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Support of the measure in an integral representation for a Nevanlinna function defined by a limit periodic continued fraction V. I. Buslaev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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    § 1. IntroductionBy means of Schur’s well-known algorithm [1] each Schur function (that is, a function holomorphic in $\mathbb D:=\{ |z|<1\}$ and taking values in $\overline{\mathbb D}:=\{ |z|\leqslant 1\}$) can be represented as a finite or an infinite Schur continued fraction
$$
\begin{equation}
\gamma_0+\cfrac{(1-|\gamma_0|^2)z}{\overline{\gamma }_0z+\cfrac{1} {\gamma_1+\cfrac{(1-|\gamma_1|^2)z}{\overline{\gamma }_1z+\cfrac{1}{\gamma_2+\dotsb}}}}
\end{equation}
\tag{1}
$$
with coefficients In (1) and throughout, a bar over a numerical quantity denotes complex conjugation; it also extends to subscripts, so that $\overline{\gamma }_n:=\overline{\gamma_n}=\operatorname{Re} \gamma_n-i\operatorname{Im} \gamma_n$. Schur [1] showed that a result of converse type, namely the following statement, also holds. Schur’s Theorem. Let (1) be a continued fraction such that the coefficients $\gamma_0, \gamma_1,\dots$ satisfy (2). Then the limit Geronimus [2] complemented Schur’s theorem by the following result. Geronimus’s Theorem. Let (1) be a continued fraction such that the coefficients $\gamma_0,\gamma_1,\dots$ satisfy (2) and the condition Then the measure $\sigma$ associated with the Schur function $f(z)$ on the right-hand side of (3) has the support satisfying We do not present the definitions of the measures associated with Schur functions (for instance, see [3]), but note that the statement of Geronimus’s theorem is equivalent to the limit Schur function $f(z)$ in (3) admitting no holomorphic continuation across an arbitrarily small arc of $\mathbb T$. In [4] we proved the following result. Theorem 1. Let (1) be a continued fraction such that $|\gamma_n|\neq 1$ for $n=0,1,\dots$, and let $\gamma_0,\gamma_1,\dots$ be a limit periodic sequence with period $q$, that is, let there exist limits Let
$$
\begin{equation*}
\Gamma =\Gamma (\gamma ^1,\dots ,\gamma ^q):=\biggl\{ z\in\overline{\mathbb C}\colon \frac{I^2(z)}{J(z)}\in [0,4]\biggr\},
\end{equation*}
\notag
$$
where $I(z)$ and $J(z)$ are the trace and determinant of the matrix product
$$
\begin{equation*}
\begin{pmatrix} 1 & \overline{\gamma }^1z\\ \gamma ^1 & z \end{pmatrix}\times\dots\times \begin{pmatrix} 1 & \overline{\gamma }^qz\\ \gamma ^q & z \end{pmatrix}.
\end{equation*}
\notag
$$
Then the following assertions hold. $1^\circ$. The continued fraction (1) converges to a meromorphic function $f(z)$ in $\mathbb C\setminus \Gamma $ locally uniformly in the spherical metric in $\mathbb C\setminus (\Gamma \cup \Xi )$, where $\Xi$ is a finite set of cardinality at most $2(q-1)q$. $2^\circ$. The meromorphic function $f(z)$ in $\mathbb C\setminus \Gamma $ satisfies $f(z)=(\overline{f}(z^{-1}))^{-1}$ and has no (single-valued) meromorphic extension to any set of the form $(\mathbb C\setminus \Gamma)\cup \{ |z- z_0|< \varepsilon \}$, where $z_0\in \Gamma $ and $\varepsilon >0$. $3^\circ$. If $|\gamma ^p|<1$ for all $p=1,\dots ,q$, then $\Gamma \subset\mathbb T$. In assertion $2^\circ$ of Theorem 1 and throughout, we set $\overline{f}(z):=\overline{f(\overline{z})}$. Recall that a Nevanlinna function is a holomorphic function in the upper half-plane $\mathbb C_+:=\{\operatorname{Im} z>0\}$ that takes values in the closure of $\overline{\mathbb C}_+$. An integral representation for Nevanlinna functions is well known [5], namely the following result holds. Riesz–Herglotz Theorem. A function $G(z)$ is a Nevanlinna function if and only if there exists a nonnegative measure $\varsigma $ with support on the extended real line $\overline{\mathbb R}:=\{ \operatorname{Im} z=0\}\cup\{\infty\}$ such that Note that the right-hand side of (4) defines a holomorphic function $G(z)$ outside the support of $\varsigma $ that satisfies
$$
\begin{equation*}
G(\overline{z})=\overline{G(z)}, \qquad z\in \mathbb C\setminus \operatorname{supp} \varsigma.
\end{equation*}
\notag
$$
Let $d_0,d_1,\dots$ be a sequence of points in $\mathbb C_+$, and let denote a linear fractional map, which takes $\mathbb C_+$ to $\mathbb D$ if $d\in\mathbb C_+$. In [3] we presented a modified version of Schur’s algorithm that assigns to an arbitrary Nevanlinna function a finite or infinite continued fraction of the form
$$
\begin{equation}
\delta_{-1}+\cfrac{(\delta_{-1}-\overline{\delta}_{-1})\psi_{d_0}(z)} {-\psi_{d_0}(z)+\cfrac{1}{\delta_0+\cfrac{(1-|\delta_0|^2)\psi_{d_{1}}(z)}{\overline{\delta }_0\psi_{d_{1}}(z)+\cfrac{1} {\delta_1+\cfrac{(1-|\delta_1|^2)\psi_{d_{2}}(z)}{\overline{\delta }_1\psi_{d_{2}}(z)+\cfrac{1}{\delta_2+\dotsb }}}}}}
\end{equation}
\tag{5}
$$
with coefficients $\operatorname{Im} \delta_{-1}\geqslant 0$, $|\delta_n|\leqslant 1$, $n=0,1,\dots$ . The following theorem was also proved. Theorem 2. Let $d_0,d_1,\dots$ be points in a compact set $E$ in $\mathbb C_+$, and let $\delta_{-1},\delta_0,\dots$ be the coefficients of a multipoint (at $d_0,d_1,\dots$) continued fraction (5) that satisfy Then the following results hold. $1^\circ$. The even convergents $\pi_0(z)\equiv \delta_{-1},\pi_2(z),\dots$ of the fraction (5) are Nevanlinna functions that converge to a Nevanlinna function $G(z)$ locally uniformly in $\mathbb C_+$; the odd convergents $\pi_1(z)\equiv \overline{\delta }_{-1},\pi_3(z),\dots$ converge to the function $\overline{G}(z):=\overline{G(\overline{z})}$ locally uniformly in $\mathbb C_-:=\{ \operatorname{Im} z<0\}$; an analogue of Schur’s algorithm at the points $d_0,d_1,\dots$, as applied to $G(z)$, returns the original continued fraction (5). $2^\circ$. If the points $d_0,d_1,\dots$ have a limit distribution such that
$$
\begin{equation}
\frac{\xi_{d_0}+\dots +\xi_{d_{n-1}}}{n} \xrightarrow[n\to\infty]{*} \zeta,
\end{equation}
\tag{7}
$$
where $\xi_{d}$ is the Dirac measure at $d$ and $\zeta$ is a probability measure with support on $E$, and if the coefficients $\delta_{-1},\delta_0,\dots$ of the fraction (5) satisfy
$$
\begin{equation}
\varlimsup_{n\to\infty}\prod_{k=0}^n(1-|\delta_k|)^{1/n}=1,
\end{equation}
\tag{8}
$$
then the support of the measure $\varsigma $ in the Riesz–Herglotz integral representation (4) for $G(z)$ coincides with $\overline{\mathbb R}$. In condition (7) of Theorem 2 we let $\xrightarrow{*}$ denote weak convergence of measures. Thus, (7) means that
$$
\begin{equation*}
\lim_{n\to\infty }\frac{v(d_0)+\dots +v(d_{n-1})}{n}=\int v(z)\,d\zeta (z),
\end{equation*}
\notag
$$
where $v(z)$ is any continuous function on the Riemann sphere $\overline{\mathbb C}$. To deduce a result of the modified version of Geronimus’s theorem, it was also shown in [3] that assertion $2^\circ$ of Theorem 2 also holds if the pair of conditions (7) and (8) is replaced by
$$
\begin{equation*}
\lim_{n\to\infty}d_n =d\in\mathbb C_+\quad \text{and} \quad \varlimsup_{n\to\infty}\prod_{k=0}^n(1-|\delta_k|^2)^{1/n}=1.
