\gamma]$. More precisely,
$$
\Bbb Z_{[\gamma ]} \equaldef \{n \in \Bbb Z : (\hat \sg)^n[\gamma ] = 
[\gamma]\}. 
$$
\item [4] Let
$$
{\hat G}_q = \{[\gamma ]\in \hat G : \Bbb Z_{[\gamma ]}\ \text {has
finite index in}\ \Bbb Z\}.
$$
\item [5] Let $G_q$ be the {\it annihilator} of ${\hat G}_q$, i.e.
$$
G_q \equaldef \text {Ann}\ {\hat G}_q \equaldef \cap_{[\gamma ]\in \hat 
G}\text {ker}\ \gamma.
$$
Then $G_q$ is in fact a closed normal $\sg$ invariant subgroup of
$G$. Let
\item [6] $\bG = G/G_q$,
\item [7] $\bar \sg$ denote the automorphism induced by $\sg$ on $\bar G$. 
\item [8] Let $\pi_q : G\rightarrow G/G_q$ be the canonical homomorphism 
(which is a continuous and open map). A typical element of $\bG$ will be 
denoted by
$$
\bar g \equaldef \pi_q(g).
$$
\item [9] Let $\bnu$ denote the normalized Haar measure on $\bG$. 
Observe that $\pi_q^*(\nu) = \bnu$.
\item [10] Given a $\vp \in \cyg$ we shall set
$$
\bar \vp = \pi_q \circ \vp.
$$
Thus $\vp \rightarrow \bar \vp$ defines a continuous map from $\cyg 
\rightarrow \cybg$.
\item [11] Let $\bar S_{\bar \vp} (\bar g,y) = (\bsg [\bar \vp (y)g],Ty)$, 
where $\vp \in C(Y,G)$.
\endroster
\endproclaim

\medskip
  
	The subgroup $G_q$ is the {\it obstruction} to $\sg$ being 
equicontinuous. More precisely, in the language of topological dynamics
$(\bar G,\bar \sg )$ is a {\it maximal equicontinuous factor} of the 
dynamical system $(G,\sg )$. The following theorem essentially allows
one to reduce the proof to the case when $\sg$ is equicontinuous. 

\medskip

\proclaim {2.2 Proposition} With the notation as above, given $\vp \in 
\cyg$, the dynamical system $(G\times Y,S_\vp,\nu\times m)$ is ergodic 
if and only if the system $(\bar G\times Y,\bar S_{\bar \vp},\bar
\nu\times m)$ is ergodic. 
\endproclaim

\medskip

	This Proposition implies that
$$
C_{erg}(Y,G) = \{\vp \in C(Y,G) : (\bar G\times Y,\bar S_{\bar
\vp},\bar \nu \times m)\ \text {is ergodic}\}.
$$
We want to show that this set is residual. We will in fact prove a
slightly stronger result, of which Theorem 1.3 is a direct consequence.

	To state this result, we first introduce some notation. If
$f\in\ltbg$, we use $\av{f}$ to denote the average of $f$ over $\bG$
with respect to $\bnu$, so $\av{f}=\int_{\bG}f(g)\,d\bnu (g)$.  If $f$
belongs to $\ltbgy$ and $y\in Y$, we use $f^y$ to denote the function
$\bG\ni \bar g\mapsto f(\bar g,y)\in\cc$.  Then $f^y\in\ltbg$ for
almost all $y\in Y$, and 
$\int_{Y}\|f^y\|_{L^2}^2dm(y) \le\|f\|_{L^2}^2$.

	For $f \in \ltbgy$ set
$$
\hat f(\bar g,y) = \av {f} \equaldef \int_{\bG}f(\bar g,y)d\bnu(\bar g).
$$
Then $\hat f$ is in fact the orthogonal projection of $f$ on the space
of square-integrable functions on $\bG \times Y$ that are functions of
$y$ only.

We use 
$$
\align
{\Cal H}&=\ltbgy\,,\\
{\Cal H}_0 &= \{f\in {\Cal H}:\int_{\bG \times Y}f\,d(\bnu \times m) = 
0\}\,,\\
{\Cal H}_{0,av} &= \{f\in {\Cal H}:\hat f\equiv 0\}\,.
\endalign
$$
Given $\vp \in \cyg$, define a unitary operator $\hat {\bar S}_{\bar
\vp}$ on $\Cal H$ by setting
$$
\hat {\bar S}_{\bvp}f = f\circ {\bar S}_{\bvp} = f\circ {\bar S}_{\vp
\circ \pi_q}\,.\tag 2.1
$$
Furthermore, let
$$
W_{n,\bvp} = \frac {1}{n}\sum\limits_{i =
0}^{n-1}\hat {\bar S}^i_{\bvp}\,. \tag 2.2
$$
We will then prove the following

\medskip

\proclaim{Theorem 2.3} Let $(Y,S,m)$ be a discrete dynamical system and 
$G$ be a compact metric group. Suppose that
\roster
\item[1] $(Y,S,m)$ is ergodic and aperiodic and 
\item[2] $(G,\sigma )$ is admissible.
\endroster
Then the set
$$
C_{erg,av}(Y,G) = \{\vp \in \cyg :\lim_{n\to\infty}W_{n,\bvp}f=0
\ \text {for all}\ f\in{\Cal H}_{0,av}\}, \tag{2.3}
$$
is a residual subset of $\cyg$. 
\endproclaim

\medskip\noindent{\bf PROOF THAT THEOREM 2.3 IMPLIES THEOREM 1.3.}\ 
The assumptions of Theorem 1.3 allow us to apply Theorem 2.3. Let
$\vp\in C_{erg,av}(Y,G)$. If $f\in{\Cal H}_0$, then  $\tilde f\equaldef
f-\hat f\in{\Cal H}_{0,av}$. Hence $W_{n,\bvp}\tilde f\to 0$ as
$n\to\infty$. Since $S$ is ergodic, and $\int_Y h(y)\,dm(y)=0$, 
$W_nh=\frac{1}{n}\sum_{i=0}^{n-1}h\circ S^i$ converge to $0$, where $h(y)\equaldef
\langle f^y\rangle$. Observe that
$$
W_nh = W_{n,\bvp}\hat f.
$$
Therefore $W_{n,\bvp}f = W_{n,\bvp}\tilde f + W_{n,\bvp}\hat f \to 0$
as $n\to\infty$. Since this is true for all $f\in{\Cal H}_0$, it
follows that $(\bG \times Y,\bar S_{\bvp} ,\bnu \times m)$ is
ergodic. Thus Proposition 2.2 shows that  $C_{erg,av}(Y,G)\subseteq
C_{erg}(Y,G)$ and consequently $C_{erg}(Y,G)$ is residual. \qid

