Clark Measures for Rational Inner Functions
CCLARK MEASURES FOR RATIONAL INNER FUNCTIONS
KELLY BICKEL † , JOSEPH A. CIMA, AND ALAN A. SOLA Abstract.
We analyze the fine structure of Clark measures and Clark isometries asso-ciated with two-variable rational inner functions on the bidisk. In the degree ( n,
1) case,we give a complete description of supports and weights for both generic and exceptionalClark measures, characterize when the associated embedding operators are unitary, andgive a formula for those embedding operators. We also highlight connections betweenour results and both the structure of Agler decompositions and study of extreme pointsfor the set of positive pluriharmonic measures on 2-torus. Introduction
A bounded analytic function φ : D d → C is said to be inner if | φ ( ζ ) | = 1 for almostevery ζ ∈ T d , where D is the unit disk and T is the unit circle. In the one-variable case,each inner function ψ defines a class of positive Borel measures { σ α } α ∈ T on T that satisfy1 − | ψ ( z ) | | α − ψ ( z ) | = (cid:90) T − | z | | ζ − z | dσ α ( ζ ) , for z ∈ D . These measures have a number of important applications and properties; among otherresults, they are the spectral representing measures for rank 1 unitary perturbationsof certain compressed shift operators and via Alexandrov’s theorem, they disintegrateLebsegue measure, see [11, 16] for comprehensive introductions to this classical theory.Generalizations of these measures to the polydisk D d were recently studied by E. Doubtsovin [12]; other multivariate generalizations of Clark theory can be found in [3, 18]. In thispaper, we obtain precise information about both the two-variable Clark measures on thebidisk defined in [12] and associated isometries, in the setting of two-variable rationalinner functions.1.1. Notation and Setup.
To define Clark measures on the bidisk, we need some no-tation. Denote the Poisson kernel on D by P z ( ζ ) = P ( z, ζ ) := (1 − | z | )(1 − | z | ) | ζ − z | | ζ − z | , for z ∈ D , ζ ∈ T , and the Cauchy kernel for the bidisk by C w ( z ) = C ( z, w ) = 1(1 − z w )(1 − z w ) , for z ∈ D , w ∈ D . Date : January 5, 2021.2020
Mathematics Subject Classification.
Primary 28A25, 28A35; Secondary 32A08, 47A55.
Key words and phrases.
Clark measure, rational inner function, unitary embedding. † Research supported in part by National Science Foundation DMS grant a r X i v : . [ m a t h . F A ] J a n BICKEL, CIMA, AND SOLA
Recall that C acts as the reproducing kernel for the Hardy space H ( D ), which consistsof all analytic functions f : D → C satisfying the norm boundedness condition (cid:107) f (cid:107) H := sup The body of this paper begins with Section 2, which providessome information about the Clark measures σ α associated to a general RIF φ . Specifically,Theorem 2.1 gives a simple proof that σ α cannot possess any point-masses (a fact notedearlier in [28]), and the section also gives further information about the closed set(2) C α := { ζ ∈ T : ˜ p ( ζ ) = αp ( ζ ) } , which contains the support of σ α . BICKEL, CIMA, AND SOLA From Section 3 onward, we study RIFs φ = ˜ pp with deg φ = ( n, . In a sense, these arethe simplest two-variable RIFs, but the constructions from [8, 9] and the examples in ourSection 5 show that they can still be quite complicated. First, note that for these RIFs,(3) p ( z ) = p ( z ) + z p ( z )is a polynomial of degree at most ( n, 1) that does not vanish on D ,˜ p ( z ) := z ˜ p ( z ) + ˜ p ( z ) , where each ˜ p i ( z ) = z n p i (1 / ¯ z ) , the polynomials p , ˜ p share no common factors, and p has at most n distinct zeros on T . InSubsection 3.1, we recall some important properties about the model spaces and formulasassociated to