\end{equation*}
\notag
$$
§ 2. Statement of the main resultIn this paper we consider the convergence properties of the continued fraction (5) with limit periodic points $d_0,d_1,\dots$ and limit periodic coefficients $\delta_{-1},\delta_0,\dots$ . We prove the following result. Theorem 3. Let $d_0,d_1,\dots$ be points in $\mathbb C_+$ that have periodic limits Then $\Gamma$ is a union of at most $q$ closed intervals of $\overline{\mathbb R}$, and the continued fraction (5) converges in $\overline{\mathbb C}\setminus (\Gamma \cup \Xi )$ locally uniformly in the spherical metric, where $\Xi $ is a finite set of cardinality at most $4q^2$, to a function $\mathbf G(z)$ defined by the right-hand side of (4) for a measure $\varsigma$ satisfying where $\widetilde{\Gamma}$ is the set of poles of $\mathbf G(z)$.§ 3. Auxiliary lemmasIn the proof of Theorem 3 we use some auxiliary lemmas established previously. Lemma 1. Let Then the limit
$$
\begin{equation*}
\lim_{n\to\infty} (S_1\circ\dots\circ S_{n} )(w,z)=f(z), \qquad (w,z)\in\Lambda ,
\end{equation*}
\notag
$$
exists uniformly on compact subsets of $\Lambda $, where $f(z)$ is a meromorphic function in $\Omega\setminus \Gamma$ independent of $w$. In particular, the limit exists locally uniformly in the spherical metric in $\Omega\setminus (\Gamma \cup \Xi )$, where $\Xi =\bigcup_{p=1}^{m}\{h^p(z)= 0\}$.Here and throughout, by the coefficient matrix of a linear fractional transformation $S(w)=(aw+b)/(cw+d)$ we mean the matrix $S=\begin{pmatrix} a & b\\c & d \end{pmatrix}$. Note that the ratio $(\operatorname{Tr} S)^2/{\det S}=(a+d)^2/(ad-bc)$ is independent of the possible multiplication of the matrix of $S$ by a nonzero scalar. Before stating the next lemma recall that the transfinite diameter $\mathbf d_{v }{K}$ of the compact set ${K}\subset\overline{\mathbb C}$ in the external field $v$, where $v$ is a continuous real function on ${K}$, is the quantity
$$
\begin{equation}
\mathbf d_{v }{K}:=\lim_{n\to\infty}\biggl (\max_{z_1,\dots ,z_n\subset {K}}\prod_{1\leqslant q<r\leqslant n}|z_q-z_r|e^{-(v (z_q)+v (z_r))}\biggr)^{2/((n-1)n)}.
\end{equation}
\tag{14}
$$
The expression after the limit sign in (14) is nonnegative and nonincreasing with $n$, so the limit exists. If $K\subset \mathbb C$ and $v(z)\equiv 0$ (that is, there is no external field), then $\mathbf d_{v }{K}=\mathbf d{K}$, where $\mathbf d{K}$ is the standard transfinite diameter of the compact set $K\subset \mathbb C$ (for details about transfinite diameters of compact sets in external fields, see [7]). From the standpoint of applications (see [8]–[11]), of main interest are external fields of the form $v(z)=-\mathscr V ^\lambda (z)$ on $K$, where
$$
\begin{equation}
\mathscr V ^\lambda (z):=-\int\log |z-t|\,d\lambda (t)
\end{equation}
\tag{15}
$$
is the logarithmic potential of the unit positive Borel measure $\lambda$ with support outside $K$. Lemma 2. Let $\mathbf E$ be a compact set in $\overline{\mathbb C}$, let $\infty\in \mathbf E$ and $\mathbf G(z)\in H(\mathbf E)$, and let $\{ \mathbf \Phi_n(z)\}_{n=1}^\infty$ be a sequence of holomorphic functions in $\overline{\mathbb C}\setminus \mathbf E$ such that locally uniformly in $\overline{\mathbb C}\setminus \mathbf E$. Then for each compact set $\Gamma \subset \overline{\mathbb C}$ disjoint from $\mathbf E$, consisting of a finite number of continua and such that $\mathbf G(z)$ has meromorphic extensions to all connected components of $\overline{\mathbb C}\setminus \Gamma $ intersecting $\mathbf E$, the inequality holds, where $\mathbf d_{\varphi}\Gamma $ is the transfinite diameter of $\Gamma $ in the external field $\varphi (z)$. Inequality (17) also holds when $\infty\not\in \mathbf E$, but the additional assumption $\varphi (z)=-\mathscr V ^\lambda (z)$ is fulfilled, where $\mathscr V ^\lambda (z)$ is the logarithmic potential, defined by (15), of the unit positive Borel measure $\lambda$ with support on $\mathbf E$. In the particular case when $G(z)=\sum_{n=0}^\infty a_nz^{-n}$ is a holomorphic function in a neighbourhood of $z=\infty$ and $\Phi_n(z)\equiv 1$, inequality (17) coincides with
$$
\begin{equation*}
\varlimsup_{n\to\infty}|\det(a_{j+k-1})_{j,k=1,\dots ,n}|^{1/n^2}\leqslant \mathbf d\Gamma ,
\end{equation*}
\notag
$$
which is the well-known Pólia theorem [12] on estimates for Hankel determinants of a meromorphic function in terms of the standard transfinite diameter of the compact set such that the function has a meromorphic extension outside this set. In the general case Lemma 2 was proved in [13]. We will use inequality (17) here in the case when $\Gamma$ is (see (13)) the preimage of a line segment (whose standard transfinite diameter is well known to be a quarter of its length) under a rational map, and $\varphi (z)=-\mathscr V ^\mu (z)$, where the measure $\mu$ is concentrated at poles of this rational map. In this case (and even in a slightly more general situation when $\Gamma$ is the preimage of a compact set with known standard transfinite diameter with respect to a rational map) we have the following result established in [14]. Lemma 3. Let $\mathbf K$ be a compact set in $\mathbb C$, let $R(z)=P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are monic polynomials of degree $k$ and $m$, respectively, without common zeros, and let In the special case when
$$
\begin{equation*}
m=0, \qquad n=\deg P(z), \qquad \Gamma :=\{ z\in\mathbb C\colon P(z)\in \mathbf K\}\quad\text{and} \quad \varphi (z)\equiv 0
\end{equation*}
\notag
$$
equality (18) coincides with the well-known identity
$$
\begin{equation*}
\mathbf d\bigl\{ z\in\mathbb C\colon P(z)\in \mathbf K\bigr\} = (\mathbf d \mathbf K )^{1/n}
\end{equation*}
\notag
$$
for the (standard) transfinite diameters of polynomial preimages of compact sets. It is easy to see that in another extreme case, when $m=n$, we do not require that the polynomial $P(z)$ in Lemma 3 be monic. In the calculations of the left-hand side of (17) we use Lemma 2 in [3]. Lemma 4. Let Then assertion $1^\circ$ of Theorem 2 holds, that is, the even convergents $\pi_{2n}(z)={P_{2n}(z)}/{Q_{2n}(z)}$ of (19) are Nevanlinna functions which converge locally uniformly in $\mathbb C_+$ to a Nevanlinna function $G(z)$; the odd convergents $\pi_{2n+1}(z)={P_{2n+1}(z)}/{Q_{2n+1}(z)}$ converge to the function $\overline{G}(z):=\overline{G(\overline{z})}$ locally uniformly in $\mathbb C_-$. For all $n=1,2,\dots$,
$$
\begin{equation}
\det\biggl (\frac{1}{2\pi i}\oint_{\mathbf E_n}\frac{\mathbf G(z)z^{l+j-2}}{\prod_{k=0}^{n-1}(z-d_k)(z-\overline{d}_k)}\,dz\biggr )_{l,j=1,\dots ,n}= \prod_{k=0}^{n-1}\frac{\rho_{k}}{|Q_{2k}(d_{k})|^2(d_{k}-\overline{d}_{k})} ,
\end{equation}
\tag{20}
$$
where $\displaystyle \oint_{\mathbf E_n}$ is the integral over a system of contours in $\mathbb C_+\cup\mathbb C_-$ that encircle the points in the set
$$
\begin{equation}
\begin{gathered} \, \notag \mathbf E_n:=\{ d_0, \overline{d}_0,\dots ,d_{n-1},\overline{d}_{n-1}\}, \qquad \mathbf G(z):=\begin{cases} G(z),& z\in\mathbb C_+, \\ \overline{G}(z),&z\in\mathbb C_-, \end{cases} \\ \rho_{0} :=\delta_{-1}-\overline{\delta }_{-1}\quad\textit{and} \quad \rho_k:=\rho_{0} \prod_{j=0}^{k-1}(1-|\delta_j|^2), \quad k=1,2,\dots\,. \end{gathered}
\end{equation}
\tag{21}
$$
In addition, andApart from the above results of Lemma 4, in the proof of Theorem 3 we also need an analogue of inequality (23) that is not stated in terms of the continued fraction (19), but in terms of its ‘remainder’
$$
\begin{equation}
\delta_0+\cfrac{(1-|\delta_0|^2)(z-d_1)}{\overline{\delta}_0(z-d_1) +\cfrac{z-\overline{d}_1}{\delta_1+\dotsb}}.
