Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

Abstract

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty \}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.

1. Introduction

There is a vast theory of orthogonal polynomials with respect to measures on $\mathbb{C}$ and their root asymptotics, exemplified by the Ullman–Stahl–Totik theory of regularity. Let $\mu$ be a compactly supported probability measure and $\{p_n\}_{n=0}^\infty$ the corresponding orthonormal polynomials, obtained by the Gram–Schmidt process from $\{z^n\}_{n=0}^\infty$ in $L^2(d\mu )$. Then

$$\begin{equation} \liminf _{n\to \infty } \lvert p_n(z) \rvert ^{1/n} \ge e^{G_\mathsf{E}(z,\infty )} {\tag{1.1}} \end{equation}$$

for $z$ outside the convex hull of $\operatorname {supp}\mu$, where $\mathsf{E}$ is the essential support of $\mu$ and $G_\mathsf{E}$ denotes the potential theoretic Green function for the domain $\overline{\mathbb{C}} \setminus \mathsf{E}$; if that domain is not Greenian, one takes $G_\mathsf{E} = +\infty$ instead. For measures compactly supported in $\mathbb{R}$, this theory can be interpreted in terms of self-adjoint operators. In particular, for any bounded half-line Jacobi matrix

$$\begin{equation*} J = \begin{pmatrix} b_1 & a_1 & & \\ a_1 & b_2 & a_2 & \\ & a_2 & \ddots & \ddots \\ & & \ddots & \end{pmatrix} \end{equation*}$$

with $a_\ell > 0$, $b_\ell \in \mathbb{R}$,

$$\begin{equation} \limsup _{n\to \infty } \left( \prod _{\ell =1}^n a_\ell \right)^{1/n} \le \operatorname {Cap} \sigma _{\mathrm{ess}}(J), {\tag{1.2}} \end{equation}$$

where $\operatorname {Cap}$ denotes logarithmic capacity. For both of these universal inequalities, the case of equality (and existence of limit) is called Stahl–Totik regularity 27; the theory originated with the case $\mathsf{E} = [-2,2]$, first studied by Ullman 30.

We extend aspects of this theory to the setting of rational functions with poles in $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty \}$. One motivation for this is the search for a more conformally invariant theory. Statements such as 1.1, 1.2 rescale in obvious ways with respect to affine transformations (automorphisms of $\mathbb{C}$) which preserve $\mathbb{R}$, so it is obvious that an affine pushforward of a Stahl–Totik regular measure is Stahl–Totik regular. However, the point $\infty$ has a very special role throughout the theory: for a Möbius transformation $f$ which does not preserve $\infty$, $p_n \circ f$ are rational functions with a pole at $f^{-1}(\infty )$, and $f(J)$ as defined by the functional calculus is not a finite band matrix. Thus, it is a nontrivial question whether a Möbius pushforward of a Stahl–Totik regular measure is Stahl–Totik regular.

The set of Möbius transformations which preserve $\overline{\mathbb{R}}$ is the semidirect group product $\mathrm{PSL}(2,\mathbb{R}) \rtimes \{ \mathrm{id}, z\mapsto -z \}$, whose normal subgroup $\mathrm{PSL}(2,\mathbb{R})$ corresponds to the orientation preserving case. Denote by $f_* \mu$ the pushforward of $\mu$, defined by $(f_* \mu )(A) = \mu (f^{-1}(A))$ for Borel sets $A$. As an example of our techniques, we obtain the following:

Theorem 1.1.

Let $f \in \mathrm{PSL}(2,\mathbb{R}) \rtimes \{ \mathrm{id}, z\mapsto -z \}$. If $\mu$ is a Stahl–Totik regular measure on $\mathbb{R}$ and $\infty \notin \operatorname {supp}(f_* \mu )$, then the pushforward measure $f_*\mu$ is also Stahl–Totik regular.

However, we will mostly work in the more general setting when multiple poles on $\overline{\mathbb{R}}$ are allowed, which arises naturally in the spectral theory of self-adjoint operators. Denote by $T_{f,d\mu }$ the multiplication operator by $f$ in $L^2(d\mu )$. The matrix representation for $T_{x,d\mu (x)}$ in the basis of orthogonal polynomials is a Jacobi matrix, and through this classical connection, the theory of orthogonal polynomials is inextricably linked to the spectral theory of Jacobi matrices. In this matrix representation, resolvents $T_{({\mathbf{c}}-x)^{-1},d\mu (x)}$ are not finite-diagonal matrices. However, in a basis of orthogonal rational functions with poles at ${\mathbf{c}}_1, \dots , {\mathbf{c}}_g, \infty$, the multiplication operators $T_{({\mathbf{c}}_1-x)^{-1},d\mu (x)}$, …, $T_{({\mathbf{c}}_g-x)^{-1},d\mu (x)}$, $T_{x,d\mu (x)}$ all have precisely $2g+1$ nontrivial diagonals. The corresponding matrix representations are called GMP matrices; they were introduced by Yuditskii 32.

Self-adjoint operators and their matrix representations are an important part of this work, so we choose to present the theory in a more self-contained way, using self-adjoint operators from the ground up; this has similarities with 22. Some proofs could be shortened by using orthogonal polynomials with respect to varying weights 27, Chapter 3, but some facts rely on the precise structure obtained by the periodically repeating sequence of poles.

We should also compare this to the case of CMV matrices: for a measure supported on the unit circle, Stahl–Totik regularity is still defined in terms of orthogonal polynomials, but the CMV basis 422 is given in terms of positive and negative powers of $z$, i.e., orthonormal rational functions with poles at $\infty$ and $0$. The symmetries in that setting lead to explicit formulas for the CMV basis in terms of the orthogonal polynomials; it is then a matter of calculation to relate the exponential growth rate of the CMV basis to that of the orthogonal polynomials, and to interpret regularity in terms of the CMV basis. In our setting, there is no such symmetry and no formula for orthonormal rational functions in terms of orthonormal polynomials.

In order to state our results in a conformally invariant way, we will use the following notations and conventions throughout the paper. The measure $\mu$ will be a probability measure on $\overline{\mathbb{R}}$. We denote by $\operatorname {supp}\mu$ its support in $\overline{\mathbb{R}}$, and we consider its essential support (the support with isolated points removed), denoted

$$\begin{equation*} \mathsf{E} = \operatorname {ess\,supp}\mu . \end{equation*}$$

We will always assume that $\mu$ is nontrivial; equivalently, $\mathsf{E} \neq \emptyset$.

Fix a finite sequence with no repetitions, ${\mathbf{C}} = ( {\mathbf{c}}_1,\dots , {\mathbf{c}}_{g+1})$ with ${\mathbf{c}}_k \in \overline{\mathbb{R}} \setminus \operatorname {supp}\mu$ for all $k$. Consider the sequence $\{ r_n \}_{n=0}^\infty$ where $r_0 = 1$ and for $n = j (g+1) + k$, $1\le k \le g+1$,

$$\begin{equation} r_n(z) = \begin{cases} \frac{1}{( {\mathbf{c}}_k - z)^{j+1}} & {\mathbf{c}}_k \in \mathbb{R}, \\ z^{j+1} & {\mathbf{c}}_k = \infty . \end{cases} {\tag{1.3}} \end{equation}$$

Applying the Gram–Schmidt process to this sequence in $L^2(d\mu )$ gives the sequence of orthonormal rational functions $\{ \tau _n \}_{n=0}^\infty$ whose behavior we will study. We note that the special case $\operatorname {supp}\mu \subset \mathbb{R}$, $g=0$, ${\mathbf{C}} = (\infty )$ corresponds to the standard construction of orthonormal polynomials associated to the measure $\mu$ (note that, since we denote by $\operatorname {supp}\mu$ the support in $\overline{\mathbb{R}}$, the statement $\operatorname {supp}\mu \subset \mathbb{R}$ implies that $\mu$ is compactly supported in $\mathbb{R}$), and our first results are an extension of the same techniques.

The first result is a universal lower bound on the growth of $\{ \tau _n \}_{n=0}^\infty$ in terms of a potential theoretic quantity. If $\mathsf{E}$ is not a polar set, we use the (potential theoretic) Green function for the domain $\overline{\mathbb{C}} \setminus \mathsf{E}$, denoted $G_\mathsf{E}$, and we define

$$\begin{equation} {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}) = \begin{cases} \frac{1}{g+1} \sum _{k=1}^{g+1} G_\mathsf{E}(z,{\mathbf{c}}_k) & \mathsf{E}\text{ is not polar,} \\ +\infty & \mathsf{E}\text{ is polar.} \end{cases} {\tag{1.4}} \end{equation}$$

Theorem 1.2.

For all $z \in \overline{\mathbb{C}} \setminus \overline{\mathbb{R}}$,

$$\begin{equation*} \liminf _{n\to \infty } \lvert \tau _{n}(z) \rvert ^{1/n} \ge e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}. \end{equation*}$$

This is a good place to point out that our current setup is not related to the recent paper 13, in which the behavior was compared to a Martin function at a boundary point of the domain. Here, the behavior is compared to a combination of Green functions 1.4, all the poles are in the interior of the domain $\overline{\mathbb{C}} \setminus \mathsf{E}$, and the difficulty comes instead from the multiple poles.

Another universal inequality for orthonormal polynomials comes from comparing their leading coefficients to the capacity of $\mathsf{E}$. In our setting, the analog of the leading coefficient must be considered in a pole-dependent way. Denote

$$\begin{equation*} {\mathcal{L}}_n = \mathrm{span} \{ r_\ell \mid 0 \le \ell \le n\}. \end{equation*}$$

By the nature of the Gram–Schmidt process, there is a $\kappa _n > 0$ such that

$$\begin{equation*} \tau _n - \kappa _n r_n \in {\mathcal{L}}_{n-1}. \end{equation*}$$

The Gram–Schmidt process can be reformulated as the $L^2(d\mu )$-extremal problem

$$\begin{equation} \kappa _n = \max \left\{\operatorname {Re}\kappa : f=\kappa r_n +h, h\in {\mathcal{L}}_{n-1}, \|f\|_{L^2(d\mu )}\leq 1\right\}. {\tag{1.5}} \end{equation}$$

By strict convexity of the $L^2$-norm, these $L^2$-extremal problems have unique extremizers given by $f = \tau _n$, and $\kappa _n$ is explicitly characterized as a kind of leading coefficient for $\tau _n$ with respect to the pole at ${\mathbf{c}}_k$ where $n = j (g+1) + k$, $1 \le k \le g+1$. Below, we will also relate the constants $\kappa _n$ to off-diagonal coefficients of certain matrix representations.

The growth of the leading coefficients $\kappa _n$ will be studied along sequences $n=j(g+1)+k$ for a fixed $k$, and bounded by quantities related to the pole ${\mathbf{c}}_k$. If $\mathsf{E}$ is not a polar set, it is a basic property of the Green function that the limits

$$\begin{equation*} \gamma _\mathsf{E}^k= \begin{cases} \lim _{z\to {\mathbf{c}}_k}(G_\mathsf{E}(z,{\mathbf{c}}_k)+\log |z-{\mathbf{c}}_k|),&{\mathbf{c}}_k\ne \infty ,\\ \lim _{z\to {\mathbf{c}}_k}(G_\mathsf{E}(z,{\mathbf{c}}_k)-\log |z|),& {\mathbf{c}}_k=\infty \end{cases} \end{equation*}$$

exist. Note that if ${\mathbf{c}}_k = \infty$, $\gamma _\mathsf{E}^k$ is precisely the Robin constant for the set $\mathsf{E}$. We further define constants $\lambda _k$ by

$$\begin{equation} \log \lambda _k = \begin{cases} \gamma _\mathsf{E}^k + \sum _{\substack{1\leq \ell \leq g+1\\ \ell \neq k}}G_\mathsf{E}({\mathbf{c}}_k,{\mathbf{c}}_\ell ) & \mathsf{E}\text{ is not polar,} \\ +\infty & \mathsf{E}\text{ is polar.} \end{cases} {\tag{1.6}} \end{equation}$$

Theorem 1.3.

For all $1 \le k \le g+1$, for the subsequence $n(j)=j(g+1)+k$,

$$\begin{equation} \liminf _{j \to \infty } \kappa _{n(j)}^{1/n(j)} \geq \lambda _k^{1/ (g+1)}. {\tag{1.7}} \end{equation}$$

Theorem 1.4.

The following are equivalent:

(i)

For some $1 \le k \le g+1$, for the subsequence $n(j)=j(g+1)+k$,$$\begin{equation*} \lim _{j\to \infty } \kappa _{n(j)}^{1/n(j)} = \lambda _k^{1 / (g+1)}; \end{equation*}$$

(ii)

For all $1 \le k \le g+1$, for the subsequence $n(j)=j(g+1)+k$,$$\begin{equation*} \lim _{j\to \infty } \kappa _{n(j)}^{1/n(j)} = \lambda _k^{1 / (g+1)}; \end{equation*}$$

(iii)

$$\begin{equation*} \lim _{n\to \infty } \left( \prod _{\ell =1}^{g+1} \kappa _{n+\ell } \right)^{1/n} = \left( \prod _{k=1}^{g+1} \lambda _k \right)^{1/(g+1)}; \end{equation*}$$

(iv)

For q.e. $z\in \mathsf{E}$, we have $\limsup _{n\to \infty } |\tau _{n}(z)|^{1/n} \leq 1$;

(v)

For some $z\in {\mathbb{C}}_+$, $\limsup _{n\to \infty } |\tau _{n}(z)|^{1/n} \leq e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$;

(vi)

For all $z\in {\mathbb{C}}$, $\limsup _{n\to \infty } |\tau _{n}(z)|^{1/n} \leq e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$;

(vii)

Uniformly on compact subsets of ${\mathbb{C}}\setminus {\mathbb{R}}$, $\lim _{n\to \infty } |\tau _{n}(z)|^{1/n}= e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$.