\medskip

\noindent{\bf PROOF OF THEOREM 2.3.}\ Given $f \in \Cal 
H_{0,av}$, $\eps > 0$ and $\bar n\in \nn$, define a 
set $E(f,\eps,\bar n)$ as follows:
$$
E(f,\eps,\bar n) = \{\vp\in\cyg : ||W_{n,\bvp}f||_2<\eps\ \text{\rm for 
some}\ n\in\nn\ \text{\rm such that}\ n>\bar n\}\,.\tag 2.4
$$

\medskip

\proclaim {Lemma 2.4} Let $\vp\in \cyg$, and let ${\Cal F}$ be a dense 
subset of $\Cal H_{0,av}$. If $\vp\in E(f,\frac {1}{n},\overline n)$ for
all $f\in{\Cal F}$, $n\in\nn$, $\overline n\in\nn$, then $\vp\in 
C_{erg,av}(Y,G)$. 
\endproclaim
\demo{Proof} Fix an $f\in{\Cal F}$. By the $L^2$ ergodic theorem, the 
sequence
$W_{n,\bvp}f$ converges in ${\Cal H}$ as $n\rightarrow \infty$ 
to some $f^*\in \Cal H$. Since $\vp\in E(f,\frac{1}{n},\overline n)$ for
all $n,\overline n\in\nn$, $f^*$ must be the zero function.  So
$W_{n,\bvp}f\to 0$ for every $f\in{\Cal F}$. Since ${\Cal F}$ is dense in
$\Cal H_{0,av}$, and $\|W_{n,\bvp}\|\le 1$ for all $n$, it follows that
$W_{n,\bvp}f\rightarrow 0$ as $n\rightarrow\infty$ for all $f\in{\Cal
H}_{0,av}$, so $\vp\in C_{erg,av}(Y,G)$. \qid
\enddemo

\medskip

In view of Lemma 2.4, Theorem 1.3 will follow from the Baire 
category theorem---together with the facts that (a) $C(Y,G)$ is a
complete metric space, because $G$ is a compact metric group, (b) $
C(G\times Y)\cap {\Cal H}_{0,av}$ is dense ${\Cal H}_{0,av}$, and (c)
$C(Y\times G)$ is separable--- if we prove

\medskip

\proclaim {Lemma 2.5} Let $f\in \cbgy \cap{\Cal H}_{0,av}$.
Then $E(f,\frac{1}{n},\overline n)$ is open and dense in $\cyg$. 
\endproclaim

\medskip

The openness of the set $E(f,\eps,\bar n)$ is easily checked and we
shall establish its density in the next section. The proof of density
involves a series of constructions begining with an application of the
following general result about approximating the integral of a function
over a compact group by average of its translates. First we introduce
the necessary notation.

	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 $C(G)$. 
Furthermore, $\tau_{g_1}\tau_{g_2}=\tau_{g_1g_2}$ whenever $g_1,g_2\in 
G$.

\medskip

\proclaim {Lemma 2.6} Let $G$ be a compact metric group. If 
$\cak\subseteq \cyg$ is compact and $\eta>0$,  then there exists a positive 
integer $P$, and members $\gamma_1,\ldots,\gamma_{P}$ of $G$, such that
\def\refholds{2.5}
$$
\Big\|\av{h}-\frac {1}{P}\sum_{r=1}^{P}\tau_{\gamma_r^{-
1}}h\Big\|_{sup}\le\eta\,\,\,\text{\rm  for all}\,\,h\in\cak\,.
\tag{2.5}
$$
\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 
$$
\Big\|\av{h}-\frac {1}{P}\sum_{r=1}^{P}\tau_{\gamma_r^{-
1}}h\Big\|_{sup}\le\eta\,\,\,\text{\rm  for all}\,\,h\in\cak\,.
\tag {2.6}
$$ 
\qid
\enddemo

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

\medskip

	So 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] an $f\in \cbgy\cap{\Cal H}_{0,av}$;
\item [2] a $\vp\in \cyg$;
\item [3] a positive number $\eps$;
\item [4] $\tilde U$-a neighbourhood of $e_G$-the
identity of $G$.
\item [5] a positive integer $\bar n$.
\endroster
For the given $\vp$ and $\eps$ let $\delta$ be as in Property B. Let $U
= \{g\in \tilde U : d(g,e_G) < \delta\}$ and set
$$
{\Cal V}(\vp,U) = \{\psi\in \cyg : \psi(y)\vp(y)^{-1}\in U\ \text {for
all}\ y\in Y\}.
$$
We will then show that ${\Cal V}(\vp,U)\cap E(f,\eps,\bar 
n)\neq \emptyset$. If this is true for arbitrary $\tilde U$ and $\vp$, it
will follow that $E(f,\eps,\bar n)$ is dense in $\cyg$, concluding our
proof.

For $\vp\in \cyg$, the iterates $\hat S_\vp^{j}$ of $\hat S_\psi$ 
are given by
$$
\hat S_\vp^{j}f(g,y)=f(\vp (y,j)\sg^j(g),S^{j}y)\,,\tag 3.1
$$
where
$$
\vp (y,j)= \sg \big(\vp (S^{j-1}y)\big)\cdot\sg^2\big(\vp
(S^{j-2}y)\big) \cdot\cdot \cdot \cdot \cdot \sg^j\big(\vp
(y)\big)\,\,\, \text{\rm for}\,\,\,j>0, \tag 3.2
$$
and we let $\vp (y,0)=e_G$. Note that $\vp $ satisfies the following
`cocycle identity'
\def\refcocycle{3.3}
$$
\vp (y,m+n) = \vp (S^my,n){\sg}^n[\vp (y,m)],\qquad m,n \in \nn. \tag 3.3
$$
Identities \thetag {3.1,3.2} remain valid when $\vp, S$ and $\sg$ 
are replaced by $\bvp,\bar S$ and $\bar \sg$ respectively.