\end{equation}
\tag{24}
$$
Note that using the scheme of the proof of Lemma 4 (presented in the proof of Lemma 2 in [3]) we can prove all analogues of the assertions of Lemma 4 stated in terms of the fraction (24). However, in the proof of Theorem 3 we only need an analogue of inequality (23), which is easier to derive as a consequence of Lemma 4. Corollary. Given a continued fraction (24), let the points $d_1,d_2,\dots$ lie in $\mathbb C_+$ and the coefficients $\delta_{0},\delta_{1},\dots$ be less than 1 in modulus. Then
$$
\begin{equation}
Q^*_{2n}(z)\neq 0, \qquad z\in\mathbb C_+, \quad n=0,1,\dots ,
\end{equation}
\tag{25}
$$
where $Q^*_{2n}(z)$ is the denominator of the $2n$th convergent of (24). In fact, denoting by $P^*_{2n}(z)$ the numerator of the $2n$th convergent of (24), from the form of the fractions (19) and (24) we obtain the equality
$$
\begin{equation*}
\frac{P_{2n+2}(z)}{Q_{2n+2}(z)} =\delta_{-1}+\cfrac{(\delta_{-1}-\overline{\delta }_{-1})(z-d_0)}{-(z-d_0) +\cfrac{z-\overline{d}_0}{\frac{P^*_{2n}(z)}{Q^*_{2n}(z)}}},
\end{equation*}
\notag
$$
which shows that
$$
\begin{equation*}
P_{2n+2}(z)=-\overline{\delta }_{-1}(z-d_0)P^*_{2n}(z)+ \delta_{-1}(z-\overline{d}_0)Q^*_{2n}(z)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Q_{2n+2}(z)=-(z-d_0)P^*_{2n}(z)+ (z-\overline{d}_0)Q^*_{2n}(z).
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
Q^*_{2n}(z)=\frac{Q_{2n+2}(z)\bigl (\frac{P_{2n+2}(z)}{Q_{2n+2}(z)}-\overline{\delta }_{-1}\bigr)}{(\delta_{-1}-\overline{\delta }_{-1})(z-\overline{d}_0)}.
\end{equation*}
\notag
$$
This equality, (23) and the inclusions $z\in\mathbb C_+$, $d_0\in\mathbb C_+$, $\delta_{-1}\in\mathbb C_+$ and ${P_{2n+2}(z)}/{Q_{2n+2}(z)} \in\overline{\mathbb C}_+$ yield (25). § 4. Proof of Theorem 3For brevity, throughout this section by a ‘theorem’ without further specifications we will always mean Theorem 3. First we prove some results supplementing the theorem and required for its proof. Namely, we show that
$$
\begin{equation}
\deg I(z)\leqslant q, \qquad I(\overline{z})=\overline{I(z)}, \qquad I(d^p)\neq 0, \quad p=1,\dots ,q,
\end{equation}
\tag{26}
$$
and
$$
\begin{equation}
\mathbf d_{\varphi}\Gamma =\prod_{p=1}^{q}\frac{(1-|\delta ^p|^2)^{1/2q}}{|I(d_p)|^{1/q^2}},
\end{equation}
\tag{27}
$$
where $\mathbf d_{\varphi}\Gamma $ is the transfinite diameter of the compact set $\Gamma$ in the external field
$$
\begin{equation}
\varphi (z)=\frac {1}{2q}\sum_{p=1}^{q}(\log |z-d_p|+\log |z-\overline{d}_p|).
\end{equation}
\tag{28}
$$
Let $\mathscr S ^p(z)$, $p=1,\dots ,q$, be the matrices defined by equality (11) in the statement of the theorem. Using induction on $k=1,\dots ,q$ we easily see that their products $ (\mathscr S^{1}\times\dots \times \mathscr S ^{k} )(z)$, $k=1,\dots ,q$, have the following form:
$$
\begin{equation*}
(\mathscr S ^{1}\times\dots \times \mathscr S ^{k} )(z)= \begin{pmatrix} A_k(z) & B_k(z) \\ \overline{B}_k(z) & \overline{A}_k(z) \end{pmatrix}, \qquad k=1,\dots ,q,
\end{equation*}
\notag
$$
where $A_k(z)$ and $B_k(z)$ are polynomials of degree at most $k$. Hence for $k=q$ and (12) we obtain
$$
\begin{equation*}
I(z):=\operatorname{Tr} (\mathscr S ^{1}\times\dots \times \mathscr S ^{q} )(z)= (A_q+\overline{A}_q )(z)=\overline{I}(z), \qquad \deg I(z)\leqslant q.
\end{equation*}
\notag
$$
Thus we have proved the first and second assertions in (26). Going over to the inequalities $I(d^p)\neq 0$, $p=1,\dots ,q$ (that is, to the proof of the last assertion in (26)), for $p=1,\dots ,q$ and $n=-1,0,1,2,\dots $ we set
$$
\begin{equation}
\begin{gathered} \, \notag d^{nq+p} :=d^{p}, \qquad \delta ^{nq+p} :=\delta ^{p} , \qquad \mathscr S ^{nq+p} (z):=\mathscr S ^p(z) , \\ S^{2nq+2p}(z):= \begin{pmatrix} 0 & z-\overline{d}^{p-1} \\ 1 & \delta ^{p-1}, \end{pmatrix}\ \ \text{and}\ \ S^{2nq+2p+1}(z):= \begin{pmatrix} 0 & (1-|\delta ^{p-1} |^2)(z-d ^{p}) \\ 1 & \overline{\delta } ^{p-1}(z-d ^{p}) \end{pmatrix}, \end{gathered}
\end{equation}
\tag{29}
$$
where $d^{p}$, $\delta ^{p}$ and $\mathscr S ^p(z)$ were defined in (9), (10) and (11), respectively; $d^0:=d^q$. Setting
$$
\begin{equation}
\mathscr L ^{p}(z):= (S^{p+1}\times\dots \times S^{p+2q} )(z), \qquad p=1,2,\dots ,
\end{equation}
\tag{30}
$$
we show that for the polynomials $I(z)$ and $J(z)$ defined by (12) we also have the relations
$$
\begin{equation}
I(z)=\operatorname{Tr}\mathscr L ^p(z)\quad\text{and} \quad J(z)=\det\mathscr L ^p(z), \quad p=1,2,\dots\,.