Definition 1.5.

The measure $\mu$ is ${\mathbf{C}}$-regular if it obeys one (and therefore all) of the assumptions of Theorem 1.4.

In this terminology, Stahl–Totik regularity is precisely $(\infty )$-regularity, i.e., ${\mathbf{C}}$-regularity for the special case $\operatorname {supp}\mu \subset \mathbb{R}$, $g=0$, ${\mathbf{C}} = (\infty )$. Theorems 1.2, 1.3, 1.4 are closely motivated by foundational results for Stahl–Totik regularity. A new phenomenon appears through the periodicity with which poles are taken in 1.3 and the resulting subsequences $n(j) = j(g+1)+k$: since $\kappa _n$ is a normalization constant for $\tau _n$, it is notable that control of $\kappa _n$ along a single subsequence $n(j) = j(g+1)+k$ in Theorem 1.4i provides control over the entire sequence. This phenomenon doesn’t have an exact analog for orthogonal polynomials, where $g=0$. We will also see below that this is essential in order to characterize the regularity of a GMP matrix using only the entries of the matrix itself and not its resolvents.

Moreover, we show that the regular behavior described by Theorem 1.4 is independent of the set of poles ${\mathbf{C}}$:

Theorem 1.6.

Let ${\mathbf{C}}_1, {\mathbf{C}}_2$ be two finite sequences of elements from $\overline{\mathbb{R}} \setminus \operatorname {supp}\mu$, not necessarily of the same length. Then $\mu$ is ${\mathbf{C}}_1$-regular if and only if it is ${\mathbf{C}}_2$-regular.

Corollary 1.7.

Let $\operatorname {supp}\mu \subset \mathbb{R}$. Let ${\mathbf{C}}$ be a finite sequence of elements from $\overline{\mathbb{R}} \setminus \operatorname {supp}\mu$. Then $\mu$ is ${\mathbf{C}}$-regular if and only if it is Stahl–Totik regular.

Thus, Theorem 1.4 should not be seen as describing equivalent conditions for a new class of measures, but rather a new set of regular behaviors for the familiar class of Stahl–Totik regular measures.

We consistently work with poles on $\overline{\mathbb{R}}$ since our main interest is tied to self-adjoint problems. Some of our results are in a sense complementary to the setting of 27, Section 6.1, where poles are allowed in the complement of the convex hull of $\operatorname {supp}\mu$, and the behavior of orthogonal rational functions is considered with respect to a Stahl–Totik regular measure. Due to this, it is natural to expect that these results hold more generally, for measures on $\mathbb{C}$ and general collections of poles and Möbius transformations. Moreover, in our setup the poles are repeated exactly periodically, but we expect this can be generalized to a sequence of poles which has a limiting average distribution. Related questions for orthogonal rational functions were also studied by 310.

As noted in 27, Section 6.1, poles in the gaps of $\operatorname {supp}\mu$ can cause interpolation defects in the problem of interpolation by rational functions. In our work, these interpolation defects show up as possible reductions in the order of the poles. For example, consider ${\mathbf{C}} = (\infty ,0)$. Then, by construction, $\tau _{2j+1}$ is allowed a pole at $0$ of order at most $j$. However, if $\mu$ is symmetric with respect to $z\mapsto -z$, the functions $\tau _n$ will have an even/odd symmetry. Since $\tau _{2j+1}$ contains a nontrivial multiple of $z^{j+1}$, it follows that $\tau _{2j+1}(z) = (-1)^{j+1} \tau _{2j+1}(-z)$. By this symmetry, the actual order of the pole at $0$ is $j+1-k$ for some even $k$, so it cannot be equal to $j$ (it will follow from our results that in this case, the order of the pole is $j-1$). The same effect can be seen for the pole at $\infty$ for ${\mathbf{C}} = (0,\infty )$. In the polynomial case, this does not occur: $p_n$ always has a pole at $\infty$ of order exactly $n$.

We will consider at once the distribution of zeros of $\tau _n$ and the possible reductions in the order of the poles. We will prove that all zeros of $\tau _n$ are real and simple, and that $n-g \le \deg \tau _n \le n$. We define the normalized zero counting measure

$$\begin{equation*} \nu _n = \frac{1}{n} \sum _{w: \tau _n(w)=0} \delta _w. \end{equation*}$$

Although we normalize by $n$, $\nu _n$ may not be a probability measure: however $1 - g/n \le \nu _n(\overline{\mathbb{R}}) \le 1$. Therefore, normalizing by $\deg \tau _n$ instead of by $n$ would not affect the limits as $n\to \infty$.

We will now describe the weak limit behavior of the measures $\nu _n$ as $n\to \infty$. To avoid pathological cases, we assume that $\mathsf{E}$ is not polar; in that case, denoting by $\omega _\mathsf{E}( dx,w)$ the harmonic measure for the domain $\overline{\mathbb{C}} \setminus \mathsf{E}$ at the point $w$, we define the probability measure on $\mathsf{E}$,

$$\begin{equation*} \rho _{\mathsf{E},{\mathbf{C}}} = \frac{1}{g+1} \sum _{j=1}^{g+1} \omega _\mathsf{E}(\mathrm{d} x,{\mathbf{c}}_j). \end{equation*}$$

The results below describe weak limits of measures in the topology dual to $C(\overline{\mathbb{R}})$.

Theorem 1.8.

Let $\mu$ be a probability measure on $\overline{\mathbb{R}}$. Assume that $\mathsf{E}$ is not a polar set.

(a)

If $\mu$ is ${\mathbf{C}}$ regular, then $\operatorname *{w-lim}_{n\to \infty } \nu _n = \rho _{\mathsf{E},{\mathbf{C}}}$.

(b)

If $\operatorname *{w-lim}_{n\to \infty } \nu _n =\rho _{\mathsf{E},{\mathbf{C}}}$, then $\mu$ is ${\mathbf{C}}$ regular or there exists a polar set $X \subset \mathsf{E}$ such that $\mu (\overline{\mathbb{R}} \setminus X) = 0$.

We now turn to matrix representations of self-adjoint operators. Fix a sequence ${\mathbf{C}} = ({\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1})$ such that ${\mathbf{c}}_{k_\infty } = \infty$ for some $1\leq k_\infty \leq g+1$. A half-line GMP matrix 32 is the matrix representation for multiplication by $x$ in the basis $\{\tau _n \}_{n=0}^\infty$ for this sequence ${\mathbf{C}}$; its matrix elements are

$$\begin{equation*} A_{mn} = \int \overline{\tau _m(x)} x \tau _n(x) \,d\mu (x). \end{equation*}$$

The condition that ${\mathbf{c}}_{k_\infty } = \infty$ for some $k_\infty$ guarantees that $A_{mn} = 0$ for $\lvert m - n \rvert > g+1$, so these matrix elements generate a bounded operator $A$ on $\ell ^2(\mathbb{N}_0)$ such that $A_{mn} = \langle e_m, A e_n \rangle$, where $(e_n)_{n=0}^\infty$ denotes the standard basis of $\ell ^2(\mathbb{N}_0)$. We say that $A \in \mathbb{A}({\mathbf{C}})$.

GMP matrices have the property that some of their resolvents are also GMP matrices; namely, for any $k \neq k_\infty$, $({\mathbf{c}}_k - A)^{-1} \in \mathbb{A}(f({\mathbf{C}}))$ where $f$ is the Möbius transform $f: z \mapsto ({\mathbf{c}}_k - z)^{-1}$ and $f({\mathbf{C}}) = (f({\mathbf{c}}_1), \dots , f({\mathbf{c}}_{g+1}))$.

Note that the special case $g=0$, ${\mathbf{C}} = (\infty )$ gives precisely a Jacobi matrix. A Jacobi matrix is said to be regular if it is obtained by this construction from a regular measure; analogously, we will call a GMP matrix regular if it is obtained from a regular measure. Just as regularity of a Jacobi matrix can be characterized in terms of its off-diagonal entries, we will show that regularity of a GMP matrix can be characterized in terms of its entries in the outermost nontrivial diagonal. We will also obtain a GMP matrix analog of the inequality 1.2.

The GMP matrix has an additional block matrix structure; in particular, for a GMP matrix with ${\mathbf{c}}_{k_\infty } = \infty$, on the outermost nonzero diagonal $m=n - g-1$, the only nonzero terms appear for $n = j(g+1)+k_\infty$, and those are strictly positive. Thus, we denote

$$\begin{equation} \beta _j = \langle e_{j(g+1)+k_\infty }, Ae_{(j+1)(g+1)+k_\infty } \rangle . {\tag{1.8}} \end{equation}$$

Theorem 1.9.

Fix a probability measure $\mu$ with $\operatorname {supp}\mu \subset \mathbb{R}$ and a sequence ${\mathbf{C}} = ({\mathbf{c}}_1,\dots , {\mathbf{c}}_{g+1})$ with ${\mathbf{c}}_k = \infty$. Then

$$\begin{equation} \limsup _{j \to \infty } \left( \prod _{\ell =1}^j \beta _\ell \right)^{1/ j } \le \lambda _{k_\infty }^{-1}. {\tag{1.9}} \end{equation}$$

Moreover, the measure $\mu$ is Stahl–Totik regular if and only if

$$\begin{equation} \lim _{j \to \infty } \left( \prod _{\ell =1}^j \beta _\ell \right)^{1/ j } = \lambda _{k_\infty }^{-1}. {\tag{1.10}} \end{equation}$$

The proof will use a relation between the sequence $\{\beta _j\}_{j=1}^\infty$ and the constants $\{\kappa _{j(g+1)+k_\infty }\}_{j=1}^\infty$. In particular, the characterization of regularity in Theorem 1.9 is made possible by the characterization of regularity in terms of the subsequence $\{ \kappa _{j(g+1)+k}\}_{j=1}^\infty$ for any single $k$. Theorem 1.9 also corroborates the perspective that regularity of the measure is the fundamental notion which manifests itself equally well in many different matrix representations.

Since the resolvents $({\mathbf{c}}_k - A)^{-1}$ are also GMP matrices and their measures are pushforwards of the original measure, they are also regular GMP matrices; in this sense, Theorem 1.9 provides $g+1$ criteria for regularity, one corresponding to each subsequence $n(j) = j(g+1)+{k}$, $1\leq k\leq g+1$.

As an application of this theory, we show that it provides a proof of a theorem for Jacobi matrices originally conjectured by Simon 23. Let $\mathsf{E} \subset \mathbb{R}$ be a compact finite gap set,

$$\begin{equation} \mathsf{E}=[\mathbf{b}_0,\mathbf{a}_0]\setminus \bigcup _{k=1}^g (\mathbf{a}_k,\mathbf{b}_k), {\tag{1.11}} \end{equation}$$

and denote by ${\mathcal{T}}_\mathsf{E}^+$ the set of almost periodic half-line Jacobi matrices with $\sigma _{\mathrm{ess}}(J) = \sigma _\mathrm{ac}(J) = \mathsf{E}$ 514. Through algebro-geometric techniques and the reflectionless property, this class of Jacobi matrices has been widely studied for their spectral properties and quasiperiodicity (see also 2631 for more general spectral sets). They also provide natural reference points for perturbations, which is our current interest. On bounded half-line Jacobi matrices $J$, we consider the metric

$$\begin{equation} d(J, \tilde{J}) = \sum _{k=1}^\infty e^{-k} ( \lvert a_k - \tilde{a}_k \rvert + \lvert b_k - \tilde{b}_k \rvert ). {\tag{1.12}} \end{equation}$$

On norm-bounded sets of Jacobi matrices, convergence in this metric corresponds to strong operator convergence. However, instead of distance to a fixed Jacobi matrix $\tilde{J}$, we will consider the distance to ${\mathcal{T}}_\mathsf{E}^+$,

$$\begin{equation*} d(J, {\mathcal{T}}_\mathsf{E}^+) = \inf _{\tilde{J} \in {\mathcal{T}}_\mathsf{E}^+} d(J, \tilde{J}) = \min _{\tilde{J} \in {\mathcal{T}}_\mathsf{E}^+} d(J, \tilde{J}). \end{equation*}$$

Denote by $S_+$ the right shift operator on $\ell ^2(\mathbb{N}_0)$, $S_+ e_n = e_{n+1}$. The condition $d( (S_+^*)^m J S_+^m, {\mathcal{T}}_\mathsf{E}^+) \to 0$ as $m\to \infty$ is called the Nevai condition. For $\mathsf{E} = [-2,2]$, this corresponds simply to the commonly considered condition $a_n \to 1$, $b_n \to 0$ as $n\to \infty$ 18. In general, as a consequence of 21, the Nevai condition implies regularity. The converse is false; however:

Theorem 1.10.