such RIFs. For example, such RIFs possess a specific Agler decomposition or sums of squares formula of the form(4) p ( z ) p ( w ) − ˜ p ( z )˜ p ( w ) = (1 − z ¯ w ) n (cid:88) j =1 R j ( z ) R j ( w ) + (1 − z ¯ w ) Q ( z ) Q ( w )where R , . . . , R n , Q ∈ C [ z , z ], deg R j ≤ ( n − , , and deg Q ≤ ( n, . In Subsection 3.2, we study some preliminary objects, which are key in analyzing boththe Clark measures σ α and the isometric operators J α associated to φ . Those objects aredetailed in the following definition: Definition 1.1. Fix φ = ˜ p/p with deg φ = ( n, and α ∈ T . Define the following: • The points ( τ , λ ) , . . . , ( τ m , λ m ) are the zeros of p on T . Here, ≤ m ≤ n . • B α is the rational function B α ( z ) := ˜ p ( z ) − αp ( z ) αp ( z ) − ˜ p ( z ) , where any common factors of the numerator and denominator have been cancelled. • E α and L k are the sets in T defined by E α := { ( ζ, B α ( ζ )) : ζ ∈ T } and L k = { τ k } × T for k = 1 , . . . , m . • W α is the function on T defined by W α ( ζ ) := | p ( ζ ) | − | p ( ζ ) | | ˜ p ( ζ ) − αp ( ζ ) | . Lastly, we say α ∈ T is an exceptional value for φ if there is a k such that φ ∗ ( τ k , λ k ) = α and α ∈ T is a generic value for φ otherwise. Subsection 3.3 contains our first main result, the following complete characterizationof the Clark measures σ α associated to a given degree ( n, 1) RIF φ : Theorem 1.2. For α ∈ T , the Clark measure σ α satisfies (cid:90) T f ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T f ( τ k , ζ ) dm ( ζ ) for all f ∈ L ( σ α ) , where dν α = W α dm , the functions B α , W α are from Definition 1.1,and the constants c αk are nonzero (and positive) if and only if φ ∗ ( τ k , λ k ) = α . LARK MEASURES FOR RIFS 5 When they are non-zero, the constants c αk can be obtained from the formula in (17).We should note that although Clark measures can often be computed in the one-variablecase, very is known in the two-variable setting. Indeed, this theorem can be viewed asa significant generalization of Example 4.3 in [12], which described σ α for the specificdegree (1 , 1) RIF given in (1), and of Example 3 in [25], which includes the case where φ ( z ) = z b ( z ), for b a finite Blaschke product with b (0) ∈ R . Because of its length, theproof of Theorem 1.2 is broken into two pieces, Propositions 3.8 and 3.9.In Section 3.4, we prove our other main result, a formula for the isometry J α : K φ → L ( σ α ) and an exact characterization of when it is unitary: Theorem 1.3. Fix α ∈ T . i. For each each f ∈ K φ , ( J α f )( ζ ) = f ∗ ( ζ ) for σ α -a.e. ζ ∈ T . ii. J α : K φ → L ( σ α ) is unitary if and only if α is a generic value for φ . This result is in contrast to the one-variable case, where J α is always unitary. Here,by computing the non-tangential values of φ at its finite number of singularities, thistheorem allows us to easily identify whether a given J α is unitary. Part (i) is true in theone-variable setting and follows from a famous (and more general) result of Poltoratskiabout normalized Cauchy transforms, see [30] and [11, Theorem 10.3.1]. Thus, our resultcan be viewed as a partial two-variable analogue of Poltoratski’s result. Again, due tolength, we break the proof into two pieces, Propositions 3.11 and 3.12.Section 4 connects our ( n, 1) results to two related areas of study. First, in Theorem 4.1,we connect Theorem 1.2 to the theory of Agler decompositions and use our results about σ α and J α to establish formulas for some of the polynomials R j in (4). Then, we observethat this study of Clark measures can be put into a more general context. Specifically,let P ( T ) denote the set of Borel probability measures on T equipped with the topologyof