\end{equation}
\tag{31}
$$
Since for all $p,k=1,2,\dots $ the sequences $\mathscr L ^{p}(z)$ and $\mathscr L ^{k}(z)$ can be obtained from one another by cyclic permutations of the matrices involved, it is sufficient to prove (31) for $p=1$. Note that
$$
\begin{equation}
\begin{gathered} \, S^{2k}(z)= (\widehat{S}^{2k}\times \breve{S}^{2k} )(z), \\ \text{where } \widehat{S}^{2k}(z):=\begin{pmatrix} 0 & z-\overline{d}^{k-1} \\1 & 0\end{pmatrix}\text{ and } \breve{S}^{2k}(z):=\begin{pmatrix} 1 & \delta ^{k-1} \\0 & 1\end{pmatrix}, \end{gathered}
\end{equation}
\tag{32}
$$
and in view of the definition of $S^{2k+1}(z)$ in (29) and the definition of $\mathscr S ^{k}(z)$ in (11)
$$
\begin{equation}
\begin{aligned} \, &(\breve{S}^{2k}\times S^{2k+1}\times \widehat{S}^{2k+2} )(z) \notag \\ &\qquad=\begin{pmatrix} z-d^{k} & \delta ^{k-1}(z-\overline{d}^{k}) \\ \overline{\delta }^{k-1}(z-d^{k}) & z-\overline{d}^{k} \end{pmatrix} =\mathscr S ^{k}(z),\qquad k=1,\dots ,q. \end{aligned}
\end{equation}
\tag{33}
$$
Hence for $p=1$ equality (30) can be written as
$$
\begin{equation*}
\mathscr L ^{1}(z)= (\widehat{S}^2\times \breve{S}^2\times S^3\times \widehat{S}^4\times \dots \times S^{2q-1}\times \widehat{S}^{2q}\times \breve{S}^{2q}\times S^{2q+1} )(z),
\end{equation*}
\notag
$$
with the right-hand side that (since $\widehat{S}^2=\widehat{S}^{2q+2}(z)$) is a cyclic permutation of the product
$$
\begin{equation*}
\bigl((\breve{S}^2\times S^3\times \widehat{S}^4)\times \dots \times (\breve{S}^{2q}\times S^{2q+1}\times \widehat{S}^{2q+2})\bigr )(z),
\end{equation*}
\notag
$$
which coincides (for $k=1,\dots ,q$) with the product $ (\mathscr S ^1\times \dots\times\mathscr S ^{q} )(z)$ by (33). Hence (12) yields (31). Now, on the basis of (31) we show that for all $p=1,\dots ,q$ we have
$$
\begin{equation}
I(z)=P^p_{2q-1}(z) - \delta ^{p-1}Q^p_{2q-1}(z)+Q^p_{2q}(z),
\end{equation}
\tag{34}
$$
where (for fixed $ p=1,\dots ,q$) $P_n^p(z)$ and $Q_n^p(z)$, $n=0,1,\dots $, are the numerator and denominator of the $n$th convergent $\pi_n^p(z)=P_n^p(z)/Q_n^p(z)$ of the periodic continued fraction
$$
\begin{equation}
\delta ^{p-1}+\cfrac{(1-|\delta ^{p-1}|^2)(z-d ^{p})}{\overline{\delta}^{p-1}(z-d ^{p}) +\cfrac{z-\overline{d} ^{p}}{\delta ^{p}+\cfrac{(1-|\delta ^{p}|^2)(z-d ^{p+1})}{\overline{\delta}^{p}(z-d ^{p+1})+\cfrac{z-\overline{d} ^{p+1}}{\delta ^{p+1}+\dotsb}}}}.
\end{equation}
\tag{35}
$$
Note that the fraction (35) is a special case (for $\delta_{0}=\delta ^{p-1}, \delta_{1}=\delta ^{p},\dots $ and $d_{1}=d^{p}$, $d_2=d^{p+1}$, $\dots$) of the fraction (24) (defined before the corollary to Lemma 4). Hence by the corollary to Lemma 4 (see (25))
$$
\begin{equation}
Q^p_{2n}(z)\neq 0, \qquad p=1,\dots ,q, \quad n=0,1,\dots , \quad z\in\mathbb C_+.
\end{equation}
\tag{36}
$$
Note also that the initial conditions and three-term relations for $n=1,2,\dots$ for the fraction (35) have the following form:
$$
\begin{equation}
P^p_{0}(z)\equiv \delta ^{p-1}, \quad P^p_{1}(z)=(z-d^{p}) , \quad Q^p_{0}(z)\equiv 1 , \quad Q^p_{1}(z)= \overline{\delta }^{p-1}(z-d^{p}),
\end{equation}
\tag{37}
$$
$$
\begin{equation}
P^p_{2n}(z)=\delta ^{p+n-1} P^p_{2n-1}(z) +(z-\overline{d} ^{p+n-1})P^p_{2n-2}(z),
\end{equation}
\tag{38}
$$
$$
\begin{equation}
P^p_{2n+1}(z)=(z-d^{p+n})\overline{\delta }^{p+n-1} P^p_{2n}(z) +(z-d^{p+n})(1-|\delta ^{p+n-1} |^2)P^p_{2n-1}(z),
\end{equation}
\tag{39}
$$
$$
\begin{equation}
Q^p_{2n}(z)=\delta ^{p+n-1} Q^p_{2n-1}(z) +(z-\overline{d} ^{p+n-1})Q^p_{2n-2}(z)
\end{equation}
\tag{40}
$$
and
$$
\begin{equation}
Q^p_{2n+1}(z)=(z-d^{p+n})\overline{\delta }^{p+n-1} Q^p_{2n}(z) +(z-d^{p+n})(1-|\delta ^{p+n-1} |^2)Q^p_{2n-1}(z) ).