If $\mathsf{E} \subset \mathbb{R}$ is a compact finite gap set and $J$ is a regular Jacobi matrix with $\sigma _{\mathrm{ess}}(J) = \mathsf{E}$, then

$$\begin{equation} \lim _{N\to \infty }\frac{1}{N} \sum _{m=1}^N d( (S_+^*)^m J S_+^m, {\mathcal{T}}_\mathsf{E}^+) = 0. {\tag{1.13}} \end{equation}$$

The condition 1.13 is described as the Cesàro–Nevai condition; it was first studied by Golinskii–Khrushchev 15 in the OPUC setting with essential spectrum equal to $\partial \mathbb{D}$. Theorem 1.10 was conjectured by Simon 23 and proved in the special case when $\mathsf{E}$ is the spectrum of a periodic Jacobi matrix with all gaps open by using the periodic discriminant and techniques from Damanik–Killip–Simon 7 to reduce to a block Jacobi setting. It was then proved by Krüger 17 by very different methods under the additional assumption $\inf _{n} a_n > 0$. While this is a common assumption in the ergodic literature, regular Jacobi matrices do not always satisfy it: 22, Example 1.4 can easily be modified to give a regular Jacobi matrix with spectrum $[-2,2]$ and $\inf a_n = 0$. We prove Theorem 1.10 in full generality by applying Simon’s strategy and, instead of the periodic discriminant and techniques from 7, using the Ahlfors function, GMP matrices, and techniques of Yuditskii 32.

For the compact finite gap set $\mathsf{E} \subset \mathbb{R}$, among all analytic functions $\overline{\mathbb{C}} \setminus \mathsf{E} \to \mathbb{D}$ which vanish at $\infty$, the Ahlfors function $\Psi$ takes the largest value of $\operatorname {Re} (z \Psi (z))\rvert _{z=\infty }$. The Ahlfors function has precisely one zero in each gap, denoted ${\mathbf{c}}_k \in (\mathbf{a}_k, \mathbf{b}_k)$ for $1 \le k \le g$, a zero at ${\mathbf{c}}_{g+1} = \infty$, and no other zeros; see also 25, Chapter 8. In particular, for the finite gap set $\mathsf{E}$, this generates a particularly natural sequence of poles ${\mathbf{C}}_\mathsf{E} = ({\mathbf{c}}_1, \dots , {\mathbf{c}}_g, \infty )$.

The Ahlfors function was used by Yuditskii 32 to define a discriminant for finite gap sets,

$$\begin{equation} \Delta _\mathsf{E}(z)=\Psi (z)+\frac{1}{\Psi (z)}. {\tag{1.14}} \end{equation}$$

This function is not equal to the periodic discriminant, but it has some similar properties and it is available more generally (even when $\mathsf{E}$ is not a periodic spectrum). Namely, $\Delta _\mathsf{E}$ extends to a meromorphic function on $\overline{\mathbb{C}}$ and $(\Delta _\mathsf{E})^{-1}([-2,2]) = \mathsf{E}$. It was introduced by Yuditskii to solve the Killip–Simon problem for finite gap essential spectra. In fact, the discriminant is a rational function of the form

$$\begin{align} \Delta _\mathsf{E}(z) =\lambda _{g+1} z + d + \sum _{k=1}^g\frac{\lambda _k}{{\mathbf{c}}_k-z} {\tag{1.15}} \end{align}$$

for some $d \in \mathbb{R}$; in particular, we will explain that the constants $\lambda _j > 0$ in 1.15 match the general definition 1.6.

As a first glimpse of our proof of Theorem 1.10, we note that it uses the following chain of implications. Starting with a regular Jacobi matrix with essential spectrum $\mathsf{E}$, by a change of one Jacobi coefficient, which does not affect regularity, we can assume that ${\mathbf{c}}_k \notin \operatorname {supp}\mu$ (Lemma 7.1). Under this assumption, regularity of the Jacobi matrix implies regularity of the corresponding GMP matrix $A$ and the resolvents $({\mathbf{c}}_k - A)^{-1}$, $k=1,\dots , g$, which can be characterized in terms of their coefficients by Theorem 1.9. By properties of the Yuditskii discriminant, this further implies regularity of the block Jacobi matrix $\Delta _\mathsf{E}(A)$. Let us briefly recall that a block Jacobi matrix is of the form

$$\begin{equation} \mathbf{J} = \begin{bmatrix} {\mathfrak{w}}_0 & {\mathfrak{v}}_0 & & & & \\ {\mathfrak{v}}_0^* & {\mathfrak{w}}_{1}& {\mathfrak{v}}_1 & &&\\ & {\mathfrak{v}}_1^* & {\mathfrak{w}}_2 & {\mathfrak{v}}_2 & &\\ & & {\mathfrak{v}}_2^* & \ddots &\ddots & \\ & & & \ddots & & \end{bmatrix}, {\tag{1.16}} \end{equation}$$

where ${\mathfrak{v}}_j$ and ${\mathfrak{w}}_j$ are $d \times d$ matrices, ${\mathfrak{w}}_j = {\mathfrak{w}}_j^*$, and $\det {\mathfrak{v}}_j \ne 0$ for each $j$. Type 3 block Jacobi matrices have each ${\mathfrak{v}}_j$ lower triangular and positive on the diagonal. An extension of regularity to block Jacobi matrices was developed by Damanik–Pushnitski–Simon 8; in particular, $\mathbf{J}$ is regular for the set $[-2,2]$ if $\sigma _{\mathrm{ess}}(\mathbf{J}) = [-2,2]$ and

$$\begin{equation} \lim _{n\to \infty }\left(\prod _{j =1}^n \lvert \det {\mathfrak{v}}_j \rvert \right)^{1/n}=1. {\tag{1.17}} \end{equation}$$

This chain of arguments will result in Lemma 1.11:

Lemma 1.11.

Let $J$ be a regular Jacobi matrix, $\mathsf{E} = \sigma _{\mathrm{ess}}(J)$ a finite gap set, and ${\mathbf{C}}_\mathsf{E}$ the corresponding sequence of zeros of the Ahlfors function. Assuming ${\mathbf{c}}_k \notin \sigma (J)$ for $1\le k\le g$, denote by $A$ the GMP matrix corresponding to $J$ with respect to the sequence ${\mathbf{C}}_\mathsf{E}$. Then $\Delta _\mathsf{E}(A)$ is a regular type 3 block Jacobi matrix with essential spectrum $[-2,2]$.

With Lemma 1.11, it will follow that $\mathbf{J} = \Delta _\mathsf{E}(A)$ obeys a Cesàro–Nevai condition. That Cesàro–Nevai condition will imply 1.13 by a modification of arguments of 32. The strategy is clear: just as 32 uses a certain square-summability in terms of ${\mathfrak{v}}_j, {\mathfrak{w}}_j$ to prove finiteness of $\ell ^2$-norm of $\{ d((S_+^*)^m J S_+^m, {\mathcal{T}}_\mathsf{E}^+) \}_{m=0}^\infty$, we will use Cesàro decay in terms of ${\mathfrak{v}}_j, {\mathfrak{w}}_j$ to conclude the Cesàro decay 1.13. This can be expected due to a certain locality in the dependence between the terms of the series considered; this idea first appeared in 23 in the setting of periodic spectra with all gaps open. However, some care is needed, since the locality is only approximate in some steps; this is already visible in 1.12. Also, substantial modifications are needed throughout the proof due to the possibility of $\liminf \lVert {\mathfrak{v}}_j \rVert = 0$ (this cannot happen in the Killip–Simon class), which locally breaks some of the estimates. The fix is that this can only happen along a sparse subsequence, but the combination of a bad sparse subsequence and approximate locality means that we cannot simply ignore a bad subsequence once from the start; we must maintain it throughout the proof. A related issue arises with the Cesàro version of a Killip–Simon type functional. We will describe the necessary modifications to the detailed analysis in 32.

The rest of the paper will not exactly follow the order given in this section. In Section 2, we describe the behavior of our problem with respect to Möbius transformations, and we describe the distribution of zeros of the rational function $\tau _n$. In Section 3, we recall the structure of GMP matrices and relate their matrix coefficients to the quantities $\kappa _n$, and use this to provide a first statement about exponential growth of orthonormal rational functions on $\overline{\mathbb{C}} \setminus \overline{\mathbb{R}}$. In Section 4, we combine this with potential theoretic techniques to characterize limits of $\frac{1}{n} \log \lvert \tau _n \rvert$ as $n \to \infty$ and prove the universal lower bounds. In Section 5, we prove the results for ${\mathbf{C}}$-regularity and Stahl–Totik regularity. In Section 6 we describe a proof of Theorem 1.10.

2. Orthonormal rational functions and Möbius transformations

In Section 1, starting from the measure $\mu$ and sequence of poles ${\mathbf{C}}$, we defined a sequence $\{r_n\}_{n=0}^\infty$ and the orthonormal rational functions $\{ \tau _n \}_{n=0}^\infty$. In the next statement, we will denote these by $r_n(z; {\mathbf{C}})$ and $\tau _n(z;\mu ,{\mathbf{C}})$, in order to state precisely the invariance of the setup with respect to Möbius transformations.

Lemma 2.1.

If $f$ is a Möbius transformation which preserves $\overline{\mathbb{R}}$, then

$$\begin{equation} \tau _n(z;\mu ,{\mathbf{C}}) = \rho ^n \tau _n(f(z);f_* \mu , f({\mathbf{C}})), {\tag{2.1}} \end{equation}$$

where $f({\mathbf{C}}) = (f({\mathbf{c}}_1), \dots , f({\mathbf{c}}_{g+1}))$ and

$$\begin{equation*} \rho = \begin{cases} +1 & f \in \mathrm{PSL}(2,\mathbb{R}), \\ -1 & f \in \left( \mathrm{PSL}(2,\mathbb{R}) \rtimes \{ \mathrm{id}, z\mapsto -z \} \right) \setminus \mathrm{PSL}(2,\mathbb{R}). \end{cases} \end{equation*}$$

Proof.

Note that the sequence $\{r_n\}_{n=0}^\infty$ does not have this property: $r_n(z;{\mathbf{C}})$ is not equal to $\rho ^n r_n(f(z);f({\mathbf{C}}))$. However, if we denote

$$\begin{equation*} {\mathcal{L}}_n({\mathbf{C}}) = \mathrm{span} \{ r_\ell (\cdot ; {\mathbf{C}}) \mid 0 \le \ell \le n\}, \end{equation*}$$

then it suffices to have

$$\begin{equation} r_n(f(z);f({\mathbf{C}})) - c_n \rho ^n r_n(z;{\mathbf{C}}) \in {\mathcal{L}}_{n-1}({\mathbf{C}}) {\tag{2.2}} \end{equation}$$

for some constants $c_n > 0$. If 2.2 holds, then applying the Gram–Schmidt process to the sequences $\{ r_n(f(z);f({\mathbf{C}})) \}_{n=0}^\infty$ and $\{r_n(z;{\mathbf{C}})\}_{n=0}^\infty$ will give the same sequence of orthonormal functions, up to the sign change $\rho ^n$, which is precisely 2.1.

Note that if 2.1 holds for $f_1, f_2$, it holds for their composition, so it suffices to verify 2.2 for a set of generators of $\mathrm{PSL}(2,\mathbb{R}) \rtimes \{ \mathrm{id}, z\mapsto -z \}$. In particular, 2.2 is checked by straightforward calculations for affine transformations and for the inversion $f(z) = -1/z$, which implies the general statement since affine maps and inversion generate $\mathrm{PSL}(2,\mathbb{R}) \rtimes \{ \mathrm{id}, z\mapsto -z \}$.

■

Let us emphasize what Lemma 2.1 does and what it doesn’t do. Since the Möbius transformation acts on both the measure and the sequence of poles, Lemma 2.1 does not by itself prove Theorem 1.1. Lemma 2.1 can only say that if $\mu$ is Stahl–Totik regular, then $f_* \mu$ is $(f(\infty ))$-regular, which is not sufficient unless $f$ is affine. The proof of Theorem 1.1 will be more involved.

However, Lemma 2.1 provides a very useful conformal invariance for many of our proofs. This can be compared to choosing a convenient reference frame. Since potential theoretic notions such as Green functions are conformally invariant, our results will be invariant with respect to Möbius transformations. We will often use this invariance in the proofs to fix a convenient point at $\infty$.

