0$ for all $x\in U$;
\item[\bf Subcase 1.3.]$P(\bx)=0$ but $P(x_n)>0$
for a sequence $x_n\to \bx$, $x_n\in U$.
\end{description}
For Subcase 1.1, every $x\in U$ is a rest
point: $\phi^tx=x$. Also, since $\Phi$ is a
cocycle, $\Phi(x,t)$ is a group for every $x\in U$. Let $A(x)$
denote the matrix that generates this group: $\Phi(x,t)=e^{tA(x)}$,
$t\in\bbR$, $x\in U$ and note that the matrix-valued function $A$ is
continuous on $U$. Also, for $w\in\cnd$ with $\supp w\subset U$,
we have $Lw(x)=A(x)w(x)$. Since, by \eqref{1},
$\sigma(e^{tA(\bx)})\cap\bbT\neq\emptyset$,
the spectral mapping theorem
for matrices implies there is $\xi\in\bbR$ so that
$i\xi\in\sigma(A(\bx))$. Define $x_0=\bx$, choose $v$ with
$\|v\|=1$ so that $(A(x_0)-i\xi)v=0$,
and fix $N\in\bbN$. Also, choose $\eps>0$ and
$\delta>0$ so small that for every neighborhood $V\ni x_0$
with $\diam {V}<\eps$
the following holds:
\[\sup\{\|(A(x)-i\xi)v\|: x\in V\} < 1/2N,\quad V\subset U,\quad
\delta\sup\{\|A(x)-i\xi\|: x\in U\}<1/2N.\]
Apply Lemma~\ref{l3} for these $\eps$ and $\del$
to find $B\subset D\subset U$.
Take $w=\al(\cdot)v+w_0$ as indicated in Lemma~\ref{l3}. Recall,
that $\al(x)\leq 1$, $x\in X$.
Then, using the estimate (2) of the lemma, we find that:
\[\|(L-i\xi)w\|_{\cnd}=\sup_{x\in D}\|(A(x)-i\xi)w(x)\|\leq 1/N.\]
As a result, $i\xi\in\sigma_{{ap}}(L)$, and Theorem~\ref{t3} is
proved.
For Subcase 1.2, the neighborhood $U$ does not contain
rest points. By the semicontinuity of $P$
we can and will assume that there is a number $p_0>0$ such that
$P(x)\geq p_0$ for all $x\in U$.
Moreover, we note that
$\sig (\Phi (\bx, P(\bx))) \cap \bbT \neq \emptyset$.
Indeed, if $\phi^{P(\bx)}\bx=\bx$, then
$\Phi(\bx,kP(\bx))=[\Phi(\bx,P(\bx))]^k$ for $k\in\bbZ$.
Since, by \eqref{1},
the sequence $\{\|[\Phi(\bx,P(\bx))]^k\bv\|\}$ is bounded, the matrix
$\Phi(\bx,P(\bx))$ can not be hyperbolic. For
a notational convenience, denote $x_0=\bx$.
Thus, there is some
$v\in \calT_{x_0}X$ and $\xi \in\bbR$, so that
\begin{equation}
\Phi (x_0, P(x_0)) v = e^{i\xi} v,\quad \|v\| =1.
\lb{2}
\end{equation}
We will use the choice of $x_0$ and $v$ as in \eqref{2}
in what follows.
For Subcase 1.3, the Man\~e point $\bx$ is a rest point
and each $x\in U$ is a periodic point.
We claim that $\Phi(x_n,P(x_n))$ can not be hyperbolic.
To see this, we assume that $x_n$ {\it is} hyperbolic.
By the Stable Manifold Theorem (see, e.g., \cite{HPS}),
the periodic orbit through
$x_n\in U$ has a stable or an unstable manifold,
contrary to the fact that
$U$ consists entirely of periodic orbits.
For a nonhyperbolic $x_n$ we denote $x_0=x_n$, and
select $\xi\in\bbR$ and $v$ as in \eqref{2}.
Starting from the assumption that $1\in\sig_{{ap}}(T,\bC)$,
we have found in each subcase, 1.2 and 1.3,
a periodic point $x_0$ with period $P(x_0)>0$,
a neighborhood $U$ of $x_0$ consisting entirely of
periodic points whose periods are
uniformly bounded and separated from zero, as well as
a number $\xi\in\bbR$ and a vector
$v$ as in \eqref{2}. We will prove
that $\lambda:=i\xi/P(x_0)$ is an approximate eigenvalue for $L$.
Define $p=P(x_0)$ and let $\mu_1 > 0$ be a constant such that
\begin{equation}
P(y) \leq \mu_1 p,\quad y\in U.
\lb{per}
\end{equation}
Fix a natural number $N> \mu_1 p +1$.
In what follows we use the letter $\mu$ (resp., $\nu$)
with subscripts to denote
``big'' (resp., ``small'') constants that do not depend on $N$.
Select (small) constants $\nu_i<1$, $i=0,\ldots, 5$. The
required values of these constants
will be determined later. Define
$$
\mu_2 = 2\sup \{ \| \Phi (x_0, t) \| : |t| \leq \mu_1 p\},\quad R=N +
\mu_1 p.
$$
Also, fix $s>0$ so that $s < \min(p/8,1/4)$ and
\begin{equation}
\max_{|t| \leq s} \left| e^{-\frac{i\xi t}p} -1\right| \leq \nu_4,
\lb{S1}
\end{equation}
\begin{equation}
\sup_{|t| \leq s} \| \Phi (\phi^{-t} x_0, t) v-v\| \cdot \sup_{|n| \leq
3N/p} \| [ \Phi (x_0, p)]^n \| \leq \nu_2.
\lb{D3}
\end{equation}
\begin{lem}\lb{l4}
There exists $\eps > 0$ such that for every open set
$D\ni x_0$ with $\diam {D} \leq \eps$ and
every $y\in D$ the following implication holds:\newline
\centerline{If $\phi^t y \in D$ and $|t| \leq R$, then
$t\in \bigcup_{n\in\bbZ} (-s + np, s + np)$.}
\end{lem}
\begin{pf}
Suppose the lemma is false. Choose $D_k\ni x_0$ with
$\diam {D_k}\to 0$ and $y_k\in D_k$ together with $t_k\in\bbR$, $|t_k|\le
R$, so that $\phi^{t_k}y_k\in D_k$,
but $t_k\not\in\bigcup_{n\in\bbZ} (-s + np,
s + np)$. By compactness, we may assume $t_k\to t^*$, $|t^*|\le R$. By
continuity, $\phi^{t_k}y_k\to \phi^{t^*}x_0$, and, therefore,
$\phi^{t^*}x_0=x_0$. Since $t^*=np+\tau$
for some $n\in\bbZ$ and some $\tau$
with $s<\tau