	We shall denote by $\tau_{\bar g}$ the linear isometry on $C(\bG)$ 
induced by the translation by $\bar g \in \bG$. Furthermore, if $h \in 
C(\bG)$ define
$$
\hat {\bsg}^i(h)(x) = h(\bsg^i (x)) \qquad \text {for}\ x\in \bG\ \text
{and}\ i\in \nn \cup \{0\}.
$$
Then $\hat {\bsg}^i$ also defines a linear isometry from $C(\bG)$ onto 
$C(\bG)$ for each $i\in \nn \cup \{0\}$. With this notation
at hand, we begin describing the construction of a map $\psi \in {\Cal
V}(\vp,U)\cap E(f,\eps,\bar n)$ in a series of steps.

\medskip

\noindent {\it Step 1} : Let $\cak$ to be the closed convex hull of the set of 
all functions given by 
$$
\{\tau_{\bar g}\big(\hat {\bsg}^if^y\big) : \bar g\in \bG,\ i \in \nn
\cup \{0\},\ y\in Y\}.
$$
Since $\bsg$ is equicontinuous on $\bG$, $\cak$ is a compact convex subset
of $C(\bG)$. moreover it is invariant under $\hat {\bsg}$ and $\tau_{\bar
g}$ for all $\bar g \in \bG$ and $\av{h}=0$ 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}
$$
\|h-h_0\|_{sup}<\dz \tag 3.4
$$
for some $h_0\in\cak_0$. We let $\hat\kappa$ be the cardinality of 
$\cak_0$. We let 
$$
\beta=\frac{\eps}{8\hat\kappa}\ .\tag3.5
$$
Next, let $\delta_1 > 0$ be such that
$$
\text {if}\ d(\bar g_1,\bar g_2) < \delta_1,\ \text {then}\ |H(\bar g_1)
- H(\bar g_2)| < \frac {\beta}{2},\ \bar g_1,\bar g_2 \in \bG,
\tag 3.6
$$
for all $H \in \cak$.

\medskip

\noindent {\it Step 2} : Now applying Lemma 2.6 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
$$
\Big\|\frac {1}{P}\sum_{r=1}^P\tau_{\gamma_r^{-
1}}h\Big\|_{sup}\le\beta  \,\,\,\text{\rm  for all}\,\,h\in\cak\,.
\tag {3.7}
$$

\medskip

\noindent {\it Step 3} : For the given neighbourhood $U$ of $e_G$, using
Property A, we get a positive integer $\kappa$ such that for any
$\kappa$ right translates $U_0,\cdot \cdot \cdot ,U_{\kappa -1}$
 of $U$ we have,
$$
\sg \big(U_{\kappa -1}\big){\sg}^2\big(U_{\kappa - 2}\big)\cdot \cdot
\cdot \cdot \cdot \sg^{\kappa}\big(U_0\big) = G.
$$

\medskip

\noindent {\it Step 4} : Next, we choose $N = (\kappa + \mu )\lambda$ 
where $\mu$ and $\lambda$ are large positive integers, the precise choice 
of which will be described later, but for now, we make sure that
$$
d(\bsg^{i(\kappa + \mu)}(\bar g),\bar g) < \delta_1 ,\qquad \text {for
all}\ \bar g \in \bG \ \text {and for all}\ i\in \{1,2,\cdot \cdot
\cdot ,\lambda \}\,,
$$
where $\delta_1$ is as in Step 1. By the equicontinuity of
$\bsg$ such choice  of $\mu$ and $\lambda$ is possible.

\medskip

\noindent {\it Step 5} : Using Rokhlin's lemma, pick a Borel subset $\hat
E$ of $Y$ with the property that the sets $\hat E,\,S\hat 
E\,,\ldots\,,\,S^{N-1}\hat E$ are pairwise disjoint,  and 
$$
m(\hat E\cup S\hat E\cup\ldots\cup S^{N-1}\hat E)\ge 1-c_1\,,
$$
where $c_1$ is a positive constant whose choice will be described later.

\medskip

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

\medskip

\noindent {\it Step 7} : Let $\Phi^N$ be the subset of $\cyg$
consisting of all functions $\eta:Y\to G$ of the form 
\def\refofform{3.8}
$$
\eta(y)=g_{r}\cdot\sg \big(\vp(S^ry)\big)\cdot g_{r-1}\cdot
\sg^2(\vp(S^{r-1}y)\big)\cdot \cdot \cdot \cdot 
\sg^{r+1}\big(\vp(y)\big)\cdot g_0   \tag 3.8
$$
as $r$ ranges over $\{0,\ldots, N-1\}$ and $(g_0,\ldots,g_r)$ ranges
over the set of all $r+1$-tuples of members of $G$. Then $\Phi^N$
is compact, because the right-hand side of (\refofform), regarded as
a function of $y$, depends continuously on the $r+1$-tuple
$(g_0,\ldots,g_r)$, which takes values in the compact space
$G^{r+1}$. It follows that the family $\Phi^N$ is equicontinuous, so
corresponding to the neighborhood $V$ of ${\bar e}_G$ chosen in Step 6, 
there exists a constant $\omega(N,V)$ with the property that 
$\bar \pi_q\big(\eta(y)\eta(z)^{-1}\big) \in V$ whenever
$\eta\in\Phi^N$, $y,z\in Y$, and $d(y,z)\le\omega(N,V)$.