Note that this will be possible even though some objects are not conformally invariant. Some of our results compare the sequences $\kappa _n$ with the $\lambda _k$, and although those objects are not preserved under conformal transformations, both sequences are affected in a compatible way so that the inequalities and equalities are preserved. Explicitly, fix $k$ and $n = j(g+1)+k$ and a Möbius transformation $f \in \mathrm{PSL}(2,\mathbb{R})$ (a reflection can be considered separately). Let us denote a local dilation factor $f'({\mathbf{c}}_k)= \lim _{z\to {\mathbf{c}}_k} \frac{r_k(z,{\mathbf{C}})}{r_k(f(z),f({\mathbf{C}}))}$. Then, we use Lemma 2.1 to compute

$$\begin{align*} \kappa _{n(j)}&=\lim _{z\to {\mathbf{c}}_k}\frac{\tau _n(z,\mu ,{\mathbf{C}})}{r_n(z,{\mathbf{C}})}= \lim _{z\to {\mathbf{c}}_k}\frac{\tau _n(f(z),f_*\mu ,f({\mathbf{C}}))}{r_n(z,{\mathbf{C}})} =\frac{\tilde{\kappa }_{n(j)}}{f'({\mathbf{c}}_k)^{j+1}}, \end{align*}$$

where $\tilde{\kappa }_{n(j)}$ is the leading coefficient $\tau _n(z,f_*\mu ,f({\mathbf{C}})) - \tilde{\kappa }_nr_n(z,f({\mathbf{c}}_k)) \in {\mathcal{L}}_{n-1}(f({\mathbf{C}}))$. If $\mathsf{E}$ is nonpolar, the Green function is conformally invariant so we find by another computation

$$\begin{align*} \log \tilde{\lambda }_k&\coloneq \lim _{w\to f({\mathbf{c}}_k)}\left( G_{f(\mathsf{E})}(w,f({\mathbf{c}}_k)) -\log \lvert r_k(w,f({\mathbf{C}})) \rvert \right)+\sum _{\substack{1\leq \ell \leq k\\\ell \ne k}}G_{f(\mathsf{E})}(f({\mathbf{c}}_k),f({\mathbf{c}}_\ell ))\\ &=\lim _{z\to {\mathbf{c}}_k}\left( G_{\mathsf{E}}(z,{\mathbf{c}}_k) - \log \lvert r_k(z,{\mathbf{C}}) \rvert + \log \left\lvert \frac{r_k(z,{\mathbf{C}})}{r_k(f(z),f({\mathbf{C}}))} \right\rvert \right)+\sum _{\substack{1\leq \ell \leq k\\\ell \ne k}}G_{\mathsf{E}}({\mathbf{c}}_k,{\mathbf{c}}_\ell )\\ &=\log \lambda _k+\log (f'({\mathbf{c}}_k)), \end{align*}$$

where we have used that $f\in \mathrm{PSL}(2,{\mathbb{R}})\implies f'>0$ on $\overline{\mathbb{R}}$. Thus, $\tilde{\lambda }_k=f'({\mathbf{c}}_k)\lambda _k$. If $\mathsf{E}$ is polar, then $f(\mathsf{E})$ is as well. From these calculations, it becomes elementary to verify that statements such as those in Theorems 1.3, 1.4 are conformally invariant.

Note that technical ingredients of the proof, such as polynomial factorizations, give a preferred role to $\infty$ so they break symmetry. For instance, we will often use the observation that the subspace ${\mathcal{L}}_n$ can be represented as

$$\begin{equation} {\mathcal{L}}_n = \left\{ \frac{P}{R_n} \mid P \in {\mathcal{P}}_n\right\} {\tag{2.3}} \end{equation}$$

for some suitable polynomial $R_n$ with factors which account for finite poles ${\mathbf{c}}_k \neq \infty$. We will use the representation 2.3 after placing a convenient point at $\infty$. This idea is already seen in the next proof.

Lemma 2.2.

All zeros of the rational function $\tau _{n}$ are simple and lie in $\overline{\mathbb{R}}$. Moreover, $n-g \le \deg \tau _n \le n$.

Let $n=j(g+1) + k$, $1\le k \le g+1$, and denote by $I$ the connected component of ${\mathbf{c}}_k$ in $\overline{\mathbb{R}} \setminus \operatorname {supp}\mu$. Then $\tau _{n}$ has no zeros in $I$ and at most one zero in any other connected component of $\overline{\mathbb{R}} \setminus \operatorname {supp}\mu$.

Proof.

Fix $1\leq k\leq g+1$ and without loss of generality, assume ${\mathbf{c}}_k = \infty$. Then, in the representations 2.3, we can notice that $R_{n-1} = R_n$. In particular, then $\tau _{n} \in {\mathcal{L}}_n \setminus {\mathcal{L}}_{n-1}$ implies the representation $\tau _{n(j)} = \frac{P_{n}}{R_n}$ for some polynomial $P_n$ of degree $n$.

Recall that $\tau _n$, $n=k+(j-1)(g+1)$ is the unique minimizer for the extremal problem 1.5. By complex conjugation symmetry, the minimizer is real. To proceed further, we study zeros of $P_n$ by using Markov correction terms.

We say that a rational function $M$ is an admissible Markov correction term if $M > 0$ a.e. on $\mathsf{E}$ and $M(z) P_n(z) \in {\mathcal{P}}_{n-1}$. In this case, using $\langle M \tau _n, \tau _n \rangle > 0$, we see that the function $g(\epsilon ) = \lVert \tau _n - \epsilon M \tau _n \rVert ^2$ obeys

$$\begin{equation*} g'(0) = - 2 \langle M \tau _n, \tau _n \rangle < 0. \end{equation*}$$

Thus, for small enough $\epsilon > 0$, the function

$$\begin{equation*} \tilde{\tau }_n = \tau _n - \epsilon M \tau _n \end{equation*}$$

obeys $\lVert \tilde{\tau }_n \rVert _{L^2(d\mu )} < \lVert \tau _n\rVert _{L^2(d\mu )}$. Since $\tilde{\tau }_n$ is of the form $\tilde{\tau }_n=\kappa _nz^{j+1}+h(z)$ for some $h(z)\in {\mathcal{L}}_{n-1}$ and in particular has the same leading coefficient as $\tau _n$, the function $\tilde{\tau }_n / \lVert \tilde{\tau }_n \rVert _{L^2(d\mu )} \in {\mathcal{L}}_n$ contradicts extremality of $\tau _n$. In other words, for the extremizer $\tau _n$, there cannot be any admissible Markov correction terms.

Assume that $P_n$ has a nonreal zero $w\in \overline{{\mathbb{C}}}\setminus \overline{{\mathbb{R}}}$. Then, since $\tau _n$ is real, $P_n(\overline{w})=0$, so the Markov correction term $M(z;w)= \frac{1}{(z-w)(z-\bar{w})}$ would be admissible, leading to contradiction.

Assume that $P_n$ has two zeros $x_1,x_2$ in the same connected component of $\overline{{\mathbb{R}}}\setminus \operatorname {supp}\mu$; then, the Markov correction term

$$\begin{equation*} M(z;x_1,x_2)= \frac{1}{(z-x_1)(z-x_2)} \end{equation*}$$

would be admissible, leading to contradiction.

There are no zeros of $P_n$ in $I$. Otherwise, if $x \in I$ was a zero, the Markov term

$$\begin{equation*} M(z,x) = \begin{cases} \frac{1}{z-x},& x < \inf \mathsf{E}, \\ \frac{1}{x-z}, & x > \sup \mathsf{E} \end{cases} \end{equation*}$$

would be admissible.

Finally, all zeros of $P_n$ are simple: otherwise, if $x_0 \in \mathbb{R}$ was a double zero, the Markov term

$$\begin{equation*} M(z,x_0) = \frac{1}{(z-x_0)^2} \end{equation*}$$

would be admissible.

The properties of zeros of $\tau _n$ follow from those of $P_n$. There may be cancellations in the representation $\tau _n = \frac{P_n}{R_n}$, but since $P_n$ has at most a simple zero at ${\mathbf{c}}_\ell$, the only possible cancellations are simple factors $(z-{\mathbf{c}}_\ell )$, $\ell \neq k$. Thus, $n- g \le \deg \tau _n \le n$.

■

The use of Markov correction factors is standard in the Chebyshev polynomial literature and is applied here with a modification for the $L^2$-extremal problem (in the $L^\infty$-setting, singularities in $M$ are treated with a separate argument near the singularity, which would not work here).

Corollary 2.3.

The measures $\nu _n$ are a precompact family with respect to weak convergence on $C(\overline{\mathbb{R}})$. Any accumulation point $\nu = \lim _{\ell \to \infty } \nu _{n_\ell }$ is a probability measure and $\operatorname {supp}\nu \subset \mathsf{E}$.

Proof.

By Lemma 2.2, $\nu _n(\overline{{\mathbb{R}}})\leq 1$, so precompactness follows by the Banach-Alaoglu theorem. If $\nu = \lim _{\ell \to \infty } \nu _{n_\ell }$, then since $1-\frac{g}{n_\ell }\leq \nu _{n_{\ell }}(\overline{{\mathbb{R}}})\leq 1$, $\nu (\overline{{\mathbb{R}}})=1$.

Let $(\mathbf{a},\mathbf{b})$ be a connected component of $\overline{\mathbb{R}} \setminus \mathsf{E}$. Let us prove that $\nu ((\mathbf{a},\mathbf{b})) = 0$. By Möbius invariance, it suffices to assume that $(a,b)$ is a bounded subset of $\mathbb{R}$.

Fix $r\in \mathbb{N}$. As $\operatorname {supp}\mu \setminus \mathsf{E}$ is a discrete set, we have

$$\begin{align*} \#\{ x\in \operatorname {supp}\mu : \mathbf{a}+1/r< x< \mathbf{b}-1/r \}=M<\infty . \end{align*}$$

So, by Lemma 2.2, $\nu _{n_\ell }((\mathbf{a}+1/r, \mathbf{b}-1/r))\leq \frac{2M+1}{n_\ell }$ and by the Portmanteau theorem and sending $r\to \infty$, $\nu ((\mathbf{a},\mathbf{b}))=0$ and $\operatorname {supp}\nu \subset \mathsf{E}$.

■

3. GMP matrices and exponential growth of orthonormal rational functions

In this section, we consider orthonormal rational functions through the framework of GMP matrices. We begin by recalling the structure of GMP matrices 32. The GMP matrix has a tridiagonal block matrix structure, with the beginnings of new blocks corresponding to occurrences of ${\mathbf{c}}_{k_\infty } = \infty$. Explicitly,

$$\begin{align*} A= \begin{bmatrix} \tilde{B}_{0} & \tilde{A}_{0} & & & & \\ \tilde{A}_{0}^* & B_{1}& A_{1}& &&\\ & A_{1}^*& B_{2}&A_{2}& &\\ & & A_2^* & \ddots &\ddots & \\ & & & \ddots & & \end{bmatrix}, \end{align*}$$

where $\tilde{B}_0$ is a $k_\infty \times k_\infty$ matrix, $\tilde{A}_0$ is a $k_\infty \times (g+1)$ matrix. For $j\ge 0$, $A_j, B_j$ are $(g+1) \times (g+1)$ matrices; while for $j\geq 1$ these appear in $A$ unmodified in the above, $\tilde{A}_0$ and $\tilde{B}_0$ are projections of $A_0$ and $B_0$ respectively. More precisely, let $X^-$ denote the upper triangular part of a matrix $X$ (excluding the diagonal) and $X^+$ the lower triangular part (including the diagonal). Then, indexing the entries of $A_j$, $B_j$, $j\geq 0$ from $0$ to $g$, we see they are of the form

$$\begin{align} A_j={\vec{p}}_j\vec{\delta }_{0}^{\,\intercal },\quad B_j=\hat{\mathbf{C}}+({\vec{q}}_j{\vec{p}}_j^{\,\intercal })^++({\vec{p}}_j{\vec{q}}_j^{\,\intercal })^-, {\tag{3.1}} \end{align}$$

where ${\vec{p}}_j,{\vec{q}}_j \in {\mathbb{R}}^{g+1}$ , with $({\vec{p}}_j)_0 >0$ and $\hat{\mathbf{C}}={\operatorname {diag}\{0,{\mathbf{c}}_{k_\infty +1},\dots ,{\mathbf{c}}_{g+1},{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{k_\infty -1}\}}$ (with the obvious modification if $k_\infty =1$ or $k_\infty =g+1$) and $\vec{\delta }_{0}$ denotes the standard first basis vector of ${\mathbb{R}}^{g+1}$. $\tilde{A}_0$ and $\tilde{B}_0$ are projections of $A_0$ and $B_0$,

$$\begin{align*} \tilde{A}_{0}= \Pi A_0 \quad \tilde{B}_{0}=\Pi B_0\Pi ^* \end{align*}$$

with $\Pi$ the block matrix $\Pi \coloneq \left[0_{k_\infty \times (g+1-k_\infty )}\vert I_{k_\infty \times k_\infty }\right]$. We will refer to ${\{{\vec{p}}_j,{\vec{q}}_j\}}_{j\geq 1}$ as the GMP coefficients of $A$. While the precise structure will not be essential throughout the paper, we point out two things. First on the outermost diagonal of $A$ in each block there is only one nonvanishing entry, given by $({\vec{p}}_j)_0$, which is positive and which is at a different position depending on the position of $\infty$ in the sequence ${\mathbf{C}}$. And secondly, in general as a self-adjoint matrix $B_j$ could depend on $(g+1)(g+2)/2$ parameters, but we see that in fact they only depend on $2(g+1)$. This is not that surprising due to their close relation to three-diagonal Jacobi matrices. A similar phenomenon also appears for their unitary analogs 6.

Remark 1.