\end{equation}
\tag{41}
$$
From the definition (32) (for $k=p$) of the matrix $\breve{S}^{2p}(z)$, the definition (29) of $S^{2p+1}(z)$ and the initial conditions (37) we obtain
$$
\begin{equation*}
\begin{aligned} \, \breve{S}^{2p}(z)\times S^{2p+1}(z) &= \begin{pmatrix} 1 & \delta ^{p-1} \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 0 & (1-|\delta ^{p-1} |^2)(z-d ^{p}) \\ 1 & \overline{\delta } ^{p-1}(z-d ^{p}) \end{pmatrix} \\ &=\begin{pmatrix} P^p_{0}(z) & P^p_{1}(z) \\ Q^p_{0}(z) & Q^p_{1}(z) \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
Hence, from (38)–(41), using induction on $n$, for all $n=1,2,\dots $ we can easily deduce the equalities
$$
\begin{equation}
\breve{S}^{2p}(z)\times (S^{2p+1}\times \dots\times S^{2p+n})(z)= \begin{pmatrix} P^p_{n-1}(z) & P^p_{n}(z) \\ Q^p_{n-1}(z) & Q^p_{n}(z) \end{pmatrix}, \qquad p=1,\dots ,q,
\end{equation}
\tag{42}
$$
where the matrices $S^{2p+1}(z),S^{2p+2}(z),\dots $ were defined in (29). Using the definition (30) of the matrices $\mathscr L ^{2p}(z)$, from (42) for $n=2q$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \mathscr L ^{2p}(z) &= (\breve{S}^{2p})^{-1}(z)\times \breve{S}^{2p}(z)\times (S^{2p+1}\times \dots\times S^{2p+2q})(z) \\ &=(\breve{S}^{2p})^{-1}(z)\times \begin{pmatrix} P^p_{2q-1}(z) & P^p_{2q}(z) \\ Q^p_{2q-1}(z) & Q^p_{2q}(z) \end{pmatrix} \\ &= \begin{pmatrix} 1 & -\delta ^{p-1}\\ 0 & 1 \end{pmatrix}\times \begin{pmatrix} P^p_{2q-1}(z) & P^p_{2q}(z) \\ Q^p_{2q-1}(z) & Q^p_{2q}(z) \end{pmatrix} \\ &=\begin{pmatrix} P^p_{2q-1}(z)-\delta ^{p-1}Q^p_{2q-1}(z)) & P^p_{2q}(z)-\delta ^{p-1}Q^p_{2q}(z)) \\ Q^p_{2q-1}(z) & Q^p_{2q}(z) \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
Taking (31) into account this implies (34). By the initial conditions (37), for $z=d^p$ we have and
$$
\begin{equation*}
P^p_{1}(d^{p})-\delta ^{p-1}Q^p_{1}(d^{p})=(z-d^p)(1-|\delta ^{p-1}|^2)\big |_{z=d^p}=0,
\end{equation*}
\notag
$$
which implies, in view of relations (38)–(41) for $z=d^p$, the equalities
$$
\begin{equation}
P^p_{n}(d^{p})-\delta ^{p-1}Q^p_{n}(d^{p})=0, \qquad n=0,1,2,\dots\,.
\end{equation}
\tag{43}
$$
From (34), (43) (for $n=2q-1$) and (36) we deduce that
$$
\begin{equation}
I(d^{p})=P^p_{2q-1}(d^{p})-\delta ^{p-1}Q^p_{2q-1}(d^{p})+Q_{2q}^p(d^{p})=Q_{2q}^p(d^{p})\neq 0,\qquad p=1,\dots ,q.
\end{equation}
\tag{44}
$$
We have established that $I(d^{p})\neq 0$, $p=1,\dots ,q$, which completes the proof of (26). Now we prove (27). Note that by (11) and (12)
$$
\begin{equation}
J(z)=\det ((\mathscr S ^1\times\dots \times\mathscr S ^q)(z) )=\prod_{p=1}^q(1-|\delta ^p|^2)(z-d^p)(z-\overline{d}^p).
\end{equation}
\tag{45}
$$
It follows from (13) and (45) that
$$
\begin{equation*}
\Gamma =\biggl \{ z\in\overline{\mathbb C}\colon \frac{I^2(z)}{J(z)}\in [0,4]\biggr \} =\biggl \{ z\in\overline{\mathbb C}\colon \frac{I^2(z)}{\prod_{p=1}^q(z-d^p)(z-\overline{d}^p)}\in \mathbf K\biggr \},
\end{equation*}
\notag
$$
where $\mathbf K$ is the interval $\bigl[0,4\prod_{p=1}^{q}(1-|\delta ^p|^2)\bigr]$. Following the notation of Lemma 3 set
$$
\begin{equation}
R(z):=\frac{P(z)}{Q(z)}, \quad\text{where } P(z):=I^2(z)\quad\text{and} \quad Q(z):=\prod_{p=1}^q(z-d^p)(z-\overline{d}^p),
\end{equation}
\tag{46}
$$
and note that, as follows from (26), the fraction $R(z)$ is irreducible and $\deg P(z)\leqslant 2q =\deg Q(z)$. Therefore, and, as noted after the statement of Lemma 3, the assumption that the polynomial $P(z)$ in Lemma 3 is monic can be dropped. Also note that the external field $\varphi (z)$ in Lemma 3, defined by the polynomial $Q(z)$ in (46), has the expression (28). Since the standard transfinite diameter of a line segment is the quarter of its length, it follows that
$$
\begin{equation*}
(\mathbf d\mathbf K)^{1/n}=\prod_{p=1}^{q}(1-|\delta ^p|^2)^{1/(2q)},
\end{equation*}
\notag
$$
and by the definition $P(z):=I^2(z)$ (see (46)) and the equality $I(\overline{z})=\overline{I(z)}$ (see (26))
$$
\begin{equation*}
\biggl (\prod_{p=1}^{q}P(d^p)P(\overline{d}^p)\biggr )^{1/n^2} =\biggl (\prod_{p=1}^{q}I^2(d^p)I^2(\overline{d}^p)\biggr )^{1/(2q)^2}=\prod_{p=1}^{q}|I(d^p)|^{1/q^2}.
\end{equation*}
\notag
$$
Hence from Lemma 3 (see (18)) we obtain
$$
\begin{equation*}
\mathbf d_{\varphi }\Gamma =\frac{(\mathbf d \mathbf K)^{1/n}}{\bigl (\prod_{p=1}^{q}P(d^p)P(\overline{d}^p)\bigr)^{1/n^2}}=\prod_{p=1}^{q}\frac{(1-|\delta ^p|^2)^{1/2q}}{|I(d_p)|^{1/q^2}},
\end{equation*}
\notag
$$
which coincides with (27). Equality (27) is proved. Turning directly to the proof of the theorem, using Lemma 1 we prove that the continued fraction (5) converges. Let
$$
\begin{equation}
\begin{gathered} \, S_{0}(w,z):=w+\delta_{-1}, \qquad S_{1}(w,z):=\frac{(\delta_{-1}-\overline{\delta }_{-1})(z-d_0)}{w-(z-d_0)}, \\ \notag S_{2n}(w,z):=\frac{z-\overline{d}_{n-1}}{w+\delta_{n-1}}\ \ \text{and} \ \ S_{2n+1}(w,z):=\frac{(1-|\delta_{n-1}|^2)(z-d_{n})}{w+\overline{\delta }_{n-1}(z-d_{n})}, \quad n=1,2,\dots, \end{gathered}
\end{equation}
\tag{47}
$$
be the linear fractional transformations (in $w$) corresponding to the fraction (19), which is equivalent to (5). It follows from the assumptions (9) and (10) of the theorem that the coefficients of these linear fractional (in $w$) transformations $S_{n}(w,z)$ have periodic limits (with period $2q$) as $n\to\infty$ locally uniformly in $\mathbb C$. Set
$$
\begin{equation*}
S^{2p}(w,z):=\lim_{n\to\infty} S_{2nq+2p}(w,z)=\frac{z-\overline{d}^{p-1}}{w+\delta ^{p-1}}, \qquad p=1,\dots ,q,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
S^{2p+1}(w,z):=\lim_{n\to\infty} S_{2nq+2p+1}(w,z)=\frac{(1-|\delta ^{p-1} |^2)(z-d ^{p})}{w+ \overline{\delta } ^{p-1}(z-d ^{p})}, \qquad p=0,\dots ,q-1.