\medskip

\noindent {\it Step 8} : Write $J=\{0,1,\ldots,N-1\}$. Using the
continuity of maps  $S^j$, ($j\in J$), pick a positive  $c_3$ such that
\roster
\item[1] $d(S^jy,S^jz)\le c_2$ whenever $d(y,z)\le c_3$ and $j\in J$,
\item[2] $c_3\le\omega(N,V)$.
\endroster

\medskip

\noindent {\it Step 9} : We partition $Y$ into finitely many 
Borel-measurable sets $Y_1,\ldots,Y_s$ of diameter less than $c_3$, and
let $\hat E_i=\hat E\cap Y_i$ for $i=1,\ldots,s$. Write 
$I=\{1,\ldots,s\}$.  Using the fact that $m$ is regular, pick
compact sets $E_i$ such that $E_i\subseteq\hat E_i$ and $m(\hat
E_i\backslash E_i)<\frac{c_1}{sN}$. Write $E_{ij}=S^jE_i$ for $i\in I$,
$j\in J$ (so $E_{i0}=E_i$). Then the sets $E_{ij}$ are pairwise
disjoint, and
$$
m\Biggl(\bigcup_{i\in I}\bigcup_{j\in J}E_{ij}\Biggr)\ge 1-2c_1\,.
$$
We discard any $E_i$'s that have measure zero, so 
$m(E_{ij})>0$  for all $i\in I$, $j\in J$.

\medskip

\noindent {\it Step 10} : We pick points $\bar y_i\in E_i$ for each
$i\in I$, and then let $\bar y_{i,j}=S^{j}\bar y_i$, so $\bar y_{i,j}\in
E_{ij}$.

\medskip

\noindent {\it Step 11} : Now we are going to define the map $\psi$
belonging to ${\Cal V}(\vp,U)$. For this purpose, we begin by dividing the
integer interval $J = [0,2,\cdot \cdot \cdot \cdot ,N-1]$ 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
$$
\align
J_\ell^1&=\{(\ell-1)(\kappa+\mu),(\ell-1)(\kappa+\mu)+1,\ldots,
\ell\kappa+(\ell-1)\mu-1\}\,,\\
J_\ell^2&=\{\ell\kappa+(\ell-1)\mu,\ell\kappa+(\ell-1)\mu+1,
\ldots,\ell(\kappa+\mu)-1\}\,.
\endalign
$$
The map $\psi$ will satisfy
\def\refeqone{3.9}
$$
\psi (\bar{y}_{i,j})=\vp (\bar{y}
_{i,j})\,\,\,\text{\rm if}\ j\in
J_\ell^2\,,\tag 3.9
$$
and
\def\refeqtwo{3.10}
$$
\psi (\bar{y}_{i,j}) = g_{ij}\vp(\bar{y}_{i,j})\,\,\,\text{\rm
if}\ j\in J_\ell^1\,,\tag 3.10
$$
where the $g_{ij}$ are members of $U$ whose choice will be
described later. By Property B there exists a continuous map
$\theta:Y\to U$ such that $\theta (\bar y_{i,j}) = g_{ij}$ for $j\in
J_{\ell}^1$ and $\theta (\bar y_{i,j})=e_G$ for $j\in J_{\ell}^2$. If
we then define $\psi(y)=\theta(y)\vp(y)$, then $\psi$ satisfies
(\refeqone) and (\refeqtwo) and furthermore $\psi \in {\Cal V}(\vp,U)$. 

\medskip

\noindent {\it Step 12} : Now we are now going to estimate the norm
$\|W_{n,\bar \psi}f\|_{L_2}$ for large $n$. We will choose 
$n=(\kappa+\mu)\rho$, where $\rho$ is a large positive integer whose
choice will be described later, but it is such that
$\frac{\lambda}{\rho}$ is still a large positive integer.