For later reference, we provide an alternative point of view on the block structure of $A$. The structure provided above is chosen so that $c_{k_\infty }$ is at the first diagonal position of the $B$-blocks. Recall also that to these blocks we attached a column $\vec{p}$ (with positive first entry $({\vec{p}})_0 >0$) to the right and a row ${\vec{p}}_j^{\,\intercal }$ at the bottom. If, instead of viewing this as a block matrix structure with blocks of size $(g+1)\times (g+1)$, we view this structure as overlapping blocks of size $(g+2)\times (g+2)$ which overlap at the positions of $c_{k_\infty }$, then those would contain all nonvanishing entries of the GMP matrix (i.e., it would also include the vector ${\vec{p}}_j$). Moreover, the positive entries are exactly at the upper right and the lower left corner of the bigger block. Now placing the window of size $(g+1)\times (g+1)$ on the top of the bigger block corresponds to the structure presented above. We will encounter in Section 6 that in other settings it may be more natural to place the block at the lower corner, and in this case the $B$ blocks will have structure similar to 3.1.

Now the various notations for the off-diagonal blocks $A_j$, the vectors ${\vec{p}}_j$ which determine them, and the coefficients $\beta _j$ defined in 1.8 are related as

$$\begin{equation*} \beta _j = \langle e_{j(g+1)+k_\infty } , A e_{(j+1)(g+1)+k_\infty } \rangle = (A_{j})_{00} = ({\vec{p}}_{j})_0. \end{equation*}$$

The coefficients $\beta _j$ are a special case of the coefficients $\Lambda _n$ defined for $n=j(g+1)+k$, $1\leq k\leq g+1$ as

$$\begin{equation} \Lambda _n = \begin{cases} \langle e_{j(g+1)+k}, ({\mathbf{c}}_k-A)^{-1}e_{(j+1)(g+1)+k} \rangle & k\ne k_\infty , \\ \langle e_{j(g+1)+k}, Ae_{(j+1)(g+1)+k} \rangle & k=k_\infty . \end{cases} {\tag{3.2}} \end{equation}$$

Namely $\beta _j = \Lambda _{j(g+1) + k_\infty }$, and the coefficients $\Lambda _{j(g+1)+\ell }$ for $k \neq k_\infty$ instead occur as outermost diagonal coefficients for the GMP matrix $({\mathbf{c}}_k - A)^{-1}$. In our later applications to the discriminant of $A$, both the coefficients of $A$ and of its resolvents will appear, so we will work with $\Lambda _n$ throughout.

Next, we connect the coefficients 3.2 to the solutions of the $L^2$-extremal problem 1.5.

Lemma 3.1.

For all $n$,

$$\begin{align} \frac{\kappa _{n}}{\kappa _{n+g+1}}=\Lambda _n. {\tag{3.3}} \end{align}$$

Proof.

Let $n = j(g+1)+k$. By self-adjointness,

$$\begin{align*} \Lambda _n = \langle e_{n},r_k(A) e_{n+g+1}\rangle = \langle r_k \tau _{n} , \tau _{n+g+1}\rangle =\langle \kappa _{n}r_{n+g+1} +h ,\tau _{n+g+1} \rangle \end{align*}$$

for some $h\in {\mathcal{L}}_{n+g}$. By orthogonality, $\langle \tau _{n+g+1}, h\rangle = 0$, so $\langle \tau _{n+g+1},r_{n+g+1} \rangle =\frac{1}{\kappa _{n+g+1}}$ implies that

$$\begin{equation*} \Lambda _n = \langle \tau _{n+g+1}, \kappa _{n} r_{n+g+1} +h \rangle =\frac{\kappa _{n}}{\kappa _{n+g+1}}. \end{equation*}$$■

We now adapt to GMP matrices ideas from the theory of regularity for Jacobi matrices 22.

Lemma 3.2.

Let $A\in \mathbb{A}({\mathbf{C}})$. For all $j\geq 1$, $\|\vec{p}_j\|\leq \lVert A \rVert$.

Proof.

Fix $j \ge 1$ and denote $n= j (g+1) + k_\infty$. For any $\ell = 0,\dots ,g$,

$$\begin{equation*} (p_j)_\ell =\langle e_{n - g - 1+\ell },Ae_{n} \rangle = \int \tau _{n-g-1+\ell }(x) x\tau _n(x) d\mu (x). \end{equation*}$$

Since the vectors $\tau _{n-g-1+\ell }$ are orthonormal, by the Bessel inequality,

$$\begin{equation*} \lVert {\vec{p}}_j \rVert ^2 \le \int \lvert x \tau _n(x) \rvert ^2 d\mu (x) \le \lVert A \rVert ^2 \int \lvert \tau _n(x) \rvert ^2 d\mu (x) = \lVert A \rVert ^2 \end{equation*}$$

since $\lVert A \rVert = \sup _{x\in \operatorname {supp}\mu } \lvert x \rvert$.

■

Lemma 3.3.

For $z \in \mathbb{C} \setminus \mathbb{R}$,

$$\begin{equation} \liminf _{n\to \infty }\frac{1}{n}\log |\tau _{n}(z)| > 0. {\tag{3.4}} \end{equation}$$

Proof.

We adapt the proof of 22, Proposition 2.2. It suffices to prove 3.4 along the subsequences $n(j) = j(g+1) + k$, $j\to \infty$, for $1\le k \le g+1$. Moreover, due to $\overline{\mathbb{R}}$-preserving conformal invariance, it suffices to fix $k$ and prove

$$\begin{equation} \liminf _{j\to \infty }\frac{1}{n(j)}\log |\tau _{n(j)}(z)| > 0 {\tag{3.5}} \end{equation}$$

under the assumption that ${\mathbf{c}}_k = \infty$. This allows us to use the associated GMP matrix $A \in \mathbb{A}({\mathbf{C}})$.

Note that for any $m$, since $\{\tau _\ell \}_{\ell =0}^\infty$ is an orthonormal basis of $L^2(d\mu )$,

$$\begin{equation*} \sum _\ell A_{m\ell } \tau _\ell (z)= \sum _\ell \langle z \tau _m(z), \tau _\ell (z) \rangle \tau _\ell (z) = z \tau _m(z). \end{equation*}$$

This equality holds in $L^2(d\mu )$, but since all functions are rational, it also holds pointwise. Thus, if we fix $z \in \mathbb{C} \setminus \mathbb{R}$, the sequence $\vec{\varphi }= \{ \tau _\ell (z) \}_{\ell =0}^\infty$ is a formal eigensolution for $A$ at energy $z$, i.e. $(A-z)\vec{\varphi }=0$ componentwise. Since $A$ is represented as a block tridiagonal matrix, let us also write $\vec{\varphi }$ in a matching block form, as $\vec{\varphi }^\top = \begin{bmatrix} \vec{u}_0^\top & \vec{u}_1^\top & \vec{u}_2^\top & \dots \end{bmatrix}$ where

$$\begin{align*} &\vec{u}_{0}^\top =\begin{bmatrix} \tau _{0}(z)&\dots & \tau _{k-1}(z)\end{bmatrix},\quad \vec{u}_{j}^\top =\begin{bmatrix} \tau _{n(j-1)-1}(z)&\dots & \tau _{n(j)-1}(z)\end{bmatrix},\quad j\geq 1. \end{align*}$$

We also consider the projection of $\vec{\varphi }$ onto the first $j+1$ blocks,

$$\begin{equation*} \vec{\varphi }_{j}^\top = \begin{bmatrix} \vec{u}_0^\top &\dots & \vec{u}_j^\top &0 & \dots \end{bmatrix}, \end{equation*}$$

and compute $(A-z) \vec{\varphi }_j$. By the block tridiagonal structure of $A$, for $m< n(j-1)$ we have $\langle e_m,(A-z)\vec{\varphi }_{j}\rangle =0$. For $0\leq \ell \leq g$, we have

$$\begin{equation*} \langle e_{n(j-1)+\ell },(A-z)\vec{\varphi }\rangle -\langle e_{n(j-1)+\ell },(A-z)\vec{\varphi }_{j}\rangle =(p_j)_\ell \tau _{n(j)}(z) \end{equation*}$$

so that $\langle e_{n(j-1)+\ell },(A-z)\vec{\varphi }_{j}\rangle =-(p_j)_\ell \tau _{n(j)}(z)$. Moreover,

$$\begin{equation*} \langle e_{n(j)},(A-z)\vec{\varphi }_{j}\rangle =\langle e_{n(j)},A\vec{\varphi }_{j}\rangle =({\vec{p}}_j)^* u_j(z). \end{equation*}$$

For $m>n(j)$, we again have $\langle e_{m},(A-z)\vec{\varphi }_{j}\rangle =0$. In conclusion, $(A-z) \vec{\varphi }_j$ has only two nontrivial blocks,

$$\begin{align*} ((A-z)\vec{\varphi }_j )^\top =\begin{bmatrix} 0 & \dots & 0 &-({\vec{p}}_j\tau _{n(j)}(z))^\top & (({\vec{p}}_j)^* u_j)^\top & 0 & \dots \end{bmatrix}. \end{align*}$$

In particular, we can compute

$$\begin{equation} \langle \vec{\varphi }_j ,(A-z)\vec{\varphi }_j \rangle = -\vec{u}_j^*\tau _{n(j)}(z) {\vec{p}}_j. {\tag{3.6}} \end{equation}$$

Since $A$ is self-adjoint and $\vec{\varphi }_j \in \ell ^2(\mathbb{N}_0)$, by a standard consequence of the spectral theorem 29, Lemma 2.7.,

$$\begin{align*} \lvert \operatorname {Im} z \rvert \|\vec{\varphi }_{j}\|^2 \le |\langle \vec{\varphi }_{j},(A-z)\vec{\varphi }_{j}\rangle |. \end{align*}$$

Using 3.6 and the Cauchy–Schwarz inequality gives

$$\begin{align*} \lvert \operatorname {Im} z \rvert \sum _{m=0}^{j}\|\vec{u}_m \|^2 &\leq |\tau _{n(j)}(z)|\|{\vec{p}}_j\| \|\vec{u}_j\|. \end{align*}$$

By Lemma 3.2, with $C =\lvert \operatorname {Im} z \rvert / \lVert A \rVert$,

$$\begin{align} C\sum _{m=0}^{j}\|\vec{u}_m\|^2\leq |\tau _{n(j)}(z) |\|\vec{u}_j\|. {\tag{3.7}} \end{align}$$

Applying the AM-GM inequality to the right-hand side of 3.7 gives

$$\begin{equation*} |\tau _{n(j)}(z)|\|\vec{u}_j(z)\|\leq \frac{1}{2}\left( C \|\vec{u}_j(z)\|^2+C^{-1} |\tau _{n(j)}(z)|^2\right) \end{equation*}$$

which together with 3.7 implies

$$\begin{align} |\tau _{n(j)}(z)|^2\geq C^2 \sum _{m=0}^{j}\|\vec{u}_m \|^2. {\tag{3.8}} \end{align}$$

Since $|\tau _{n(j)}(z)|^2\leq \|\vec{u}_{j+1} \|^2$, this implies that

$$\begin{equation*} \sum _{m=0}^{j+1}\|\vec{u}_m \|^2\geq \left(1+C^2\right)\sum _{m=0}^{j}\|\vec{u}_m \|^2. \end{equation*}$$

Since $\lVert \vec{u}_0 \rVert \ge \lvert \tau _0(z) \rvert = 1$, this implies by induction that

$$\begin{align*} \sum _{m=0}^{j}\|\vec{u}_m \|^2\geq \left(1+ C^2\right)^j. \end{align*}$$

Combining this with 3.8 gives a lower bound on $\lvert \tau _{n(j)}(z) \rvert$ which implies 3.5.

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The estimates in the previous proof also lead to the following:

Corollary 3.4.

For any $z\in {\mathbb{C}}\setminus {\mathbb{R}}$, the quantities

$$\begin{equation*} \liminf _{j\to \infty }\frac{1}{j(g+1)+k} \log | \tau _{j(g+1)+k}(z) |, \qquad \limsup _{j\to \infty }\frac{1}{j(g+1)+k} \log | \tau _{j(g+1)+k}(z) | \end{equation*}$$

are independent of $k \in \{1,\dots , g+1\}$.

Proof.

Assume $j\ge 1$. For $k-g-1 \le \ell \le k-1$, the estimate 3.8 gives

$$\begin{equation*} |\tau _{j(g+1)+k}(z)|^2\geq C^2\|\vec{u}_j \|^2\geq C^2|\tau _{j(g+1)+\ell }(z)|^2 \end{equation*}$$

which implies

$$\begin{align} \liminf _{j\to \infty }\frac{1}{j(g+1)+k} \log | \tau _{j(g+1)+k}(z) |\geq \liminf _{j\to \infty }\frac{1}{j(g+1)+\ell } \log | \tau _{j(g+1)+\ell }(z) | {\tag{3.9}} \end{align}$$

and

$$\begin{align} \limsup _{j\to \infty }\frac{1}{j(g+1)+k} \log | \tau _{j(g+1)+k}(z) |\geq \limsup _{j\to \infty }\frac{1}{j(g+1)+\ell } \log | \tau _{j(g+1)+\ell }(z) |. {\tag{3.10}} \end{align}$$

Clearly, the right-hand sides don’t change if $\ell$ is shifted by $g+1$, so 3.9, 3.10 hold for all $k, \ell \in \{1,\dots ,g+1\}$ with $k \neq \ell$. By symmetry, since the roles of $k, \ell$ can be switched, we conclude that equality holds in 3.9, 3.10.