\end{equation*}
\notag
$$
We observe that the matrices $S^{2nq+2p}(z)$ and $S^{2nq+2p+1}(z)$, defined in (29) for $p=1,\dots ,q$ and $n=-1,0,1,2,\dots$, are the coefficient matrices of the limit linear fractional transformations $S^{2p}(w,z)$ and $S^{2p+1}(w,z)$, respectively. For $p=1,\dots, 2q$ set
$$
\begin{equation*}
\begin{gathered} \, \mathscr L ^{p}(w,z):= (S^{p+1}\circ\dots \circ S^{p+2q} )(w,z), \quad\text{where } S^{t+2q}(w,z):=S^{t}(w,z), \\ t=1,\dots, 2q, \end{gathered}
\end{equation*}
\notag
$$
and note that the matrix $\mathscr L ^{p}(z)$ defined by (30) is the coefficient matrix of $\mathscr L ^{p}(w,z)$, $p=1,\dots ,2q$. By Lemma 1 (for $w=0$ and $m$ replaced by $2q$) the limit
$$
\begin{equation}
\lim_{n\to\infty} S_{0}\circ\dots\circ S_{n}(0,z)=\mathbf G(z)
\end{equation}
\tag{48}
$$
exists uniformly in the spherical metric on compact subsets of $\overline{\mathbb C}\setminus (\check{\Gamma }\cup\Xi )$, where $\mathbf G(z)$ is a meromorphic function in $\overline{\mathbb C}\setminus \check{\Gamma }$,
$$
\begin{equation*}
\check{\Gamma }=\biggl\{z\in\overline{\mathbb C}\colon \frac{\check{I}^2(z)}{\check{J}(z)}\in [0,4]\biggr\}
\end{equation*}
\notag
$$
and the polynomials $\check{I}(z)$ and $\check{J}(z)$ are the trace and determinant, respectively, of the coefficient matrix of any transformation $\mathscr L ^{p}(w,z)$, $p=1,\dots ,2q$. From (31), in view of the above it follows that
$$
\begin{equation*}
I(z)=\operatorname{Tr} \mathscr L ^p(z)=\check{I}(z)\quad\text{and} \quad J(z)=\det \mathscr L ^p(z)=\check{J}(z),
\end{equation*}
\notag
$$
and therefore where $\Gamma$ is the compact set from the statement of the theorem defined by (13). As mentioned in Lemma 1, the set $\Xi$ consists of the points $z\in\overline{\mathbb C}$ such that $w=0$ is the repelling fixed point of at least one linear fractional (in $w$) transformation $\mathscr L ^{p}(w,z)$, $p=1,\dots ,2q$. Hence $\Xi$ is a finite set of cardinality at most $(2q)^2$ because the coefficients of the transformations $\mathscr L ^{p}(w,z)$ are easily seen to be polynomials of degree at most $2q$ of $z$ (so each equality $\mathscr L ^{p}(0,z)=0$, $p=1,\dots ,2q$, holds at $2q$ points $z\in\overline{\mathbb C}$ at most). For the continued fraction (19) the initial conditions and three-term relations have the following form for $n=1,2,\dots $:
$$
\begin{equation}
P_{0}(z)=\delta_{-1}, \quad P_{1}(z)=-\overline{\delta }_{-1}(z-d_0), \quad Q_{0}(z)=1,\quad Q_{1}(z)=-(z-d_0) ,
\end{equation}
\tag{49}
$$
$$
\begin{equation}
P_{2n} (z)=\delta_{n-1}P_{2n-1} (z)+(z-\overline {d}_{n-1})P_{2n-2} (z),
\end{equation}
\tag{50}
$$
$$
\begin{equation}
P_{2n+1} (z)=(z-d_{n})\overline{\delta}_{n-1}P_{2n} (z)+(z-d_{n})(1-|\delta_{n-1}|^2)P_{2n-1} (z),
\end{equation}
\tag{51}
$$
$$
\begin{equation}
Q_{2n} (z)=\delta_{n-1}Q_{2n-1} (z)+(z-\overline {d}_{n-1})Q_{2n-2} (z)
\end{equation}
\tag{52}
$$
and
$$
\begin{equation}
Q_{2n+1} (z)=(z-d_{n})\overline{\delta}_{n-1}Q_{2n} (z)+(z-d_{n})(1-|\delta_{n-1}|^2)Q_{2n-1} (z).
\end{equation}
\tag{53}
$$
From the definitions (47) of $S_{0}(w,z)$ and $S_1(w,z)$ and (49) we obtain
$$
\begin{equation*}
(S_{0}\circ S_1)(w,z)=\frac{P_{0}(z)w+P_{1}(z)}{Q_{0}(z)w+Q_{1}(z)},
\end{equation*}
\notag
$$
which is the first step $n=1$ of the inductive argument on $n=1,2,\dots$ that proves the equalities
$$
\begin{equation}
(S_{0}\circ \dots\circ S_n)(w,z)=\frac{P_{n-1}(z)w+P_{n}(z)}{Q_{n-1}(z)w+Q_{n}(z)}, \qquad n=1,2,\dots,
\end{equation}
\tag{54}
$$
by using relations (50)–(53). For $w=0$ equalities (54) have the form
$$
\begin{equation}
(S_{0}\circ \dots\circ S_n)(0,z)=\frac{P_{n}(z)}{Q_{n}(z)}=\pi_n(z), \qquad n=1,2,\dots\,.
\end{equation}
\tag{55}
$$
It follows from (48) and (55) that the limit exists uniformly in the spherical metric in $\overline{\mathbb C}\setminus (\Gamma \cup\Xi )$, where $\mathbf G(z)$ is a meromorphic function in $\overline{\mathbb C}\setminus \Gamma$. In addition to the limit equality (56), by Lemma 4 we have the locally uniform limit equalities
$$
\begin{equation*}
\lim_{n\to\infty}\pi_{2n}(z)=\mathbf G(z), \quad z\in\mathbb C_+\quad\text{and} \quad \pi_{2n+1}(z)=\mathbf G(z), \quad z\in \mathbb C_-,
\end{equation*}
\notag
$$
and we know that $\mathbf G(z) |_{z\in\mathbb C_+}$ is a Nevanlinna function and $\mathbf G(\overline{z})=\overline{\mathbf G(z)}$ for ${z\!\in\!\mathbb C\!\setminus\!\mathbb R}$. In particular, $\mathbf G(z)$ is holomorphic in $\mathbb C\setminus \mathbb R$ and meromorphic in ${\overline{\mathbb C}\setminus (\mathbb R\cap\Gamma)}$. Since $\mathbf G(z) |_{z\in\mathbb C_+}$ is a Nevanlinna function, by the Riesz–Herglotz theorem there exists a nonnegative measure $\varsigma $ with support $\operatorname{supp} \varsigma =:\Upsilon$ on $\overline{\mathbb R}$ such that
$$
\begin{equation}
\mathbf G(z)=\operatorname{Re} \mathbf G(i)+\int_{\Upsilon}\frac{1+uz}{u-z}\,d\varsigma (u), \qquad z\in\overline{\mathbb C}\setminus \Upsilon .
\end{equation}
\tag{57}
$$
Thus, $\mathbf G(z)$ is holomorphic in $\overline{\mathbb C}\setminus \Upsilon$ and has a meromorphic extension to ${\overline{\mathbb C}\setminus (\Upsilon \cap \Gamma )}$. Now we prove that the limits
$$
\begin{equation}
\lim_{n\to\infty}|Q_{2(nq+p)}(d_{nq+p})|^{1/n}=|Q^p_{2q}(d^{p})|, \qquad p=1,\dots ,q,
\end{equation}
\tag{58}
$$
exist, where $Q_n(z)$ and $Q_n^p(z)$ are the denominators of the $n$th convergents of the continued fractions (19) and (35), respectively, $n=0,1,\dots$ . For $d\in\mathbb C$ and $r>0$ let $U_{d,r}$ denote the disc $\{ |z-d|< r\}$. Fix a positive number $\varepsilon$ and then a positive number $r\leqslant \varepsilon$ such that
$$
\begin{equation}
|Q_{2q}^p(z)-Q_{2q}^p(d^p)|\leqslant \varepsilon \quad\text{for all } z\in U_{d^p,r}, \quad p=1,\dots ,q.
\end{equation}
\tag{59}
$$
It follows from the assumptions (9) and (10) of the theorem that there exists a positive integer $K$ such that
$$
\begin{equation}
|d_{kq+p}-d^p|\leqslant r\quad\text{and} \quad |\delta_{kq+p}-\delta ^p|\leqslant r \quad\text{for all } k\geqslant K , \quad p=1,\dots ,q.