weak- (cid:63) convergence and define P = { f ∈ Hol( D ) : (cid:60) f ( z ) > f (0 , 0) = 1 } , which is compact in the topology of uniform convergence on compact subsets of D . Let M : P → P ( T ) denote the map that takes each f ∈ P to the unique Borel probabilitymeasure µ f with f ( z ) = (cid:90) T P z ( ζ ) dµ f ( ζ ) for z ∈ D . Then both P and its image M ( P ) are compact convex sets and by the Krein-Milmantheorem, equal the closed, convex hull of their extreme points. It is also easy to showthat f is an extreme point of P if and only if µ f is an extreme point of M ( P ). In [32],Rudin posed the question“What are the extreme points of P (or equivalently, of M ( P ))?”While this question is still open, a number of interesting examples and related results(often in the n -variable situation) have been proved by Forelli [15], Knese [22], and Mc-Donald [25, 26, 27, 28]. As the Clark measures σ α are trivially in M ( P ) when φ (0) = 0,it makes sense to consider our investigations in the context of Rudin’s question and thesesubsequent results. In particular, the following is a quick corollary of Theorem 1.2 and atheorem from [22]: Corollary 1.4. Let φ = ˜ pp be a degree ( n, RIF with ˜ p (0 , 0) = 0 . If α ∈ T , then: BICKEL, CIMA, AND SOLA i. If α is an exceptional value for φ , then σ α is not an extreme point of M ( P ) . ii. If p is saturated, deg p = deg ˜ p , and α is generic for φ , then σ α is an extreme pointof M ( P ) . Part (i) of this corollary shows that in our setting, the more complicated Clark mea-sures cannot be extreme points of M ( P ). Meanwhile part (ii) coupled with our earliercharacterizations provide explicit formulas for some extreme points of M ( P ). We shouldmention that these appear somewhat related to the measures studied in [25, Example 3].In the last section, we use our results to compute the Clark measures and study the J α isometries associated to several degree ( n, 1) RIFs. First, in Example 5.1, we use ourresults to recover Example 4.3 in [12]. Then in Example 5.2, we apply our results toa more complicated degree (2 , 1) RIF with a single singularity and in Example 5.4, westudy a degree (3 , 1) RIF with two singularities. Finally, in Example 5.6, we investigatea degree (3 , 3) RIF φ with a singularity at (1 , φ ∗ (1 , 1) = − α = − φ , the operator J − is unitary.This demonstrates that, in its current form, Theorem 1.3(ii) does not extend to generalRIFs. 2. Clark measures for RIFs We begin with some remarks about Clark measures associated with general RIFs onthe bidisk, before focusing on the degree ( n, 1) case. First, the following result appearsto be known, see for instance [28, p.732] and [23], but here, we give a simple proof in theRIF case by adapting some of the arguments from the one-variable proof of [11, Theorem9.2.1]. Theorem 2.1. If φ is a nonconstant RIF on D and α ∈ T , then σ α does not possessany point masses.Proof. Without loss of generality, we will show that σ α does not possess a point mass at(1 , w = (0 , 0) yields(5) (cid:90) T − z ¯ ζ )(1 − z ¯ ζ ) dσ α ( ζ ) = 1 − φ ( z ) φ (0 , − ¯ αφ ( z ))(1 − αφ (0 , . Observe that θ ( z ) := φ ( z, z ) is a nonconstant finite Blaschke product. Then for 0 < r < z = r and multiply both sides of (5) by (1 − r ) to get(6) (cid:90) T (1 − r ) (1 − r ¯ ζ )(1 − r ¯ ζ ) dσ α ( ζ ) = (1 − r ) (1 − θ ( r ) θ (0))(1 − ¯ αθ ( r ))(1 − αθ (0)) . Observe that lim r (cid:37) (1 − r ) (1 − r ¯ ζ )(1 − r ¯ ζ ) = (cid:26) ζ = (1 , , σ α is a finite measure, the dominated convergence theorem implieslim r (cid:37) (cid:90) T (1 − r ) (1 − r ¯ ζ )(1 − r ¯ ζ ) dσ α ( ζ ) = σ α { (1 , } . LARK MEASURES FOR RIFS 7 As θ is a nonconstant finite Blaschke product, θ (1) exists and