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4. Growth rates of orthonormal rational functions

In this section, we will combine the positivity 3.4 with potential theory techniques in order to study exponential growth rates of orthonormal rational functions. Our main conclusions will be conformally invariant, but our proofs will use potential theory arguments and objects such as the logarithmic potential of a finite measure $\nu$,

$$\begin{equation*} \Phi _\nu (z) = \int \log \lvert z -x \rvert d\nu (x), \end{equation*}$$

which is well defined when $\operatorname {supp}\nu$ does not contain $\infty$.

Theorem 4.1.

Fix $1\leq k\leq g+1$ and denote by $I$ the connected component of $\overline{{\mathbb{R}}}\setminus \operatorname {supp}\mu$ containing ${\mathbf{c}}_k$. Suppose there is a subsequence $n_\ell = j_\ell (g+1) + k$ such that $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }= \nu$ and $\frac{1}{n_\ell }\log \kappa _{n_\ell } \to \alpha \in {\mathbb{R}} \cup \{-\infty ,+\infty \}$ as $\ell \to \infty$. Then uniformly on compact subsets of $(\overline{{\mathbb{C}}} \setminus \overline{{\mathbb{R}}}) \cup (I \setminus \{{\mathbf{c}}_k\})$, we have

$$\begin{align*} h(z) \coloneq \lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)|. \end{align*}$$

The function $h$ is determined by $\nu$ and $\alpha$; in particular, if ${\mathbf{c}}_k = \infty$,

$$\begin{equation} h(z) =\alpha + \Phi _{\nu }(z)- \frac{1}{g+1} \sum _{\substack{m=1\\ m\ne k}}^{g+1}\log |{\mathbf{c}}_m-z|. {\tag{4.1}} \end{equation}$$

Moreover,

(a)

$\alpha = -\infty$ is impossible;

(b)

If $\alpha = +\infty$, the limit is $h =+ \infty$;

(c)

If $\alpha \in \mathbb{R}$, the limit $h$ extends to a positive harmonic function on $\overline{\mathbb{C}}\setminus ( \mathsf{E}\cup \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1} \})$ such that$$\begin{align} h(z) & = - \frac{1}{g+1} \log \lvert {\mathbf{c}}_m - z \rvert + O(1), \quad z \to {\mathbf{c}}_m \neq \infty , {\tag{4.2}}\\ h(z) & = \frac{1}{g+1} \log \lvert z \rvert + O(1), \quad z \to {\mathbf{c}}_m = \infty . {\tag{4.3}} \end{align}$$

Proof.

By using $\overline{\mathbb{R}}$-preserving conformal invariance, we can assume without loss of generality that ${\mathbf{c}}_k = \infty$. We will use the representation 2.3 of the subspace ${\mathcal{L}}_n$. For $n= j(g+1)+k$, counting degrees of the poles leads to

$$\begin{equation*} \tau _n = \frac{P_n}{R_n}, \qquad R_n(z) = \prod _{m=1}^{k-1}({\mathbf{c}}_m-z)\prod _{\substack{m=1\\m\ne k}}^{g+1}({\mathbf{c}}_m -z)^{j}, \end{equation*}$$

with $\deg P_n = n$. This may not be the minimal representation of $\tau _n$, but by the proof of Lemma 2.2, the only possible cancellations are simple factors $({\mathbf{c}}_m - z)$ for each $m \neq k$, so we get the minimal representation $\tau _n(z)=P(z)/Q(z)$ with

$$\begin{align*} P(z)=\kappa _n\prod _{w:\tau _n(w)=0}(z-w),\quad Q(z)= \prod _{\substack{m=1\\ m\ne k}}^{g+1}({\mathbf{c}}_m -z)^{j + \delta _{m,j}}, \end{align*}$$

where $\lvert \delta _{m,j} \rvert \le 1$ for each $j$. All that matters is that $\delta _{m,j} / j \to 0$ as $j\to \infty$. It will be useful to turn this rational function representation into a kind of Riesz representation,

$$\begin{equation} \log \lvert \tau _n(z) \rvert = \log \kappa _n + n \int \log \lvert x - z \rvert d\nu _n(x) - \sum _{\substack{1\le m \le g+1 \\ m \neq k}} (j+\delta _{m,j}) \log \lvert {\mathbf{c}}_m - z \rvert . {\tag{4.4}} \end{equation}$$

Since ${\mathbf{c}}_k = \infty$, note that $K = \overline{\mathbb{R}} \setminus I$ is a compact subset of $\mathbb{R}$. Denote $\Omega = \mathbb{C} \setminus K$. For any $z \in \Omega$, the map $x\mapsto \log |x-z|$ is continuous on $K$, so $\Phi _{\nu _{n_\ell }}(z) \to \Phi _{\nu }(z)$ as $\ell \to \infty$. In fact, convergence is uniform on compact subsets of $\Omega$: since $\operatorname {supp}(\nu _{n_{\ell }})\subset K$ and $\nu _{n_\ell }(K) \le 1$ for all $\ell$, the estimate

$$\begin{equation*} \log \left\lvert \frac{x-z_1}{x-z_2} \right\rvert \le \log \left( 1 + \frac{\lvert z_1 - z_2 \rvert }{\operatorname {dist}(z_2,K)} \right) \le \frac{\lvert z_1 - z_2 \rvert }{\operatorname {dist}(z_2, K)}, \qquad z_1,z_2 \in \Omega , \end{equation*}$$

implies uniform equicontinuity of the potentials $\Phi _{n_\ell }$ on compact subsets of $\Omega$, and the Arzelà–Ascoli theorem implies uniform convergence on compacts.

Note that b follows from 4.1. By Corollary 2.3, $\operatorname {supp}\nu \subset \mathsf{E}$ and $\Phi _{\nu }(z)$ is harmonic on ${\mathbb{C}}\setminus \mathsf{E}$, so the right hand side extends to a harmonic function on ${\mathbb{C}}\setminus ( \mathsf{E}\cup \{{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$ and we denote this extension also by $h$. By Lemma 3.3, $h$ is positive on ${\mathbb{C}}_+\cup {\mathbb{C}}_-$, so $\alpha \neq -\infty$; moreover, by the mean value property, $h$ is positive on ${\mathbb{C}}\setminus ( \mathsf{E}\cup \{{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$.

The remaining asymptotic properties follow from 4.1. Under the assumption ${\mathbf{c}}_k=\infty$, $\operatorname {supp}\nu$ is a compact subset of ${\mathbb{R}}$, and $\Phi _\nu (z)=\log |z|+O(1)$, $z\to \infty$. It then follows that $h(z)=\frac{1}{g+1}\log |z|+O(1)$ as $z\to \infty$. Of course, $h(z)=-\frac{1}{g+1}\log |z-{\mathbf{c}}_m |+O(1)$ near each ${\mathbf{c}}_m \neq {\mathbf{c}}_k$.

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Theorem 4.1 motivates interest in positive harmonic functions on $\overline{\mathbb{C}} \setminus (\mathsf{E} \cup \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$. If $\mathsf{E}$ is polar, by Myrberg’s theorem 2, Theorem 5.3.8, any such function is constant. If $\mathsf{E}$ is not polar, knowing the asymptotic behavior of $h$ at the poles, positivity of $h$ improves to the following lower bound on $h$. Lemma 4.2 reflects a standard minimality property of the Green function 11, Section VII.10.

Lemma 4.2.

Assume that $\mathsf{E}$ is a nonpolar closed subset of $\overline{{\mathbb{R}}}$. Let $h$ be a positive superharmonic function on $\overline{{\mathbb{C}}}\setminus ( \mathsf{E}\cup \{ {\mathbf{c}}_1,\dots , {\mathbf{c}}_{g+1} \} )$. Suppose $h(z)+\frac{1}{g+1}\log |z-{\mathbf{c}}_k|$ has an existent limit at ${\mathbf{c}}_k$ for each finite ${\mathbf{c}}_k$, and $h(z)-\frac{1}{g+1}\log |z|$ has an existent limit at $\infty$ if one of the ${\mathbf{c}}_k=\infty$. Then

$$\begin{align} h(z)\geq {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}) {\tag{4.5}} \end{align}$$

for $z\in \overline{\mathbb{C}}\setminus \mathsf{E}$. For $1\leq k\leq g+1$, define

$$\begin{align*} \alpha _k= \begin{cases} \lim _{z\to {\mathbf{c}}_k}(h(z)+\frac{1}{g+1}\log |z-{\mathbf{c}}_k|),&{\mathbf{c}}_k\ne \infty ,\\ \lim _{z\to \infty }(h(z)-\frac{1}{g+1}\log |z|),& {\mathbf{c}}_k=\infty . \end{cases} \end{align*}$$

Then

$$\begin{align} \alpha _k\geq \frac{\log \lambda _k}{g+1}. {\tag{4.6}} \end{align}$$

Proof.

We will use a stronger, q.e. version of the maximum principle 20, Thm 3.6.9. Define

$$\begin{equation*} \tilde{h}(z)\coloneq {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})-h(z), \end{equation*}$$

which is bounded at ${\mathbf{c}}_k$ for $1\leq k\leq g+1$ and so extends to a subharmonic function on $\overline{{\mathbb{C}}}\setminus \mathsf{E}$. Since ${\mathcal{G}}_\mathsf{E}$ vanishes q.e. on $\mathsf{E}$, we have for q.e. $t\in \mathsf{E}$,

$$\begin{equation*} \limsup _{z\to t} \tilde{h}(z)=-\liminf _{z\to t} h(z)\leq 0. \end{equation*}$$

Now we show $\tilde{h}$ is bounded above on $\overline{\mathbb{C}}\setminus \mathsf{E}$. Let $\mathcal{U}$ be a union of small neighborhoods containing the points ${\mathbf{c}}_k$ in $\overline{{\mathbb{C}}}\setminus \mathsf{E}$. By the definition of the Green function, ${\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})$ defines a harmonic and bounded function on $\overline{\mathbb{C}}\setminus (\mathsf{E}\cup \mathcal{U})$. That is, there exists $M$ such that for all $z\in \overline{\mathbb{C}}\setminus (\mathcal{U}\cup \mathsf{E})$ we have

$$\begin{align*} {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\leq M. \end{align*}$$

Since $h\geq 0$, it follows on $\overline{\mathbb{C}}\setminus (\mathcal{U}\cup \mathsf{E})$ that

$$\begin{align*} \tilde{h}(z)= {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})-h(z)\leq {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\leq M. \end{align*}$$

On the other hand, by properties of the Green functions we have

$$\begin{align*} \frac{\log \lambda _k}{g+1}= \begin{cases} \lim _{z\to {\mathbf{c}}_k}({\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})+\frac{1}{g+1}\log |z-{\mathbf{c}}_k|),&{\mathbf{c}}_k\ne \infty ,\\ \lim _{z\to \infty }({\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})-\frac{1}{g+1}\log |z|),& {\mathbf{c}}_k=\infty . \end{cases} \end{align*}$$

Then, by assumption, for $1\leq k\leq g+1$, $\tilde{h}(z)=\frac{\log \lambda _k}{g+1}-\alpha _k+o(1)$ as $z\to {\mathbf{c}}_k$ and, in particular, the difference is bounded in a small neighborhood of ${\mathbf{c}}_k$. Thus, $\tilde{h}$ is bounded above on $\overline{{\mathbb{C}}}\setminus \mathsf{E}$.

So, by the maximum principle $\tilde{h}\leq 0\implies {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\leq h(z)$ on $\overline{\mathbb{C}}\setminus \mathsf{E}$. Since $0\geq \lim _{z\to {\mathbf{c}}_k}\tilde{h}(z)=\frac{\log \lambda _k}{g+1}-\alpha _k$, we have 4.6.

■

Lemma 4.3.

Under the same assumptions as Lemma 4.2, the following are equivalent:

(i)

Equality in 4.6 for all $k$ with $1\leq k\leq g+1$

(ii)

Equality in 4.6 for a single $k$ with $1\leq k\leq g+1$

(iii)

Equality holds in 4.5

Proof.

i$\implies$ ii is trivial. Suppose then ii; with the notation of Lemma 4.2, by assumption, $\tilde{h}({\mathbf{c}}_k)=0$ and $\tilde{h}$ achieves a global maximum. By the maximum principle for subharmonic functions 20, Theorem 2.3.1, $\tilde{h}\equiv 0$ on $\overline{{\mathbb{C}}}\setminus \mathsf{E}$. Finally, if iii holds, then evaluating $\tilde{h}({\mathbf{c}}_k)$ for each $1\leq k\leq g+1$ yields i.

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We will now prove Theorems 1.2 and 1.3.

Proof of Theorem 1.2.