\end{equation}
\tag{60}
$$
For $k\geqslant K$, $ p=1,\dots ,q$ set
$$
\begin{equation}
V_n^{k,p}(z):=\frac{Q_{2(kq+p)+n}(z)}{Q_{2(kq+p)}(z)}, \qquad n=0,1,\dots ,
\end{equation}
\tag{61}
$$
and note that
$$
\begin{equation}
Q_{2(kq+p)}(z)\neq 0 \quad\text{and} \quad \biggl |\frac{\overline{Q}_{2(kq+p)}(z)}{Q_{2(kq+p)}(z)}\biggr |\leqslant 1 \quad\text{for } z\in\mathbb C_+
\end{equation}
\tag{62}
$$
(the first inequality in (62) follows from (23) in Lemma 4, and the second follows from the first because all zeros of $Q_{2(kq+p)}(z)$ lie outside $\mathbb C_+$). From relations (52) and (53) (for $n$ replaced by $kq+p+n$), for the denominators of convergents of (19) we obtain the following relations for $n=1,2,\dots$:
$$
\begin{equation}
V_{2n}^{k,p}(z) =\delta_{kq+p+n-1}V_{2n-1}^{k,p}(z) +(z-\overline {d}_{kq+p+n-1})V_{2n-2}^{k,p}(z)
\end{equation}
\tag{63}
$$
and
$$
\begin{equation}
\begin{aligned} \, V_{2n+1}^{k,p}(z) &=(z-d_{kq+p+n})\overline{\delta}_{kq+p+n-1} V_{2n}^{k,p}(z) \notag \\ &\qquad+(z-d_{kq+p+n})(1-|\delta_{kq+p+n-1}|^2)V_{2n-1}^{k,p}(z) \end{aligned}
\end{equation}
\tag{64}
$$
with the initial conditions
$$
\begin{equation}
V_{0}^{k,p}(z)\equiv 1\quad\text{and} \quad V_{1}^{k,p}(z)=\frac{Q_{2(kq+p)+1}(z)}{Q_{2(kq+p)}(z)}.
\end{equation}
\tag{65}
$$
It follows from the second equality in (22) (for $n=kq+p$) and (62) that
$$
\begin{equation}
|V_{1}^{k,p}(z)|=\biggl |(z-d_{kq+p})\frac{\overline{Q}_{2(kq+p)}(z)}{Q_{2(kq+p)}(z)}\biggr| \leqslant |z-d_{kq+p}|, \qquad z\in\mathbb C_+.
\end{equation}
\tag{66}
$$
It is easy to see that for $k\geqslant K$, $p=1,\dots ,q$ and $z\in U_{d^p,r}$ the coefficients of relations (63) and (64) for the sequence $\{V_{n}^{k,p}(z)\}_{n=0}^\infty$ differ from the corresponding coefficients of relations (40) and (41) for the sequence $\{ Q_n^p(z)\}_{n=0}^\infty$ of denominators of convergents of (35) by $\varepsilon C_{d^1,\dots ,d^q}$ at most, where the positive constant $C_{d^1,\dots,d^q}$ depends only on $d^1,\dots ,d^q$. Furthermore, the initial conditions (65) for the sequence $\{ V_n^{k,p}(z)\}_{n=0}^\infty$, where $k\geqslant K$, $p=1,\dots ,q$ and $z\in U_{d^p,r}$, are different from the initial conditions $Q_0^p(z)\equiv 1$ and $Q_1^p(z)=\overline{\delta }^{p-1}(z-d^p)$ (see (37)) for the sequence $\{ Q_n^p(z)\}_{n=0}^\infty$ by $3\varepsilon$ at most. In fact, $V_{0}^{k,p}(z)-Q_0^p(z)\equiv 0$ and taking (66) into account, for $k\geqslant K$, $p=1,\dots ,q$ and $z\in U_{d^p,r}$ we have
$$
\begin{equation*}
\begin{aligned} \, |V_{1}^{k,p}(z)-Q_1^p(z)| &\leqslant |V_{1}^{k,p}(z)|+|Q_1^p(z)| \\ &\leqslant |z-d_{kq+p}|+|z-d^p|\leqslant 3r\leqslant 3\varepsilon . \end{aligned}
\end{equation*}
\notag
$$
Thus, for $n=0,1,\dots $ there exist constants $c(n)$ independent of $k$ and $p$ such that
$$
\begin{equation}
\begin{gathered} \, |V_{n}^{k,p}(z)-Q_{n}^p(z)|\leqslant c(n)\varepsilon, \\ n=0,1,\dots, \quad k\geqslant K, \quad p=1,\dots,q, \quad z\in U_{d^p,r}. \end{gathered}
\end{equation}
\tag{67}
$$
Recalling the definition (61) of $V_{2q}^{k,p}(z)$ and setting $c:=c(2q)+1$, from (67) for $n=2q$, taking (59) into account, for $k\geqslant K$, $p=1,\dots ,q$ and $z\in U_{d^p,r}$ we obtain
$$
\begin{equation}
|Q_{2q}^p(d^p)|-c\varepsilon \leqslant \biggl |\frac{Q_{2(kq+p)+2q}(z)}{Q_{2(kq+p)}(z)}\biggr |\leqslant |Q_{2q}^p(d^p)|+c\varepsilon .
\end{equation}
\tag{68}
$$
Let $n>K$. Multiplying inequalities (68) for $k=K,\dots ,n-1$ and extracting the $n$th root, for $n>K$, $p=1,\dots ,q$ and $z\in U_{d^p,r}$ we obtain
$$
\begin{equation*}
(|Q_{2q}^p(d^p)|-c\varepsilon )^{(n-K)/n}\leqslant \biggl |\frac{Q_{2(nq+p)}(z)}{Q_{2(Kq+p)}(z)}\biggr |^{1/n} \leqslant \bigl(|Q_{2q}^p(d^p)|+c\varepsilon \bigr)^{(n-K)/n}.