equals some λ ∈ T and itsderivative θ (cid:48) (1) exists and is nonzero, see [17, Lemma 7.5]. If λ (cid:54) = α , thenlim r (cid:37) (1 − r ) (1 − θ ( r ) θ (0))(1 − ¯ αθ ( r ))(1 − αθ (0)) = lim r (cid:37) (1 − r ) − λθ (0)(1 − ¯ αλ )(1 − αθ (0)) = 0 . If λ = α , thenlim r (cid:37) (1 − r ) (1 − θ ( r ) θ (0))(1 − ¯ αθ ( r ))(1 − αθ (0)) = 1 − αθ (0)¯ α − θ (0) · lim r (cid:37) − rθ (1) − θ ( r ) · lim r (cid:37) (1 − r )= α · θ (cid:48) (1) · . Equating the two sides in (6) implies that σ α { (1 , } = 0 . (cid:3) We can say a little bit more about Clark measures associated with RIFs. The papers[8, 9] include several results concerning boundary behavior of two-variable RIFs, and inparticular, the structure of their unimodular level sets. Recall from (2) that for α ∈ T and φ = ˜ p/p , C α = { ζ ∈ T : ˜ p ( ζ ) = αp ( ζ ) } . Then each C α satisifes C α = { ζ ∈ T : φ ∗ ( ζ ) = α } ∪ ( Z p ∩ T ), and as was shown in[9, Theorem 2.8], the components of C α can be locally parametrized using one-variableanalytic functions. Intuitively speaking, this implies that, for each α , the Clark measure σ α of any two-variable RIF has support contained in a one-dimensional subset of T . (Weshould mention that, technically speaking, the result in [9] was proved for φ = ˜ pp withdeg p = deg ˜ p , but that assumption does not appear to materially affect the conclusions.)As an aside, we also note that general pluriharmonic measures on T can have substantiallylarger, and even two-dimensional support, viz. [28].It should be noted that knowing that σ α is a Clark measure associated with some RIFand is supported on some set of the form { ζ ∈ T : ˜ p ( ζ ) = αp ( ζ ) } does not suffice todetermine that measure (or its associated RIF) uniquely. Indeed, one can exhibit (seeExample 5.2) two different RIFs φ = ˜ p p and φ = ˜ p p whose Clark measures are not mul-tiples of each other but are both supported on the same set { ζ ∈ T : ˜ p ( ζ ) = αp ( ζ ) } = { ζ ∈ T : ˜ p ( ζ ) = αp ( ζ ) } for some α ∈ T . Thus, in order to study Clark measuresand isometries for two-variable RIFs, we will need to perform a structural analysis of themeasures σ α that goes beyond determining their supports.3. Clark measures for Degree ( n, RIFs Throughout the rest of this paper, we let φ = ˜ p/p denote a fixed degree ( n, 1) rationalinner function for some n ≥ 1. Recall that we can decompose p as in (3). Then thepolynomials p , ˜ p share no common factors. Moreover, p has no zeros on D ∪ ( D × T ) ∪ ( T × D ) and at most n distinct zeros on T . See for example, Lemma 10.1, the proof ofCorollary 13.5, and Appendix C in [21]. Remark . For a given degree ( n, 1) RIF function φ and α ∈ T , recall the objects fromDefinition 1.1 and define the following additional objects: BICKEL, CIMA, AND SOLA • ν α is the measure on T defined by dν α := W α dm . • Q and R , . . . , R n are the polynomials given in Theorem 3.2. • b , . . . , b n are the rational functions in the disk algebra A ( D ) from Proposition 3.6. • For f ∈ K φ , h, g , . . . , g n are the H ( D ) functions given in Theorem 3.2.Finally, recall that α ∈ T is an exceptional value for φ if there is a k such that φ ∗ ( τ k , λ k ) = α and α ∈ T is a generic value for φ otherwise.3.1. Model Space Preliminaries. Clark measures are closely related to the model space K φ and so, we pause to record some known facts about K φ in the degree ( n, 1) case. Theorem 3.2. There are polynomials Q, R , . . . , R n ∈ C [ z , z ] such that deg R j ≤ ( n − , , deg Q ≤ ( n, and for z, w ∈ C , (7) p ( z ) p ( w ) − ˜ p ( z )˜ p ( w ) = (1 − z ¯ w ) n (cid:88) j =1 R j ( z ) R j ( w ) + (1 − z ¯ w ) Q ( z ) Q ( w ) . Furthermore, each R j and Q vanish at each ( τ k , λ k ) and a function f ∈ K φ if and only ifthere exist g , . . . , g n , h ∈ H ( D ) such that (8) f ( z ) = Qp ( z ) h ( z ) + n (cid:88) j =1 R j p ( z ) g j ( z ) for z ∈ D . Finally, if f ∈ K φ is written as in (8) , then (cid:107) f (cid:107) K φ = (cid:107) f (cid:107) H ( D ) = (cid:107) h (cid:107) H ( D ) + n (cid:88) j =1 (cid:107) g j (cid:107) H ( D ) . Proof. As this result is not new, we just give some intuition and references for the differentcomponents of the theorem. First, note that on H ( D ) , there are two shift operators, M z and M z , defined by ( M z i f )( z ) = z i f ( z ) for i = 1 , . Let S max1 be the maximal M z -invariant subspace of K φ , where M z is multiplication by z . Then, while not obvious, itis true that S min2 := K φ (cid:9) S max1 is invariant under M z , see [4, 5]. Let K , K denote thereproducing kernels of the two Hilbert spaces S max1 (cid:9) M z S max1 := H ( K ) and S min2 (cid:9) M z S min2 := H ( K ) , respectively. This yields the Agler decomposition1 − φ ( z ) φ ( w ) = (1 − z ¯ w ) K ( z, w ) + (1 − z ¯ w ) K ( z, w ) . Since φ is a degree ( n, 1) RIF, one can show that dim H ( K ) = 1 and dim H ( K ) = n . Let Q/p be an orthonormal basis for H ( K ) and R /p, . . . , R n /p be an orthonormal basis for H ( K ). One can show that the Q, R j are polynomials with deg Q ≤ ( n, 0) and deg R j ≤ ( n − , 1) and each Q, R j vanishes at each ( τ k , λ k ). Furthermore, K ( z, w ) = Q ( z ) Q ( w ) p ( z ) p ( w ) and K ( z, w ) = n (cid:88) j =1 R j ( z ) R j ( w ) p ( z ) p ( w ) , LARK MEASURES FOR RIFS 9 and substituting the formulas into the Agler decomposition and multiplying through bythe denominator gives (7). For the details, see for example [4, 6, 19] and the referenceswithin.The characterization of functions in K φ from (8) follows from the fact that the repro-ducing kernel k ( z, w ) of K φ satisfies k ( z, w ) = 11 − z ¯ w n (cid:88) j =1 R j ( z ) R j ( w ) p ( z ) p ( w ) + 11 − z ¯ w Q ( z ) Q ( w ) p ( z ) p ( w )and from standard properties of reproducing kernels. A proof of the formula for the normof functions in K φ can be found, for example, in Remark 2.3 in [7]. (cid:3) Support Sets and Consequences. In this subsection, we obtain some informationabout the objects from Definition 1.1 and Remark 3.1. First, recall that C α from (2)contains the support of σ α . This theorem gives additional information about C α and B α . Theorem 3.3. Let α ∈ T . i. Then B α is a finite Blaschke product. ii. Let α be a generic value of φ . Then deg B α = n and C α equals E α . iii. Let α be an exceptional value of φ and (after reordering if necessary), assume φ ∗ ( τ k , λ k ) = α for k = 1 , . . . , (cid:96) . Then deg B α = n − (cid:96) and C α is E α ∪ ( ∪ (cid:96)k =1 L k ) .Proof. Fix α ∈ T and observe that C α is the set of ζ ∈ T satisfying0 = ˜ p ( ζ ) − αp ( ζ ) = ζ (˜ p − αp )( ζ ) + (˜ p − αp )( ζ )= (˜ p − αp )( ζ ) ( ζ − /B α ( ζ )) . Too see that B α is a finite Blaschke product, observe that if r = αp − ˜ p , then ˜ r = ¯ α ˜ p − p . Since | r | = | ˜ r | on T , this implies | B α ( ζ ) | = 1 on T . As φ is nonconstant, | φ ( z , | = | ˜ p p ( z ) | < D and so, αp − ˜ p is nonvanishing on D . This implies B α is a finiteBlaschke product and the common zeros of its original numerator and denominator areexactly the zeros of ˜ p − αp in T . Denote those zeros by γ , . . . , γ (cid:96) . Then(9) ˜ p ( z ) − αp ( z ) = q ( z ) ( z − /B α ( z )) (cid:96) (cid:89) k =1 ( z − γ k ) , where q ∈ C [ z ] is the numerator of B α once common terms have been cancelled. Equation(9) shows that C α is the set of lines { γ k } × T for k = 1 , . . . , (cid:96) and E α . By Lemma 3.5(i),we can further assume that γ , . . . , γ (cid:96) are distinct.To establish deg B α , note that deg(˜ p − αp ) = n . This occurs because since | φ | < D , we have | p (0) | > | ˜ p (0) | and thus, the coefficient of the degree n term in ˜ p − αp isnonzero. Thus, deg B α = n − (cid:96) , its original degree minus the number of cancelled termsor equivalently, the number of lines of the form { γ } × T in C α .For (ii), let α be generic. By Lemma 3.4 below, C α cannot contain any lines of the form { γ } × T and so, our arguments give C α = E α and deg B α = n .For (iii), let α be exceptional and (after reordering if necessary), assume φ ∗ ( τ k , λ k ) = α for k = 1 , . . . , (cid:96) . Then Lemma 3.4 implies that L , . . . , L (cid:96) are exactly the lines of theform { γ } × T in C α . Then the above arguments imply C α = E α ∪ ( ∪ (cid:96)k =1 L k ) and deg B α = n − (cid:96). (cid:3) In the above proof, we used the following two lemmas. Lemma 3.4. For γ ∈ T , the set C α contains { γ } × T if and only if γ = τ k for some k and φ ∗ ( τ k , λ k ) = α. Proof. Observe that { γ } × T ⊆ C α if and only if φ ∗ ( γ, ζ ) ≡ α for all ζ ∈ T , except maybeat one ζ where p ( γ, ζ ) = 0 . Thus, for the forward direction, we can assume φ ∗ ( γ, ζ ) ≡ α (except maybe at one ζ ).As our assumptions imply deg ˜ p ( γ, ζ ) = 1, the polynomials p ( γ, · ) , ˜ p ( γ, · ) must share acommon factor with a zero on T , say ( z − β ) . This implies that p ( γ, β ) = ˜ p ( γ, β ) = 0.Thus, ( γ, β ) = ( τ k , λ k ) for some k. Set τ := ( τ k , λ k ) and write p ( z ) = n +1 (cid:88) j = M P j ( τ − z ) and ˜ p ( z ) = n +1 (cid:88) j = M Q j ( τ − z ) , where P j and Q j are homogeneous polynomials in z , z of degree j and M ≥ 1. Define λ = φ ∗ ( τ ). By [21, Propositions 14.3, 14.5], P M = λQ M and since ˜ p, p share no commonfactors, we can conclude that M = 1 and P contains a term cz with c (cid:54) = 0. Then α ≡ φ ( τ k , ζ ) = λc ( λ k − ζ ) c ( λ k − ζ ) = λ, so α = φ ∗ ( τ ) . Similarly, if α = φ ∗ ( τ ) for some τ = ( τ k , λ k ), the above equality andarguments imply that we also have α ≡ φ ∗ ( τ k , ζ ) for all ζ ∈ T \ { λ k } and so { τ k } × T isin C α . (cid:3) The following lemma describes finer behavior of B α and φ related to the singularities( τ k , λ k ) for k = 1 , . . . , m . Lemma 3.5. For α ∈ T , i. ˜ p − αp and ˜ p − αp do not possess repeated linear factors ( z − γ ) with γ ∈ T . ii. B α ( τ k ) = λ k for k = 1 , . . . , m . iii. If C α contains L k , then for all z ∈ D with z (cid:54) = λ k , ∂φ∂z ( τ k , z ) equals a fixednonzero constant C .Proof. By the factorization in (9), it is easy to see that for k ∈ N , ( z − γ ) k is a factorof ˜ p − αp if and only if it is a factor of ˜ p − αp . Then, by way of contradiction, assume˜ p − αp is divisible by ( z − γ ) . Then, γ = τ k for some k and for ζ ∈ T \ { λ k } , we have φ ( γ, ζ ) ≡ α and ∂φ∂z ( γ, ζ ) = p ∂ ˜ p∂z − ˜ p ∂p∂z p ( γ, ζ ) = ∂ ˜ p∂z − α ∂p∂z p ( γ, ζ ) = 0 . Fix any λ ∈ T with λ (cid:54)∈ { λ , . . . , λ m } . Then φ λ ( z ) := φ ( z , λ ) is a nonconstant finiteBlaschke product and so, φ (cid:48) λ ( γ ) (cid:54) = 0. See, for example, Lemma 7.5 in [17]. Since0 (cid:54) = φ (cid:48) λ ( γ ) = ∂φ∂z ( γ, λ ) = 0by the above argument, we obtain the requisite contradiction. LARK MEASURES FOR RIFS 11 For (ii), let τ := ( τ k , λ k ). First assume that φ ∗ ( τ ) (cid:54) = α . As ˜ p ( τ ) = p ( τ ) = 0, it followsby definition that τ ∈ C α . Thus, Theorem 3.3 implies λ k = B α ( τ k ) . Now if φ ∗ ( τ ) = α ,then as in the proof of Lemma 3.4, we can write:( p − α ˜ p )( z ) = n +1 (cid:88) j =2 ( P j − αQ j )( z − τ ) = ( τ k − z ) G ( z ) , where Q j , P j are homogeneous polynomials of degree j and G is a polynomial. Here, weused the fact