Using conformal invariance, we take ${\mathbf{c}}_k=\infty$. Fix $z\in \overline{\mathbb{C}}\setminus \overline{\mathbb{R}}$ and select a sequence $(n_\ell )_{\ell =1}^\infty$ such that

$$\begin{equation*} \liminf _{n\to \infty }\frac{1}{n}\log |\tau _n(z)|=\lim _{\ell \to \infty }\frac{1}{n_{\ell }}\log |\tau _{n_\ell }(z)|. \end{equation*}$$

By precompactness of the $(\nu _{n})$, we may pass to a further subsequence, which we denote again by $(n_\ell )_{\ell =1}^\infty$, so that $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }= \nu$ and $\frac{1}{n_\ell } \log \kappa _{n_\ell } \to \alpha$ for some $\nu$ and $\alpha$. Then for $h$ as in Theorem 4.1,

$$\begin{equation*} \lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)|=h(z), \end{equation*}$$

on ${\mathbb{C}}\setminus {\mathbb{R}}$. If $\alpha =+\infty$, then there is nothing to show. Suppose $\alpha <\infty$. If $\mathsf{E}$ is not polar we apply a of Theorem 4.1 to find $\alpha \in {\mathbb{R}}$, and we may use c of the same theorem and Lemma 4.2 to conclude.

If instead $\mathsf{E}$ is polar, by Myrberg’s theorem, $h$ is constant on ${\mathbb{C}}\setminus (\mathsf{E}\cup \{{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$. Computing the limit at ${\mathbf{c}}_k$ we see $h\equiv +\infty$. In particular, $\liminf _{n\to \infty }\frac{1}{n}\log |\tau _n(z)|=+\infty$ for $z\in {\mathbb{C}}\setminus {\mathbb{R}}$.

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Proof of Theorem 1.3.

Fix $1\leq k\leq g+1$ and assume again by conformal invariance that ${\mathbf{c}}_k = \infty$. Using precompactness of the measures $(\nu _{n})$, we find a subsequence $n_\ell =j_\ell (g+1)+k$ with

$$\begin{equation*} \lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell }=\liminf _{j\to \infty }\frac{1}{n(j)}\log \kappa _{n(j)}\eqcolon \alpha \end{equation*}$$

and $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$. If $\alpha =+\infty$, we are done. Suppose then $\alpha <\infty$, then we have by Theorem 4.1a, $\alpha \in {\mathbb{R}}$. Furthermore, if $\mathsf{E}$ is nonpolar, by c and Lemma 4.2, $h(z)\geq {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})$ on $\overline{\mathbb{C}}\setminus \mathsf{E}$. In particular, by the representation 4.1 we see that $\alpha =\lim _{z\to \infty }(h(z)-\frac{1}{g+1}\log |z|)$, and so 4.6 yields the desired inequality.

If instead $\mathsf{E}$ is polar, by Theorem 1.2, for each $z\in {\mathbb{C}}\setminus {\mathbb{R}}$,

$$\begin{equation*} h(z)=\lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }|\geq \liminf _{n\to \infty }\frac{1}{n}\log |\tau _n(z)|=+\infty , \end{equation*}$$

and so by Theorem 4.1b, $\alpha =+\infty$.

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5. Regularity

We will begin by proving a version of Theorem 1.4 for a fixed $k$.

Lemma 5.1.

Fix $k\in \{1,\dots ,g+1\}$. Along the subsequence $n(j) = j(g+1) +k$, the following are equivalent:

(i)

$\lim _{j\to \infty } \kappa _{n(j)}^{1/n(j)} = \lambda _k^{1 / (g+1)}$;

(ii)

For q.e. $z\in \mathsf{E}$, we have $\limsup _{j\to \infty } |\tau _{n(j)}(z)|^{1/n(j)} \leq 1$;

(iii)

For some $z\in {\mathbb{C}}_+$, $\limsup _{j\to \infty } |\tau _{n(j)}(z)|^{1/n(j)} \leq e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$;

(iv)

For all $z\in {\mathbb{C}}$, $\limsup _{j\to \infty } |\tau _{n(j)}(z)|^{1/n(j)} \leq e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$;

(v)

Uniformly on compact subsets of ${\mathbb{C}}\setminus {\mathbb{R}}$, $\lim _{j\to \infty } |\tau _{n(j)}(z)|^{1/n(j)}= e^{{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})}$.

Proof.

Using conformal invariance, we will assume throughout the proof that ${\mathbf{c}}_k=\infty$. First, suppose that $\mathsf{E}$ is polar. In this case ii is vacuous, and since ${\mathcal{G}}_\mathsf{E}\equiv +\infty$, iii and iv are trivially true. Since $\lambda _k = +\infty$, i follows from Theorem 1.3. As in the proof of Theorem 4.1, weak convergence of measures implies uniform on compacts convergence of their potentials. Thus, since $\nu _n$ are a precompact family, so are $\Phi _{\nu _n}$. Thus, the convergence $\lim _{j\to \infty } \frac{1}{n(j)} \log \kappa _{n(j)} = +\infty$ implies that $\lim _{j\to \infty } \frac{1}{n(j)} \log \lvert \tau _{n(j)}(z) \rvert = +\infty$ uniformly on compact subsets of $\mathbb{C} \setminus \mathbb{R}$, so v holds.

For the remainder of the proof, we will assume $\mathsf{E}$ is not polar. Moreover, we will repeatedly use the fact that if any subsequence of a sequence in a topological space has a further subsequence which converges to a limit, then the sequence itself converges to this limit. In particular, when concluding v, we apply this fact in the Fréchet space of harmonic functions on ${\mathbb{C}}\setminus {\mathbb{R}}$ with the topology of uniform convergence on compact sets.

iii$\implies$v: Given a subsequence of $n(j)=j(g+1)+k$, using precompactness of the measures $\nu _n$, we pass to a further subsequence $n_\ell =j_\ell (g+1)+k$ with $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$ and $\lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell }\eqcolon \alpha$, with $\alpha$ real or infinite. By Theorem 4.1, uniformly on compact subsets of ${\mathbb{C}}\setminus {\mathbb{R}}$,

$$\begin{equation*} h(z)=\lim _{\ell \to \infty } \frac{1}{n_\ell } \log |\tau _{n_{\ell }}(z)| \end{equation*}$$

with $h$ given by 4.1. Using the assumption, for some $z_0\in {\mathbb{C}}_+$, we have

$$\begin{align*} h(z_0)\leq \limsup _{j\to \infty }\frac{1}{n(j)}\log |\tau _{n(j)}(z_0)|<\infty . \end{align*}$$

So, by Theorem 4.1, $\alpha \in {\mathbb{R}}$ and $h$ has a harmonic extension to $\overline{\mathbb{C}}\setminus (\mathsf{E}\cup \{{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$. Furthermore, by Lemma 4.2, $h\geq {\mathcal{G}}_\mathsf{E}$. By assumption, we have the opposite inequality at $z_0\in {\mathbb{C}}_+$, and so, by the maximum principle for harmonic functions, $h={\mathcal{G}}_\mathsf{E}$ on ${\mathbb{C}}\setminus (\mathsf{E}\cup \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$, and in particular on ${\mathbb{C}}\setminus {\mathbb{R}}$. Thus, we have v.

v$\implies$iv: For $z\in \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\}$, ${\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})=+\infty$ and there is nothing to show. Fix $z\in {\mathbb{C}}\setminus \{{\mathbf{c}}_{1},\dots ,{\mathbf{c}}_{g+1}\}$ and let $n_\ell =j_\ell (g+1)+k$ be a subsequence with $\lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)|=\limsup _{j\to \infty }\frac{1}{n(j)}\log |\tau _{n(j)}(z)|$. By passing to a further subsequence, we may assume $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$, and $\lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell }\eqcolon \alpha$ where $\alpha$ is real or infinite. By the assumption, we have $h=\lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }|={\mathcal{G}}_\mathsf{E}$ on ${\mathbb{C}}\setminus {\mathbb{R}}$. So, by a and b, $\alpha \in {\mathbb{R}}$ and $h$ extends to a harmonic function on $\overline{{\mathbb{C}}}\setminus (\mathsf{E}\cup \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$. By the representation 4.1, we may extend $h$ subharmonically to $\overline{{\mathbb{C}}}\setminus \{{\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\}$. On this set, ${\mathcal{G}}_\mathsf{E}$ is also subharmonic, so, by the weak identity principle 20, Theorem 2.7.5, $h={\mathcal{G}}_\mathsf{E}$ on $\overline{\mathbb{C}}\setminus \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\}$. Thus, by the principle of descent 27, A.III, we have

$$\begin{align} \lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)|\leq h(z)={\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}) {\tag{5.1}} \end{align}$$

and iv follows.

v$\implies$ i: Given a subsequence of $n(j)=j(g+1)+k$, we use precompactness of the $\nu _n$ to pass to a further subsequence $n_\ell =j_\ell (g+1)+k$ with $\lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell }\eqcolon \alpha \in {\mathbb{R}}\cup \{ -\infty ,+\infty \}$ and $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$. Then in the notation of Theorem 4.1 and by assumption, for a $z\in {\mathbb{C}}\setminus {\mathbb{R}}$

$$\begin{align*} \lim _{\ell \to \infty }\log |\tau _{n_\ell }(z)|=h(z)={\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}). \end{align*}$$

So by Lemma 4.3, $\alpha =\frac{\log \lambda _k}{g+1}$. Thus, $\lambda _k^{1/(g+1)}$ is the only accumulation point of $\kappa _{n(j)}^{1/n(j)}$ in $\mathbb{R} \cup \{-\infty ,+\infty \}$ and we have i.

i$\implies$v: As before, we fix a subsequence of $n(j)=j(g+1)+k$ and use precompactness to pass to a further subsequence $n_\ell =j_\ell (g+1)+k$ with $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$. Then, by Theorem 4.1 and in the notation introduced there, uniformly on compact subsets of ${\mathbb{C}}\setminus {\mathbb{R}}$,

$$\begin{equation*} \lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)|=h(z), \end{equation*}$$

where $h$ is given by 4.1 with $\alpha =\frac{\log \lambda _k}{g+1}$. Thus, by Lemma 4.3ii, $h(z)={\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})$ on ${\mathbb{C}}\setminus {\mathbb{R}}$. Since the initial subsequence was arbitrary, we have v.

iv$\implies$ ii: Recalling that the Green function vanishes q.e. on $\mathsf{E}$, the claim follows.

ii$\implies$ v: Fixing a subsequence of $n(j)$, we again use precompactness to select a further subsequence $n_\ell =j_\ell (g+1)+k$ such that $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$ and $\lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell }\eqcolon \alpha$, $\alpha \in {\mathbb{R}}\cup \{ -\infty ,+\infty \}$. By the upper envelope theorem, there is a polar set $X_1\subset {\mathbb{C}}$ such that on ${\mathbb{C}}\setminus X_1$, $\limsup _{\ell \to \infty }\Phi _{\nu _{n_\ell }}=\Phi _{\nu }$. Now, we let $X_2\coloneq \{ t\in \mathsf{E}:\limsup _{n\to \infty }\frac{1}{n}\log |\tau _n(t)|>0\}$, which is polar by assumption, and $X_3\coloneq \{z\in {\mathbb{C}}: \Phi _\infty (z)=-\infty \}$, which is polar by 20, Theorem 3.5.1. Then, for a $t\in \mathsf{E}\setminus (X_1\cup X_2\cup X_3)$, we have

$$\begin{align*} \alpha \leq \limsup _{n\to \infty }\frac{1}{n}\log |\tau _{n}(t)|-\Phi _\nu (t)+\frac{1}{g+1}\sum _{\substack{m=1\\m\ne k}}^{g+1}\log |{\mathbf{c}}_m-t|<\infty . \end{align*}$$

So $\alpha \in {\mathbb{R}}$ by Theorem 4.1a. Thus, by c of the same theorem, uniformly on compact subsets of ${\mathbb{C}}\setminus {\mathbb{R}}$

$$\begin{equation*} h(z)=\lim _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(z)| \end{equation*}$$

and $h$ extends to a positive harmonic function on $\overline{{\mathbb{C}}}\setminus (\mathsf{E}\cup \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\})$ with logarithmic poles at each of the ${\mathbf{c}}_m$. So, $h- {\mathcal{G}}_\mathsf{E}$ extends to a harmonic function on $\overline{{\mathbb{C}}}\setminus \mathsf{E}$, and $h- {\mathcal{G}}_\mathsf{E}\geq 0$ there by Lemma 4.2. We now show that in fact $h={\mathcal{G}}_\mathsf{E}$ using the stronger, q.e. maximum principle.

We use the equality in 4.1 to extend $h$ to a subharmonic function on ${\mathbb{C}}\setminus \{ {\mathbf{c}}_1,\dots ,{\mathbf{c}}_{g+1}\}$. By the upper envelope theorem and the assumption again, for $t\in \mathsf{E}\setminus (X_1\cup X_2)$

$$\begin{equation*} h(t)=\limsup _{\ell \to \infty }\frac{1}{n_\ell }\log |\tau _{n_\ell }(t)|\leq 0. \end{equation*}$$

Then, for these $t$, since ${\mathcal{G}}_\mathsf{E}$ is positive, we have

$$\begin{equation*} \limsup _{\substack{z\to t\\ z\in {\mathbb{C}}\setminus \mathsf{E}}}\left( h(z)-{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\right)\leq \limsup _{\substack{z\to t\\ z\in {\mathbb{C}}\setminus \mathsf{E}}}h(z)\leq h(t)\leq 0 \end{equation*}$$

by upper semicontinuity. So, $\limsup _{\substack{z\to t\\ z\in {\mathbb{C}}\setminus \mathsf{E}}}\left( h(z)-{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\right)\leq 0$ for q.e. $t\in \mathsf{E}$.