\medskip

Now we begin estimating $\|W_{n,\bar \psi}f\|_{L_2}$. We let
$$
A=\bigcup_{j=0}^{N-n-1}\bigcup_{i=1}^s E_{ij}\,.
$$
Since
$
m\Big(\bigcup_{j=N-n}^{N-1}\,\bigcup_{i=1}^s E_{ij}\Big)\le
\frac{n}{N} = \frac {\rho}{\lambda}$ (because $m\Big(\cup_{i=1}^s
E_{ij}\Big)\le\frac{1}{N}$ for each $j$), the set $A$ satisfies
$$
m(A)\ge 1-2c_1-\frac{\rho}{\lambda}\,.
$$
Outside $\bG\times A$, the function $W_{n,\bar \psi}f$ is bounded
pointwise by $\|f\|_{sup}$, so
\def\refestione{3.11}
$$
\int_{\bG\times(Y\backslash A)}|W_{n,\bar \psi}f(\bar g)|^2d\bar \nu(\bar 
g)\le\|f\|_{sup}^2\,\Big(2c_1+\frac {\rho}{\lambda}\Big)\,.\tag 3.11
$$
If $(\bar g,y)\in \bG\times A$, then 
there exists unique $i\in I$, $j\in J$, such that $j\le N-n-1$ and 
$y\in E_{ij}$.  Write 
\def\refrepla{3.12}
$$
\overline{W}_{n,\bar \psi;i,j}f(\bar g) \equaldef W_{n,\bar \psi}f(\bar
g,\bar y_{i,j}) = \frac{1}{n}\Big(\sum_{k=0}^{n-1}f(\bar \psi (\bar
y_{i,j},k)\bsg^k(\bar g),S^{k}\bar y_{i,j})\Big)\,.\tag 3.12
$$
Since both $y$ and $\bar y_{i,j}$ belong to $E_{ij}$, the distance 
$d(y,\bar y_{i,j})$ is not larger than $c_3$.  Then $d(S^ky,S^k\bar
y_{i,j})\le c_2$ for all $k$. Therefore
\def\refimplyone{3.13}
$$
|f(\bar \psi (y,k)\bar h,S^{k}y)-f(\bar \psi (y,k)\bar h,S^{k}\bar 
y_{i,j})|\le\cz\tag
3.13
$$
for all $k$ and $\bar h\in \bG$. Now notice that for each $0 \leq k
\leq n$, the restriction of the function $y\to \psi(y,k)$ to the set
$A$ `belongs to $\Phi^N$' (i.e. equals restriction to $A$ of some member
of $\Phi^N$). Since $c_3\le\omega(N,V)$, we conclude that
$$
\pi_q\big(\psi (y,k)\bar h\big(\psi (\bar y_{i,j},k)\bar
h\big)^{-1}\big) = \bar \psi (y,k)\bar \psi (\bar y_{i,j},k)\in V,
$$
for all $y\in A$, $\bar h\in \bG$ and $0\leq k \leq n$. Therefore
\def\refimplytwo{3.14}
$$
|f(\bar \psi (y,k)\bar h,S^{k}\bar y_{i,j})-f(\bar \psi (\bar
y_{i,j},k)\bar h,S^{k}\bar y_{i,j})|\le\cz\,.\tag 3.14
$$
By taking $\bar h = \bsg^k(\bar g)$, and using (\refimplyone) and
(\refimplytwo) we have
\def\refimplythree{3.15}
$$
|\overline{W}_{n,\bar \psi;i,j}f(\bar g)-W_{n,\bar \psi}f(\bar 
g,y)|\le\czt\,\,\,\,\text{\rm whenever} \,\,\,\bar g\in \bG,y\in
E_{ij}\,.\tag 3.15
$$
Now fix an $i\in I$ and $j\in J$ such that $j\le N-n-1$.  
Then for each $k\in\{0,\ldots,n-1\}$ the point $S^k\bar y_{i,j}$ 
is equal to $\bar y_{i,j+k}$, and belongs to 
$E_{i,j+k}$. As $k$ varies from $0$ to $n-1$, the index
$j+k$ takes all values in the integer interval $\Gamma_n(j)=
\{j,j+1,\ldots,j+n-1\}$, which is a subset of $J$ because 
$j\le N-n-1$. 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
$J_{\ell}\subseteq\Gamma_n(j)$.  Then \def\refreplac{3.16}
$$
\overline{W}_{n,\bar \psi;i,j}f(\bar g) = 
\frac{1}{n}\Big(\overline{W}_{n,\bar 
\psi;i,j}^{c}f(\bar g) + \overline{W}_{n,\bar \psi;i,j}^bf(\bar
g)\Big)\,,\tag 3.16
$$
where $\overline{W}_{n,\bar \psi;i,j}^{c}f(\bar g)$, 
$\overline{W}_{n,\bar \psi;i,j}^bf(\bar g)$ are, respectively,
the contributions to the right-hand side of (\refrepla) of the terms for
which $k+j\in\Gamma_n^{c}(j)$ and those for which
$k+j\in\Gamma_n^b(j)$, so that, in particular, 
\def\refreplacc{3.17}
$$
\overline{W}_{n,\bar \psi;i,j}^{c}f(\bar g) = 
\sum_{\ell\in\Delta_n(j)}\,\,\sum_{k+j\in
J_{\ell}}f(\bar \psi (\bar y_{i,j},k)\bsg^k(\bar g),\bar
y_{i,j+k})\,,\tag 3.17
$$
and 
\def\refreplaccc{3.18}
$$
\overline{W}_{n,\bar \psi ;i,j}^{b}f(\bar g) = 
\sum_{k+j\in\Gamma_n^b(j)}f(\bar \psi (\bar y_{i,j},k)\bsg^k(\bar g),\bar
y_{i,j+k})\,.\tag 3.18
$$
Since $\Gamma_n^b(j)$ has at most $2(\kappa+\mu)$ indices, we get the 
bound
\def\refgetbo{3.19}
$$
\frac{1}{n}\Big|\overline{W}_{n,\bar \psi;i,j}^bf(\bar g)\Big|
\le\frac{2(\kappa+\mu)\|f\|_{sup}}{n}=\frac{2\|f\|_{sup}}{\rho}\,.\tag
3.19
$$

Now let us analyze the contribution to $\overline{W}_{n,\bar 
\psi;i,j}^{c}f(\bar g)$ of the terms  of an individual sum 
$$
\overline{W}_{n,\bar \psi;i,j;\ell}^{c}f(\bar g)
=\sum_{k+j\in J_{\ell}}f(\bar \psi (\bar y_{i,j},k)\bsg^k(\bar g),\bar
y_{i,j+k})=\sum_{u\in J_{\ell}}f(\bar \psi (\bar
y_{i,j},u-j)\bsg^{u-j}(\bar g),\bar 
y_{i,u})\,.
$$
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 (\refcocycle) we get
$$
\align
\overline{W}_{n,\bar \psi;i,j;\ell}^{c}f(\bar g)\! & = 
\!\!\!\!\sum_{v=0}^{\kappa+\mu-1}
f\big(\bar \psi(\bar y_{i,j},v+a_\ell\!-\!j)\bsg^{v+a_\ell\!-\!j}(\bar
g),\bar y_{i,a_\ell+v}\big) \!\\
& = \!\!\!\!\sum_{v=0}^{\kappa+\mu-1}f\big(\bar \psi(\bar y_{i,a_\ell},v)
\bsg^v[\bar \psi(\bar y_{i,j},a_\ell\!-\!j)]\bsg^{v+a_\ell\!-\!j}(\bar
g),\bar y_{i,a_\ell+v}\big) \\
& = \!\!\!\!\sum_{v=0}^{\kappa+\mu-1}f\big(\bar \psi(\bar y_{i,a_\ell},v)
\bsg^v\big[\bar \psi(\bar y_{i,j},a_\ell\!-\!j)\bsg^{a_\ell\!-\!j}(\bar 
g)\big],\bar y_{i,a_\ell+v}\big). 
\endalign 
$$
We now split the above sum once more, by separating out the 
first $\kappa$ terms from the remaining $\mu$ ones.  We get
$$
\overline{W}_{n,\bar \psi;i,j;\ell}^{c}f(\bar g)=
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,-}f(\bar g)+
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,+}f(\bar g)\,,
$$
where 
$$
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,-}f(\bar g)=\sum_{v=0}^{\kappa-1}
f\big(\bar \psi(\bar y_{i,a_\ell},v)\bsg^v\big[\bar \psi(\bar
y_{i,j},a_\ell\!-\!j)\bsg^{a_\ell\!-\!j}(\bar g)\big],\bar
y_{i,a_\ell+v}\big)\,,
$$
and
$$
\align
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,+}&f(\bar
g)=\sum_{w=0}^{\mu-1}f\big(\bar \psi(\bar
y_{i,a_\ell},\kappa+w)\bsg^{\kappa+w}\big[\bar \psi(\bar
y_{i,j},a_\ell-j)\bsg^{a_\ell\!-\!j}(\bar g)\big],\bar
y_{i,a_\ell+\kappa+w}\big)\\
&=\sum_{w=0}^{\mu-1}f\big(\bar \psi(\bar
y_{i,a_\ell+\kappa},w)\bsg^w\big[\bar
\psi (\bar y_{i,a_\ell},\kappa)\big]\bsg^{\kappa+w}\big[\bar \psi(\bar
y_{i,j},a_\ell-j)\bsg^{a_\ell\!-\!j}(\bar g)\big],\bar 
y_{i,a_\ell+\kappa+w}\big)\\
&=\sum_{w=0}^{\mu-1}f\big(\bar \psi(\bar
y_{i,a_\ell+\kappa},w)\bsg^w\big[\bar \psi (\bar
y_{i,a_\ell},\kappa)\bsg^{\kappa}[\bar \psi(\bar
y_{i,j},a_\ell-j)\bsg^{a_\ell\!-\!j}(\bar g)]\big],\bar
y_{i,a_\ell+\kappa+w}\big).
\endalign
$$
In the above computation we have used the cocycle identity (\refcocycle)
and $\bar y_{i,j}=S^j\bar y$. Once again the cocycle identity allows us to
write
$$
\bar \psi(\bar y_{i,j},a_\ell-j) = \bar \psi (\bar
y_i,a_\ell)\bsg^{(a_\ell\!-\!j)}\big[\bar \psi(\bar y_i,j)^{-1}\big]. 
$$
Hence,

\medskip

\noindent $\overline{W}_{n,\bar \psi; i,j;\ell}^{c,+}f(\bar g)$

\medskip

\noindent $\qquad = \sum_{w=0}^{\mu-1}f\big(\bar \psi(\bar
y_{i,a_\ell+\kappa},w)\bsg^w\big[\bar
\psi (\bar y_{i,a_\ell},\kappa )\bsg^{\kappa}[\bar \psi (\bar
y_i,a_\ell)\bsg^{(a_\ell\!-\!j)}[\bar \psi(\bar y_i,j))^{-1}\bar
g]]\big],\bar y_{i,a_\ell+\kappa+w}\big)$.

\medskip

\noindent Clearly, each function $\overline{W}_{n,\bar
\psi;i,j;\ell}^{c,-}f$ is pointwise bounded by
$\kappa\|f\|_{sup}$. Moreover, it is clear that the number of members of
$\Delta_n(j)$ is at most $\rho$.Therefore \def\refestt{3.20}
$$
\frac{1}{n}\Big|\sum_{\ell\in\Delta_n(j)}
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,-}f(\bar g)\Big|
\le\frac{\rho\kappa\|f\|_{sup}}{n}=
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}\,. \tag 3.20
$$
We now turn to the crucial estimate, namely, the bound for the functions
$\overline{W}_{n,\bar \psi;i,j;\ell}^{c,+}f$.  It is at this point that
we will choose the $g_{ij}$.  For this purpose, we will want to
assign to each index $\ell\in L$ and each $i\in I$ an integer
$r(\ell,i)\in\{1,\ldots,P\}$. 
We let $F_{i,\ell}$ be the function on $G$ defined by
$$
F_{i,\ell}(\bar g)=\frac{1}{\mu}\sum_{w=0}^{\mu-1}f(\bar \psi(\bar y_{i,
a_\ell+\kappa},w)\bsg^w(\bar g),\bar y_{i,a_\ell+\kappa+w}).
$$
Notice that
$$
F_{i,\ell}(\bar g)=\frac{1}{\mu}\sum_{w=0}^{\mu-1}f(\bar \vp (\bar y_{i,
a_\ell+\kappa},w)\bsg^w(\bar g),\bar y_{i,a_\ell+\kappa+w})\,,
$$
so that in fact the functions $F_{i,\ell}$ do not depend at all on $\psi$,
i.e.,  on the choice of the $g_{ij}$. Next, observe that
$$
\overline{W}_{n,\bar \psi;i,j;\ell}^{c,+}f(\bar g)=\mu F_{i,\ell}
\big(\bar \psi(\bar y_{i,a_\ell},\kappa)\bsg^{\kappa}\big[\bar \psi(\bar
y_i,a_\ell )\bsg^{a_\ell\!-\!j}[\bar \psi(\bar y_i,j)^{-1}\bar
g)]\big]\big).
$$

Clearly, each $F_{i,\ell}$ is a convex combination of elements
of the set
$$
\{\tau_{\bar g}\big(\hat {\bsg}^if^y\big) : \bar g\in \bG,\ i \in \nn
\cup \{0\},\ y\in Y\}.
$$
Hence $F_{i,\ell}$ belong to $\cak$. So for each $i,\ell$ we
can  choose  $H_{i,\ell}\in\cak_0$ such that 
$\|F_{i,\ell}-H_{i,\ell}\|_{sup}\le \cz$. We fix one such choice from now
on.