that P = αQ and deg( p − α ˜ p ) ≤ ( n, G ( τ ) = 0 . Using the proof of Theorem 3.3, we know ( z − τ k ) divides ˜ p − αp and so,( τ k − z ) G ( z ) = (˜ p − αp )( z ) ( z − /B α ( z )) = r ( z )( τ k − z ) ( z − /B α ( z )) . for some r ∈ C [ z ]. By (i), r ( τ k ) (cid:54) = 0. Dividing through by ( τ k − z ) and plugging in τ implies λ k = 1 /B α ( τ k ) = B α ( τ k ) . For (iii), by the assumptions, there is some r ∈ C [ z ] with r ( τ k ) (cid:54) = 0 such that(˜ p − αp )( z ) = ( z − τ k ) r ( z )( z − /B α ( z )) . Then for z ∈ C \ { λ k } , we have φ ( τ k , z ) ≡ α and we can use (ii) to conclude ∂φ∂z ( τ k , z ) = ∂ ˜ p∂z − α ∂p∂z p ( τ k , z ) = r ( τ k )( z − λ k ) c ( z − λ k ) = r ( τ k ) c , for some c (cid:54) = 0. (cid:3) We can also use the set C α to refine our understanding of the polynomials R , . . . , R n , Q from Theorem 3.2 as follows: Proposition 3.6. R , . . . , R n , Q satisfy the following properties: i. For ζ ∈ T , | Q ( ζ ) | = | p ( ζ ) | − | p ( ζ ) | . ii. For z ∈ D , R j ( z ) = r j ( z ) (1 − B α ( z ) z ) + z Q ( z ) b j ( z ) , for some unique r j ∈ C [ z ] with deg r j ≤ ( n − and rational b j ∈ A ( D ) . Proof. Part (i) follows from some algebra; substituting z = w = ( ζ , z ) into (7) gives | p ( ζ , z ) | − | ˜ p ( ζ , z ) | = (1 − | z | ) | Q ( ζ ) | . The left-hand-side becomes | p ( ζ ) + z p ( ζ ) | − | z p ( ζ ) + p ( ζ ) | = (1 − | z | ) (cid:0) | p ( ζ ) | − | p ( ζ ) | (cid:1) , and dividing by (1 − | z | ) gives the desired formula.For (ii), recall that ˜ p ( z ) = αp ( z ) whenever z = 1 /B α ( z ). Substituting z = 1 /B α ( z )and w = 1 /B α ( w ) into (7) gives0 = (1 − z ¯ w ) n (cid:88) j =1 R j ( z , /B α ( z )) R j ( w , /B α ( w ))+(1 − /B α ( z )1 /B α ( w )) Q ( z ) Q ( w ) , for all z , w where B α ( z ) (cid:54) = 0. We can rewrite this as B α ( z ) B α ( w ) n (cid:88) j =1 R j ( z , /B α ( z )) R j ( w , /B α ( w )) = 1 − B α ( z ) B α ( w )1 − z ¯ w Q ( z ) Q ( w ) , for z , w ∈ D . Then the right-hand-side is the reproducing kernel of the one-variablemodel space ˆ K B α := H ( D ) (cid:9) B α H ( D ) (which is composed of rational functions in A ( D ), see [16, Chapter 5]) times the Q term. To finish the proof, fix R j and by Theorem 3.2,write R j ( z ) = r j ( z ) + z q j ( z ) , for r j , q j ∈ C [ z ]. Then standard properties of reproducing kernels imply that for some b j ∈ ˆ K B α , B α ( z ) r j ( z ) + q j ( z ) = b j ( z ) Q ( z )for z ∈ D . Solving this for q j and substituting back into the formula for R j yields thedesired result. (cid:3) In what follows, we will also require information about the weight function W (10) W ( x, ζ ) = W x ( ζ ) = | p ( ζ ) | − | p ( ζ ) | | ˜ p ( ζ ) − xp ( ζ ) | for ( x, ζ ) ∈ T . Lemma 3.7. The function W from (10) satisfies the following properties: i. W is well defined and continuous on T , except possibly at the finite set of points ( α, τ k ) where α is exceptional for φ and L k ⊆ C α . ii. For α ∈ T , W α has at most a finite number of discontinuities on T , all of whichare removable. So, W α equals a bounded, continuous function m -a.e. on T . Proof. For (i), it follows from the proof of Theorem 3.3 that ˜ p ( ζ ) − xp ( ζ ) only vanishesat some ( ζ, x ) ∈ T if x is exceptional, ζ = τ k for some k , and L k ⊆ C x .For (ii), first observe that if α is generic, then (i) implies W α is continuous, andhence bounded, on T . If α is exceptional, after reordering the singularities of φ , assume L , . . . , L (cid:96) are exactly the lines in C α . Then by the proof of Theorem 3.3,(˜ p − αp )( z ) = q ( z ) (cid:96) (cid:89) k =1 ( z − τ k ) , for some q ∈ C [ z ] that is non-vanishing on T . Similarly, Theorem 3.2 and Proposition 3.6imply that for