\label{sec:ownership}

Let $(W,B)$ be a Blaschke model in the base-$0$ frame, with zeros $a_1,\dots,a_k$. They are simple, since
$(W B)'(a_j)=W(a_j)B'(a_j)$ and $(W B)'\neq0$ on the closed disk by (M3); hence they are distinct. Put $\zeta_j=W(a_j)$ and $P_j=P_{a_j}$. The base points $0,\zeta_1,\dots,\zeta_k$ are distinct zeros of $\Phi$, and each has exit
points (Lemma~\ref{lem:exit}). We say that a point $w^*$ of the circle is \emph{owned}\label{term:owned} by $a_j$ if $W(w^*)$ is an exit point of
$\zeta_j$.

\begin{theorem}[ownership]\label{thm:own}
Let $w^*$ be owned by $a_j$, and $z^*=W(w^*)$. Then:
\begin{enumerate}\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}
\item $v(w^*)<0$: $w^*$ is a pocket, of depth $x=-v(w^*)$, and $\varepsilon(w^*)=-1$;
\item $\sigma(w^*)\,x>1$ and $(\sigma(w^*)-P_j(w^*))\,x\le1$, i.e.\ $\sum_{l\neq j}P_l(w^*)\le1/x<\sigma(w^*)$: the pocket is on, and it
would not be on without the kernel of $a_j$;
\item $r_{\rm in}\le R_{\zeta_j}=|z^*|\le|W(a_j)|\prod_{l\neq j}|B_{a_l}(a_j)|\le|W(a_j)|$;
\item exit points of different base points are different; in particular a point is owned by at most one zero.
\end{enumerate}
\end{theorem}

\begin{proof}
Let $h=h_{\zeta_j}$ be as in Lemma~\ref{lem:exit} and $\beta=|h'(u^*)|$, so $h(0)=a_j$, $h(u^*)=w^*$ and
$\sigma(w^*)+1/v(w^*)=1/\beta$, where $\beta$ is the angular derivative of $h$ at $u^*$. Julia's lemma~\cite{Pommerenke1992,
Shapiro1993} at the point $0$ gives
\[
  \beta\ \ge\ \frac{|w^*-h(0)|^2}{1-|h(0)|^2}=\frac{|w^*-a_j|^2}{1-|a_j|^2}=\frac1{P_j(w^*)} .
\]
So $0<1/\beta=\sigma(w^*)+1/v(w^*)\le P_j(w^*)$, i.e.\ $1/v(w^*)\le-\sum_{l\neq j}P_l(w^*)\le0$. Since $0<|v|<\infty$ on the circle,
$v(w^*)<0$; then $1/x=\sigma-1/\beta<\sigma$ and $1/x\ge\sigma-P_j$. This is (ii) and the first part of (i). The sign: by
Lemma~\ref{lem:exit}(ii), $|W(e^{i\theta})|^2$ has a local minimum at $w^*$; $\operatorname{Im}v$ is not identically zero (that
would give $W=\mathrm{id}$ and $v=1>0$), so its zeros are isolated, and as
$\frac d{d\theta}|W(e^{i\theta})|^2=2|W'|^2\operatorname{Im}v$, $\operatorname{Im}v$ passes from negative to positive at $w^*$.
(A zero of $\operatorname{Im}v$ without a sign change is a point where $|W(e^{i\theta})|^2$ is strictly monotone.)
(iii) The first inequality is Lemma~\ref{lem:exit}. Schwarz--Pick for $f=W^{-1}\circ g_{\zeta_j}$ on $D(0,R_{\zeta_j})$, with $f(0)=a_j$,
gives $R_{\zeta_j}|f'(0)|\le1-|a_j|^2$, and $f'(0)=1/(W(a_j)B'(a_j))$, $(1-|a_j|^2)|B'(a_j)|=\prod_{l\neq j}|B_{a_l}(a_j)|$.
(iv) If base points $b$ and $c$ have a common exit point $z^*$, then $R_b=R_c=|z^*|=:R$, and $g_b$, $g_c$ both equal
$(\Phi|_{V_0})^{-1}$ on $D(0,R)\cap D(\Phi(z^*),s)$ (proof of Lemma~\ref{lem:exit}(i)); by the identity theorem $g_b=g_c$ on $D(0,R)$, so
$b=g_b(0)=g_c(0)=c$.
\end{proof}

Property (ii) alone does not identify the owner: one pocket can satisfy it for two zeros (for instance $P_1=P_2=p$, $\sigma=2p$
and $1/2<px\le1$). Uniqueness (iv) is a property of exit points.

\begin{corollary}[the count]\label{cor:own}
Let $(W,B)$ be a Blaschke model and choose one owned point for each zero. These are $k$ distinct on pockets with $\varepsilon=-1$, so the
remaining on pockets have signs adding up to $0$. If $\zeta_j$ has $m_j$ exit points ($m_j\ge1$, finite), there are at least
$\sum_j(m_j-1)$ on pockets with $\varepsilon=+1$, and at least $\sum_j(m_j-1)+1$ if some exit point $z^*$ of the base point $0$ is a
back case of Theorem~\ref{thm:T1}.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:own}(i),(ii),(iv), the owned points are distinct on pockets with $\varepsilon=-1$; by
Proposition~\ref{prop:criterion} and Theorem~\ref{thm:pocket} the signs of all on pockets add up to $-k$. Each $m_j$ is finite since
the exit points are pockets (Lemma~\ref{lem:Zfinite}). In the back case $W^{-1}(z^*)$ is a further on pocket with $\varepsilon=-1$
(Theorem~\ref{thm:T1}(iii)), distinct from the owned points by Theorem~\ref{thm:own}(iv). So the on pockets with $\varepsilon=-1$
number at least $\sum_jm_j$ (plus one in the back case), and those with $\varepsilon=+1$ at least that number minus $k$.
\end{proof}

\input{sections/sec8_1_NI_shallow}

Theorem~\ref{thm:own} gives a necessary condition for feasibility, in the converse direction of Lemma~\ref{lem:independence}. That
lemma gives a sufficient condition: in a group of pockets with signs adding up to $-1$, the kernel $P_j$ alone switches the pockets
on. Ownership gives less: each zero $a_j$ owns a pocket $w$ at a local minimum of $|W|$ with
$\sigma(w)x(w)>1\ge(\sigma-P_j)(w)x(w)$. For $k\ge2$
it does not give $P_jx>1$.