Since $h$ is upper semicontinuous on the compact set $\mathsf{E}$, there is an $M$ so that $\sup _{t\in \mathsf{E}}h(t)\leq M$. As in the above, now for any $t\in \mathsf{E}$, we have

$$\begin{equation*} \limsup _{\substack{z\to t\\ z\in {\mathbb{C}}\setminus \mathsf{E}}}\left( h(z)-{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})\right)\leq \limsup _{\substack{z\to t\\ z\in {\mathbb{C}}\setminus \mathsf{E}}}h(z)\leq h(t)\leq M. \end{equation*}$$

So, there is a neighborhood $\mathcal{U}$ of $\mathsf{E}$ with $\sup _{z\in \mathcal{U}\cap (\overline{{\mathbb{C}}}\setminus \mathsf{E})}(h-{\mathcal{G}}_\mathsf{E})\leq M+1$. Since the difference is harmonic on $\overline{{\mathbb{C}}}\setminus \mathcal{U}$, we conclude that $\sup _{z\in \overline{\mathbb{C}}\setminus \mathsf{E}}(h(z)-{\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}))<\infty$. Thus, by the maximum principle and the reverse inequality, $h={\mathcal{G}}_\mathsf{E}$ on $\overline{{\mathbb{C}}}\setminus \mathsf{E}$. Since the first sequence was arbitrary, we have v.

Since the implication iv$\implies$iii is clear, we may conclude.

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We now put the subsequences together and use Corollary 3.4 to show that regular behavior occurs for one $k$ if and only if it happens for all.

Proof of Theorem 1.4.

Applying Lemma 5.1 for all $k$ implies equivalence of conditions (ii), (iv), (v), (vi), (vii) from Theorem 1.4. By Corollary 3.4, for some $z \in \mathbb{C}_+$, the condition

$$\begin{equation*} \limsup _{j\to \infty } \frac{1}{j(g+1)+k} \log \lvert \tau _{j(g+1)+k}(z) \rvert \le {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}}) \end{equation*}$$

holds for one value of $k$ if and only if it holds for all. Due to Lemma 5.1, this immediately implies equivalence of conditions (i) and (iii) from Theorem 1.4. It remains to prove equivalence of (ii), (iii).

ii$\implies$ iii: For $n \in \mathbb{N}$ and $1\le k \le g+1$, denote by $N(n,k)$ the integer such that $n+1 \le N(n,k) \le n+g+1$ and $N(n,k) - k$ is divisible by $g+1$. Then $N(n,k) / n \to 1$ as $n\to \infty$ so ii implies $\lim _{n\to \infty } \kappa _{N(n,k)}^{1/n} = \lambda _k^{1/(g+1)}$. Taking the product over $k=1,\dots ,g+1$ gives iii.

iii$\implies$ ii: Similarly to the above, Theorem 1.3 shows that for all $k$,

$$\begin{equation} \liminf _{n\to \infty } \kappa _{N(n,k)}^{1/n} \ge \lambda _k^{1/(g+1)}. {\tag{5.2}} \end{equation}$$

Thus, if ii was false, this would mean that for some $k=m$, $\limsup _{n\to \infty } \kappa _{N(n,m)}^{1/n} > \lambda _m^{1/(g+1)}$. Taking products over $k$, we would have

$$\begin{align*} \limsup _{n\to \infty } \left( \prod _{k=1}^{g+1} \kappa _{N(n,k)} \right)^{1/n} &\ge \limsup _{n\to \infty } \kappa _{N(n,m)}^{1/n} \liminf _{n\to \infty } \Biggl ( \prod _{\substack{1 \le k \le g+1 \\ k \neq m}} \kappa _{N(n,k)} \Biggr )^{1/n}\\ & > \left( \prod _{k=1}^{g+1} \lambda _k \right)^{1/(g+1)} \end{align*}$$

(the last step again uses 5.2 for all $k\neq m$). This would contradict iii, so the proof is complete.

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We now prove a seemingly special case of Corollary 1.7.

Proposition 5.2.

Assume that the sequence ${\mathbf{C}}$ contains $\infty$. Then $\mu$ is Stahl–Totik regular if and only if it is ${\mathbf{C}}$-regular.

Proof.

Let us assume that $\mu$ is ${\mathbf{C}}$-regular and let $p_n$ denote the orthonormal polynomial with respect to $\mu$. Fix $z\in \mathbb{C}$. Since $\infty$ is in ${\mathbf{C}}$, $p_n\in {\mathcal{L}}_{n(g+1)}$, so the orthonormal polynomials can be expressed on the basis of orthonormal rational functions as

$$\begin{align*} p_n(z)=\sum _{m=0}^{n(g+1)}c_m\tau _m(z),\quad \sum _{m=0}^{n(g+1)}|c_m|^2=1. \end{align*}$$

Thus, in particular, $|c_\ell |\leq 1$ and we get

$$\begin{equation} |p_n(z)|\leq (1+n(g+1)) \sup _{0\leq m\le n(g+1)}|\tau _m(z)|. {\tag{5.3}} \end{equation}$$

By Theorem 1.4, for q.e. $z\in \mathsf{E}$,

$$\begin{equation} \limsup _{\ell \to \infty } \frac{1}{\ell }\log \lvert \tau _\ell (z) \rvert \le 0. {\tag{5.4}} \end{equation}$$

Thus, for q.e. $z\in \mathsf{E}$, 5.3 implies

$$\begin{equation} \limsup _{n \to \infty }\frac{1}{n}\log |p_n(z)|\leq 0. {\tag{5.5}} \end{equation}$$

Thus, $\mu$ is Stahl–Totik regular.

Conversely, assume that $\mu$ is Stahl–Totik regular. For $n = j(g+1) + k$, the polynomial $R_n$ is a divisor of $R_{g+1}^{j+1}$, so we can write $\tau _n = \frac{P_n}{R_{g+1}^{j+1}}$ where $\deg P_n \le n+g$. For any $\epsilon > 0$ there exists a polynomial $Q_\epsilon$ such that $1 - \epsilon \le Q_\epsilon R_{g+1} \le 1+ \epsilon$ on $\mathsf{E}$. Thus,

$$\begin{equation} \lvert \tau _n(z) \rvert \le (1-\epsilon )^{-j-1} \lvert P_n(z) Q_\epsilon ^{j+1}(z) \rvert {\tag{5.6}} \end{equation}$$

and $\lVert P_n Q_\epsilon ^{j+1} \rVert \le (1+\epsilon )^{j+1}$ since $\tau _n$ is normalized. Since $P_n Q_\epsilon ^{j+1}$ is a polynomial of degree at most $n+g+ (j+1) \deg Q_\epsilon$, similarly to the above, representing it in the basis of polynomials shows

$$\begin{equation} \lvert P_n(z) Q_\epsilon ^{j+1}(z) \rvert \le (1+\epsilon )^{j+1}(n+g+1+(j+1)\deg Q_\epsilon ) \sup _{0 \le m \le n+g+(j+1)\deg Q_\epsilon } \lvert p_n(z) \rvert . {\tag{5.7}} \end{equation}$$

Since $n+g + (j+1) \deg Q_\epsilon = O(n)$ as $n\to \infty$, the supremum in 5.7 grows subexponentially whenever 5.5 holds. By 5.6, this implies

$$\begin{equation*} \limsup _{n\to \infty } \frac{1}{n} \log \lvert \tau _n(z) \rvert \le \frac{1}{g+1}\log \left(\frac{1+\epsilon }{1-\epsilon }\right). \end{equation*}$$

Since $\epsilon > 0$ is arbitrary, we conclude that 5.5 implies 5.4, so 5.4 holds q.e. on $\mathsf{E}$.

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From this seemingly special case, Theorem 1.6 and Corollary 1.7 follow easily:

Proof of Theorem 1.6.

By applying a conformal transformation, the special case shows that $\mu$ is ${\mathbf{C}}_1$-regular if and only if it is $({\mathbf{c}}_k)$-regular for any single ${\mathbf{c}}_k$ in ${\mathbf{C}}_1$. By applying this twice, we conclude that if ${\mathbf{C}}_1$, ${\mathbf{C}}_2$ have a common element, then $\mu$ is ${\mathbf{C}}_1$-regular if and only if it is ${\mathbf{C}}_2$-regular.

By applying that conclusion twice, we will finish the proof. Namely, for arbitrary ${\mathbf{C}}_1$, ${\mathbf{C}}_2$, choose a sequence ${\mathbf{C}}_3$ which has common elements with both ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$. Then $\mu$ is ${\mathbf{C}}_1$-regular if and only if it is ${\mathbf{C}}_3$-regular if and only if it is ${\mathbf{C}}_2$-regular.

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Proof of Corollary 1.7.

The result follows by taking ${\mathbf{C}}_2=(\infty )$ in Theorem 1.6.

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Proof of Theorem 1.1.

By Lemma 2.1, $f_* \mu$ is Stahl–Totik regular if and only if $\mu$ is $(f^{-1}(\infty ))$-regular, and by Corollary 1.7, this is equivalent to Stahl–Totik regularity of $\mu$.

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Proof of Theorem 1.8.

(a) We note that by Corollary 1.7 we may use Theorem 1.4. Fix $1\leq k\leq g+1$, and use conformal invariance to assume ${\mathbf{c}}_k=\infty$. Given a subsequence of $n(j)=j(g+1)+k$, we use precompactness to pass to a further subsequence $n_\ell =j_\ell (g+1)+k$ with $\operatorname *{w-lim}_{\ell \to \infty }\nu _{n_\ell }=\nu$. We write

$$\begin{align} {\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})=\Phi _{\rho _{\mathsf{E},\mathbf{C}}}(z)+\frac{1}{g+1}\log \lambda _k-\frac{1}{g+1}\sum _{\substack{m=1\\m\ne k}}^{g+1}\log |z-{\mathbf{c}}_m| {\tag{5.8}} \end{align}$$

which we will use to show $\Phi _{\nu }=\Phi _{\rho _{\mathsf{E},\mathbf{C}}}$. By ii, we may apply Theorem 4.1 with $\alpha =\frac{1}{g+1}\log \lambda _k$. Then, vii yields $h={\mathcal{G}}_\mathsf{E}$ off the real line, and thus the equality between the representations 4.1 and 5.8 gives $\Phi _{\nu }(z)=\Phi _{\rho _{\mathsf{E},\mathbf{C}}}(z)$ on ${\mathbb{C}}\setminus {\mathbb{R}}$. By the weak identity principle, this equality extends to ${\mathbb{C}}$. Applying the distributional Laplacian to both sides gives $\nu =\rho _{\mathsf{E},\mathbf{C}}$. Thus, $\operatorname *{w-lim}_{n\to \infty }\nu _n= \rho _{\mathsf{E},\mathbf{C}}$.

(b) The main ingredient is a variant of Schnol’s theorem; for any $n$, $\int \lvert \tau _n \rvert ^2 \ d\mu = 1$, so

$$\begin{equation*} \sum _{n=1}^\infty n^{-2} \int \lvert \tau _n \rvert ^2 \ d\mu < \infty . \end{equation*}$$

By Tonelli’s theorem, it follows that $\sum _{n=1}^\infty n^{-2} \lvert \tau _n \rvert ^2 < \infty$ $\mu$-a.e., so there exists a Borel set $B\subset \mathbb{C}$ with $\mu (\mathbb{C}\setminus B)=0$ such that

$$\begin{equation} \limsup _{n\to \infty } \frac{1}{n} \log \lvert \tau _n(z) \rvert \le 0, \qquad \forall z \in B. {\tag{5.9}} \end{equation}$$

Suppose $\mu$ is not regular. Then, by Theorem 1.4ii, there is a $1\leq k\leq g+1$ with

$$\begin{equation*} \limsup _{j\to \infty }\frac{1}{n(j)}\log \kappa _{n(j)} > \frac{1}{g+1}\log \lambda _k. \end{equation*}$$

Using conformal invariance, we may assume ${\mathbf{c}}_k=\infty$, and we can pass to a subsequence $n_\ell = j_\ell (g+1)+ k$ such that $\alpha \coloneq \lim _{\ell \to \infty }\frac{1}{n_\ell }\log \kappa _{n_\ell } > \frac{1}{g+1}\log \lambda _k$, where $\alpha \in {\mathbb{R}}\cup \{+\infty \}$ by Theorem 4.1a. Since $\operatorname *{w-lim}_{n\to \infty }\nu _n=\rho _{\mathsf{E},{\mathbf{C}}}$, by comparing 4.1 and 5.8, we have for $z\in {\mathbb{C}}\setminus {\mathbb{R}}$,

$$\begin{equation} \lim _{\ell \to \infty }\frac{1}{n_\ell }\log \lvert \tau _{n_\ell }(z) \rvert ={\mathcal{G}}_\mathsf{E}(z,{\mathbf{C}})+d, {\tag{5.10}} \end{equation}$$

where