\end{equation*}
\notag
$$
Since $d_{nq+p}\in U_{d^p,r}$ for $n\geqslant K$ (see (60)), the above inequalities hold for ${z=d_{nq+p}}$, and for $ p=1,\dots ,q$, letting $n\to\infty$, they yield
$$
\begin{equation*}
\begin{aligned} \, |Q_{2q}^p(d^p)|-c\varepsilon &\leqslant \varliminf_{n\to\infty} \biggl |\frac{Q_{2(nq+p)}(d_{nq+p})}{Q_{2(Kq+p)}(d_{nq+p})}\biggr |^{1/n} =\varliminf_{n\to\infty} |Q_{2(nq+p)}(d_{nq+p}) |^{1/n} \\ &\leqslant \varlimsup_{n\to\infty} |Q_{2(nq+p)}(d_{nq+p}) |^{1/n}= \varlimsup_{n\to\infty} \biggl |\frac{Q_{2(nq+p)}(d_{nq+p})}{Q_{2(Kq+p)}(d_{nq+p})}\biggr |^{1/n} \\ &\leqslant |Q_{2q}^p(d^p)|+c\varepsilon . \end{aligned}
\end{equation*}
\notag
$$
Hence, as $\varepsilon$ can be arbitrary, for $ p=1,\dots ,q$ we obtain
$$
\begin{equation*}
\varliminf_{n\to\infty} |Q_{2(nq+p)}(d_{nq+p}) |^{1/n}= \varlimsup_{n\to\infty} |Q_{2(nq+p)}(d_{nq+p}) |^{1/n}=|Q_{2q}^p(d^p)|,
\end{equation*}
\notag
$$
which shows that the limits (58) exist. Now, taking (23) into account, from (58) and (44) we see that the limit
$$
\begin{equation}
\lim_{n\to\infty}\prod_{k=0}^{n-1}|Q_{2k}(d_k)|^{2/n^2}=\prod_{p=1}^{q} |Q_{2q}^p(d^p) |^{1/q^2}=\prod_{p=1}^{q} |I(d^p) |^{1/q^2}
\end{equation}
\tag{69}
$$
exists, and it is straightforward from the conditions on the points $d_0,d_1,\dots$ and coefficients $\delta_{-1},\delta_0,\dots$ (see (9) and (10)) that the following limits exist:
$$
\begin{equation}
\begin{gathered} \, \lim_{n\to\infty}\biggl |\prod_{k=0}^{n-1}\frac{\delta_{-1}-\overline{\delta }_{-1}}{d_{k} -\overline{d}_{k}}\biggr |^{1/n^2}=1, \\ \lim_{n\to\infty}\biggl |\prod_{k=1}^{n-1}\prod_{j=0}^{k-1}(1-|\delta_j|^2)\biggr |^{1/n^2}=\prod_{p=1}^{q}(1-|\delta ^p|^2)^{1/(2q)}. \end{gathered}
\end{equation}
\tag{70}
$$
Using the notation of Lemma 4, which is consistent with our notation here, from (20) and (21), taking (69), (70) and (27) into account we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to\infty}\biggl |\det\biggl (\frac{1}{2\pi i}\oint_{\mathbf E_n}\frac{\mathbf G(z)z^{l+j-2}}{\prod_{k=0}^{n-1}(z-d_k)(z-\overline{d}_k)}\,dz\biggr )_{l,j=1,\dots ,n}\biggr|^{1/n^2} \\ \notag &\qquad =\lim_{n\to\infty}\biggl |\prod_{k=0}^{n-1}\frac{(\delta_{-1}-\overline{\delta }_{-1})}{|Q_{2k}(d_{k})|^2(d_{k}-\overline{d}_{k})}\biggr |^{1/n^2} \biggl |\prod_{k=1}^{n-1}\prod_{j=0}^{k-1}(1-|\delta_j|^2)\biggr |^{1/n^2} \\ &\qquad =\prod_{p=1}^{q}\frac{ (1-|\delta ^p|^2 )^{1/2q}}{|I(d_p)|^{1/q^2}} =\mathbf d_\varphi (\Gamma ). \end{aligned}
\end{equation}
\tag{71}
$$
To prove the inclusion $\Gamma\subseteq\Upsilon$, where $\Upsilon =\operatorname{supp} \varsigma$ (see (57)), assume the converse: suppose that $\Gamma\not\subseteq\Upsilon$. Then $\Gamma\setminus\Upsilon$ contains a point $z_0$. Since the distance of $z_0$ to $\Upsilon$ is positive, apart from $z_0$, by the definition (13) of $\Gamma$, $\Gamma\setminus\Upsilon$ also contains an analytic arc. Hence we have the strict inequality
$$
\begin{equation}
\mathbf d_{\varphi}(\Gamma \cap\Upsilon ) < \mathbf d_{\varphi}\Gamma.
\end{equation}
\tag{72}
$$
Let $\mathbf E\subset\mathbb C$ be a compact subset of $\mathbb C_+\cup\mathbb C_-$ containing all points in the sequences $\{ d_n\}_{n=0}^\infty$ and $\{ \overline{d}_n\}_{n=0}^\infty$ with some neighbourhoods. Such a compact set exists because by assumption $d_0,d_1,\dots$ lie in $\mathbb C_+$ and have periodic limits (9) in $\mathbb C_+$. Since $\Upsilon\subset \overline{\mathbb R}$, it follows that $\mathbf E\cap (\Gamma\cap \Upsilon )=\varnothing$. As noted after equality (57), the function $\mathbf G(z)$ is holomorphic in $\mathbb C\setminus \Upsilon$ (in particular, on $\mathbf E$) and has a meromorphic extension to $\mathbb C\setminus (\Gamma \cap \Upsilon )$. Also note that for all $n=0,1,\dots$ the functions
$$
\begin{equation}
\mathbf \Phi_n(z):=\prod_{k=0}^{n-1}\bigl((z-d_k)(z-\overline{d}_k)\bigr)^{-1}
\end{equation}
\tag{73}
$$
are holomorphic in $\overline{\mathbb C}\setminus \mathbf E$, and by (9) they satisfy the condition
$$
\begin{equation*}
-\frac{1}{2n}\log |\mathbf \Phi_n(z)|\rightrightarrows \varphi (z)=\frac {1}{2q}\sum_{p=1}^{q}(\log |z-d_p|+\log |z-\overline{d}_p|)
\end{equation*}
\notag
$$
locally uniformly in $\overline{\mathbb C}\setminus \mathbf E$, so that they satisfy condition (16) in Lemma 2 for the function $\varphi (z)$ defined by (28) and expressible in the form $\varphi (z)=-\mathscr V ^\lambda (z)$, where $-\mathscr V ^\lambda (z)$ is the logarithmic potential of the unit positive Borel measure $\lambda =\frac {1}{2q}\sum_{p=1}^q(\xi_{d^p}+\xi_{\overline{d}^p})$ (recall that $\xi_d$ is the Dirac measure at $d$). Thus the assumptions of Lemma 2 hold for the compact sets $\mathbf E\subset \mathbb C$ and ${\Gamma \cap\Upsilon}$, the function $\mathbf G(z)\in H(\mathbf E)$ and the sequence of functions (73), which satisfy condition (16) in Lemma 2 for the limit function $\varphi (z)=-\mathscr V ^\lambda (z)$ on the right-hand side of (16). Hence by Lemma 2 we have (17), from where, taking (72) into account we deduce the strict inequality
$$
\begin{equation*}
\varlimsup_{n\to\infty}\biggl |\det\biggl (\oint_{\mathbf E}\frac{\mathbf G(z)(z)z^{l+j-2}}{\prod_{k=0}^{n-1}(z-d_k)(z-\overline{d}_k)}\,dz\biggr )_{l,j=1,\dots ,n}\biggr |^{1/n^2}\leqslant \mathbf d_{\varphi}(\Gamma \cap\Upsilon) < \mathbf d_{\varphi}\Gamma,
\end{equation*}
\notag
$$
which contradicts (71). Thus, the assumption $\Gamma\not\subseteq\Upsilon$ leads to a contradiction, and therefore $\Gamma\subseteq\Upsilon$. It follows from the inclusion $\Upsilon\subseteq\overline{\mathbb R}$, the definition (13) of the compact set $\Gamma$ and the inclusion $\Gamma\subseteq\Upsilon$ proved above that $\Gamma =\bigsqcup_{j=1}^k\Gamma_j$, where $\Gamma_1,\dots ,\Gamma_k$ is a finite set of disjoint closed intervals of $\overline{\mathbb R}$. It also follows from the definition (13) that for all $j=1,\dots ,k$ the interval $\Gamma_j$ is nondegenerate and contains at least one point from $\bigl\{ z\in\overline{\mathbb C}\colon {I^2(z)}/{J(z)}=0\bigr\}$. Since $\Gamma_1,\dots ,\Gamma_k$ are disjoint and $\deg I^2(z)\leqslant 2q=\deg J(z)$ (see (26) and (45)), so that the set $\bigl\{ z\in\overline{\mathbb C}\colon {I^2(z)}/{J(z)}=0\bigr\}$ has at most $q$ geometrically distinct points, we have $k\leqslant q$. The function $\mathbf G(z)$ is holomorphic outside $\Upsilon$ and meromorphic outside $\Gamma\subseteq\Upsilon$, and so $\Upsilon =\Gamma\cup\widetilde{\Gamma}$, where $\widetilde{\Gamma}$ is the set of poles of $\mathbf G(z)$. In combination with (57), this yields the result of the theorem. Theorem 3 is proved. 
Citation in format AMSBIB
\Bibitem{Bus25} |
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