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 nonnegative 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 
\def\refsays{3.21}
$$
\align
\bar \psi (\bar y_{i,a_\ell},\kappa)\cdot & {\bsg}^\kappa [\bar \psi(\bar
y_i,a_\ell)] \\
& = \bsg [\bar \psi(\bar y_{i,a_\ell+\kappa-1})]{\bsg}^2
[\bar \psi(\bar y_{i,a_\ell+\kappa-2})]\cdot \cdot \cdot {\bsg}^\kappa
[\bar \psi (\bar y_{i,a_\ell})]{\bsg}^\kappa [\bar \psi (\bar y_i,a_\ell
)] \\
& =  \bsg [{\bar g}_{i,a_\ell+\kappa-1}\cdot {\bar z}_{\kappa-1}]\cdot
{\bsg}^2[{\bar g}_{i,a_\ell+\kappa-2}\cdot {\bar z}_{\kappa-2}]
\cdot \cdot \cdot \cdot {\bsg}^k[{\bar g}_{i,a_\ell}\cdot {\bar
z_0}]\,,\tag 3.21
\endalign
$$
where $z_k=\vp(\bar y_{i,a_\ell+k})$ for $0<k<\kappa$, $z_0=\vp(\bar
y_{i,a_\ell})\cdot\psi(\bar y_i,a_\ell)$, (and recall that $\bar z =
\pi_q(z),\ \bar \psi = \psi \circ \pi_q$). 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 3 guarantees that 
$\sg [Uz_{\kappa-1}]\cdot {\sg}^2[Uz_{\kappa-2}]\cdot \cdot \cdot \cdot
{\sg}^\kappa [U z_0] = G$. Since the map $\pi_q:G\rightarrow G/G_q$ is
onto, we can choose $g_{i0},g_{i1},\cdot \cdot \cdot ,g_{i,\kappa-1}$
such that 
\def\refsaysss{3.22}
$$
\bar \psi (\bar y_{i,a_\ell},\kappa)\cdot{\bsg}^\kappa [\bar \psi(\bar
y_i,a_\ell)] = \gamma_{r(\ell,i)}\,. \tag 3.22
$$
Then, for any $i,j$, 
$$
\align
\sum_{\ell\in\Delta_n(j)}\overline{W}_{n,\bar
\psi;i,j;\ell}^{c,+}f(\bar g)& = \mu 
\sum_{\ell\in\Delta_n(j)}F_{i,\ell}\big(\gamma_{r(\ell,i)}{\bsg}^{\kappa 
+ a_\ell -j}[\bar \psi(\bar y_i,j)^{-1}\bar g]\big) \\ 
& = \mu\sum_{H\in\cak_0}\,\,\sum_{\ell\in A(i,H)\cap\Delta_n(j)}
\tau_{\gamma_{r(\ell,i)}^{-1}}F_{i,\ell}\big({\bsg}^{\kappa + a_\ell
-j}[\bar \psi(\bar y_i,j)^{-1}\bar g]\big).
\endalign
$$
Let
$$
{\Cal W}^\sg_{n,\bar \psi;i,j}(\bar g) = 
\mu\sum_{H\in\cak_0}\sum_{\ell\in A(i,H)\cap\Delta_n(j)}
\tau_{\gamma_{r(\ell,i)}^{-1}}H({\bsg}^{a_\ell}(\bar g)).
$$
Now fix a $\bar g \in \bG$ and set ${\bar g}^*_{i,j} = {\bsg}^{\kappa -
j}[\bar \psi (\bar y_i,j)^{-1}\bar g]$. 
Since $\|H-F_{i,\ell}\|_{sup}\le\dz$ whenever $\ell\in A(i,H)$, we have
the bound
$$
\Big|{\Cal W}^\sg_{n,\bar \psi;i,j}({\bar g}^*_{i,j})-
\sum_{\ell\in\Delta_n(j)}\overline{W}_{n,\bar
\psi;i,j;\ell}^{c,+}f(\bar g)\Big|
\le\frac{\mu\rho\eps}{16}\,,
$$
using the fact that $\Delta_n(j)$ has at most $\rho$ members. Observing
that $\frac {\mu \rho}{n} \leq 1$, we get 
\def\refcombone{3.23}
$$
\frac{1}{n}\Big|{\Cal W}^\sg_{n,\bar \psi;i,j}({\bar g}^*_{i,j})-
\sum_{\ell\in\Delta_n(j)}\overline{W}_{n,\bar
\psi;i,j;\ell}^{c,+}f(\bar g)\Big|
\le\dz\,, \tag 3.23
$$
because $n=(\kappa+\mu)\rho$. 

We now turn to the  task of estimating ${\Cal W}^\sg_{n,\bar \psi;i,j}$.
First, let
$$
{\Cal W}_{n,\bar \psi;i,j}(\bar g) = \mu\sum_{H\in\cak_0}\sum_{\ell\in
A(i,H)\cap\Delta_n(j)} \tau_{\gamma_{r(\ell,i)}^{-1}}H(\bar g).
$$
We write 
$$
{\Cal W}^\sg_{n,\bar \psi;i,j}(\bar g)=\mu\sum_{H\in\cak_0}{\Cal
W}^\sg_{n,\psi;i,j;H}(\bar g) \qquad \text {and}\qquad {\Cal W}_{n,\bar 
\psi;i,j}(\bar g)=\mu\sum_{H\in\cak_0}{\Cal W}_{n,\bar \psi;i,j;H}(\bar g),
$$
where 
\def\refsumfi{3.24}
$$
\align
{\Cal W}^\sg_{n,\bar \psi;i,j;H}(\bar g)&=\sum_{\ell\in
A(i,H)\cap\Delta_n(j)}\tau_{\gamma_{r(\ell,i)}^{-1}}H({\bsg}^{a_\ell}\bar 
g)\ \text {and} \\ 
{\Cal W}_{n,\bar \psi;i,j;H}(\bar g)&=\sum_{\ell\in
A(i,H)\cap\Delta_n(j)} \tau_{\gamma_{r(\ell,i)}^{-1}}H(\bar g). \tag{3.24}
\endalign
$$
Since $\Delta_n(j)$ has at most $\rho$ members and $a_\ell = \ell (\kappa
+ \mu)$, our choice of $\kappa + \mu$ in Step 4 gives
$$
\big|{\Cal W}^\sg_{n,\bar \psi;i,j;H}(\bar g) - {\Cal W}_{n,\bar
\psi;i,j;H}(\bar g)\big| < \frac {\rho \beta }{2}\quad \text
{for all}\ \bar g\in \bG,\ \text {and}\ \ H\in \cak.
$$
Again, since the cardinality of ${\Cal K}_0$ is $\hat \kappa$ and $\frac {\mu 
\rho}{n} \leq 1$, we get
$$
\big|{\Cal W}^\sg_{n,\bar \psi;i,j}(\bar g) - {\Cal W}_{n,\bar
\psi;i,j}(\bar g)\big| < \frac {\eps }{16}\quad \text
{for all}\ \bar g\in \bG.
$$
Combining this with (\refcombone) yields
\def\refcombtwo{3.25}
$$
\frac{1}{n}\Big|\sum_{\ell\in\Delta_n(j)}\overline{W}_{n,\bar
\psi;i,j;\ell}^{c,+}f(\bar g) - \Cal W_{n,\bar \psi;i,j}(\bar
g)\Big| \leq \cz.
\tag 3.25
$$
Now we estimate $\Cal W_{n,\bar \psi;i,j}(\bar g)$. Recall that 
$$
{\Cal W}_{n,\bar \psi;i,j}(\bar g) = \mu\sum_{H\in\cak_0}\sum_{\ell\in
A(i,H)\cap\Delta_n(j)} \tau_{\gamma_{r(\ell,i)}^{-1}}H(\bar g).
$$
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 (\refsumfi) is equal to 
$q_{i,j,H}\sum_{r=1}^{P}\tau_{\gamma_r^{-1}}H(\bar g)$,
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
$$
\Big|{\Cal W}_{n,\bar \psi;i,j;H}(\bar g)\Big|\le P\beta
q_{i,j,H}+p_{i,j,H}\|f\|_{sup}\,.
$$
Each number $q_{i,j,H}$ is bounded by $\frac{\rho}{P}$, and 
$p_{i,j,H}\le P$. So 
$$
\Big|{\Cal W}_{n,\bar \psi;i,j;H}(\bar g) 
\Big|\le \rho\beta + P\|f\|_{sup}\,. 
$$
Therefore
\def\refcombthree{3.26}
$$
\|{\Cal W}_{n,\psi;i,j}\|_{sup}\le \mu\hat\kappa\rho\beta + \mu\hat\kappa 
P\|f\|_{sup}\,.\tag
3.26 
$$ 
If we combine (\refcombtwo)and (\refcombthree), we find
\def\refcombfour{3.27}
$$
\frac{1}{n}\Big\|\sum_{\ell\in\Delta_n(j)}\overline{W}_{n,\bar 
\psi;i,j;\ell}^{c,+}f(\bar g) 
\Big\|_{sup}\le\cz+\frac{\mu\hat\kappa\rho\beta}{n}+
\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,, \tag 3.27
$$
We now use (\refestt) and get 
$$
\frac{1}{n}\Big|\overline{W}_{n,\bar \psi;i,j}^{c}\Big|_{sup}
=\frac{1}{n}\Big|\sum_{\ell\in\Delta_n(j)}
\overline{W}_{n,\bar \psi;i,j;\ell}^{c}\Big|_{sup}
\le\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\cz+
\frac{\mu\hat\kappa\rho\beta}{n}+\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,. 
$$
Then (\refgetbo) implies
\def\refcombfive{3.28}
$$
|\overline{W}_{n,\bar \psi;i,j}(\bar g)|\le\frac{2\|f\|_{sup}}{\rho}+
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\cz+
\frac{\mu\hat\kappa\rho\beta}{n}+\frac{\mu\hat\kappa P}{n}
\|f\|_{sup}\,. \tag 3.28
$$
Finally, we use (\refimplythree)  and get the pointwise estimate
\def\refimplyseven{3.29}
$$
|W_{n,\bar \psi}f(\bar g,y)|\le\czt+\frac{2\|f\|_{sup}}{\rho}+
\frac{\kappa\|f\|_{sup}}{\kappa+\mu}+\cz + 
\frac{\mu\hat\kappa\rho\beta}{n}+
\frac{\mu\hat\kappa P}{n}\|f\|_{sup}\,, \tag 3.29
$$
valid  whenever $\bar g\in G$ and $y\in A$. Now note that depending on 
the initial given data (i.e. $f,\eps,\vp ,\tilde U$ and $\bar n$ we
have so far chosen $\kappa ,\hat \kappa$ and $P$. Next, we choose
$\rho$ and $\mu$ such that
$$
n= (\kappa+\mu)\rho>\bar n\,,\quad \frac
{\|f\|_{sup}}{\rho} \le \dz \,,\quad
\frac {\kappa \|f\|_{sup}}{\mu} \le \cz\,,\quad\text{\rm and}\ \ 
\frac {P\hat \kappa \|f\|_{sup}}{\rho} \le \cz\,.
$$
Then (\refimplyseven) implies that 
\def\refimplyeight{3.30}
$$
|W_{n,\bar \psi}f(\bar g,y)|\le\frac{7\eps}{8}\quad\text{\rm 
whenever}\ \ g\in G\ \text{\rm and}\ y\in A\,.\tag 3.30
$$
This, together with (\refestione), implies that
$$
\int_{\bG\times Y}|W_{n,\bar  
\psi}f|^2\le\frac{49\eps^2}{64}+\Big(2c_1+\frac{\rho}{\lambda}\Big)
\|f\|_{sup}^2\,.
$$
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 $\|W_{n,\bar \psi}f\|_{L^2}\!<\!\eps$. 

Since $n>\bar n$, we have proved that $\psi\in E(f,\eps,\bar
n)$. We have already checked that $\psi\in{\Cal V}(\vp,U)$. Hence
$$
\vp \in E(f,\eps,\bar n)\cap{\Cal V}(\vp,U)\,,
$$
concluding the proof of Lemma 2.5 and subsequently the proof of Theorem 
2.3 as well.\qid

\medskip

\head References
\endhead

\medskip

\noindent [E] R. Ellis : The construction of minimal discrete flows,
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\noindent [KNe] H. Keynes and M. Nerurkar : Ergodicity in affine skew 
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\noindent [KN1] H. Keynes and D. Newton : Minimal $(G,\sigma )$
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\noindent [KN2] H. Keynes and D. Newton : Ergodicity in $(G,\sigma )$
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\noindent [KN3] H. Keynes and D. Newton : Minimality for
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\noindent [NS] M. Nerurkar, H. Sussmann : Construction of minimal 
cocycles arising from specific differential equations, Israel Journal
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\enddocument 
\end