ζ ∈ T , | p ( ζ ) | − | p ( ζ ) | = | Q ( ζ ) | = (cid:96) (cid:89) k =1 | ζ − τ k | | r ( ζ ) | , for some r ∈ C [ z ]. Thus, for ζ (cid:54) = τ , . . . τ (cid:96) , W α ( ζ ) = (cid:12)(cid:12)(cid:12) rq ( ζ ) (cid:12)(cid:12)(cid:12) . This shows that the only possible singularities W α could have on T are at τ , . . . , τ (cid:96) andany such singularities must be removable. (cid:3) Clark Measure Formulas. Recall that the Clark measures associated to φ arecharacterized via Theorem 1.2. We split the proof into two propositions; the first considersgeneric values for φ and the second considers exceptional values for φ . Proposition 3.8. Let α ∈ T be a generic value for φ . Then for all f ∈ L ( σ α ) , (11) (cid:90) T f ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) dν α ( ζ ) , where dν α = W α dm and B α , W α are from Definition 1.1. LARK MEASURES FOR RIFS 13 Proof. For each fixed z ∈ D , define the function ψ αz ( z ) := α + φ ( z , z ) α − φ ( z , z ) for z ∈ D . Recall that φ has no singularities on T × D . Furthermore, since α is generic, if φ ( ζ , z ) = α for ζ ∈ T , then z = 1 /B α ( ζ ) ∈ T . Thus, ψ αz ∈ A ( D ) and for all z ∈ D , (cid:60) (cid:0) ψ αz ( z ) (cid:1) = (cid:90) T − | z | | z − ζ | (cid:60) (cid:0) ψ αz ( ζ ) (cid:1) dm ( ζ ) . If z ∈ D and ζ ∈ T , then the computation in the proof of Proposition 3.6(i) gives | p ( ζ, z ) | − | ˜ p ( ζ, z ) | = (1 − | z | ) (cid:0) | p ( ζ ) | − | p ( ζ ) | (cid:1) , which one can use to obtain (cid:60) (cid:0) ψ αz ( ζ ) (cid:1) = 1 − | φ ( ζ, z ) | | α − φ ( ζ, z ) | = | p ( ζ, z ) | − | ˜ p ( ζ, z ) | | αp ( ζ, z ) − ˜ p ( ζ, z ) | = 1 − | z | | z − B α ( ζ ) | | p ( ζ ) | − | p ( ζ ) | | αp ( ζ ) − ˜ p ( ζ ) | . This implies that (cid:60) (cid:0) ψ αz ( z ) (cid:1) = (cid:90) T − | z | | z − ζ | − | z | | z − B α ( ζ ) | | p ( ζ ) | − | p ( ζ ) | | αp ( ζ ) − ˜ p ( ζ ) | dm ( ζ )= (cid:90) T P z ( ζ, B α ( ζ )) W α ( ζ ) dm ( ζ ) . Thus, by the definition of σ α , we can conclude that (cid:90) T P z ( ζ ) dσ α ( ζ ) = (cid:90) T P z ( ζ, B α ( ζ )) dν α ( ζ ) . Since finite linear combinations of Poisson functions P z are dense in C ( T ), this formulaextends to all functions in C ( T ). Since W α is bounded by Proposition 3.7, the formulaextends to f ∈ L ( σ α ) by Lemma 3.10. (cid:3) Let us now consider the exceptional α values for φ . Proposition 3.9. Let α ∈ T be an exceptional value for φ and (after re-ordering thezeros of p on T if necessary and applying Theorem 3.3), assume C α = E α ∪ ( ∪ (cid:96)k =1 L k ) .Then for all f ∈ L ( σ α ) , (12) (cid:90) T f ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) dν α ( ζ ) + (cid:96) (cid:88) k =1 c αk (cid:90) T f ( τ k , ζ ) dm ( ζ ) , where dν α = W α dm , B α , W α are from Definition 1.1, and c α , . . . , c α(cid:96) > .Proof. Recall that σ α is supported on C α = E α ∪ ( ∪ (cid:96)k =1 L k ). Since the L k are disjoint and E α ∩ L k = { ( τ k , B α ( τ k )) } , which has σ α measure 0, we only need to show (cid:90) T f ( ζ ) χ E α ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) W α ( ζ ) dm ( ζ )(13) (cid:90) T f ( ζ ) χ L k ( ζ ) dσ α ( ζ ) = c αk (cid:90) T f ( τ k , ζ ) dm ( ζ )(14) for f ∈ L ( σ α ) and k = 1 , . . . , (cid:96) , where χ E denotes the characteristic function of aset E ⊆ T . Then Lemma 3.7 implies W α is bounded on T (with at most (cid:96) removablesingularities) and by Lemma 3.10, we need only establish (13) and (14) for f ∈ C ( T ).To ease notation, throughout this proof, for any F defined on T , we will use F α todenote F α ( ζ ) := F ( ζ, B α ( ζ )) and F k to denote F k ( ζ ) := F ( τ k , ζ ) . Part 1. We first prove (13). To that end, fix a small (cid:15) > S (cid:15) = { ζ ∈ T : min k | ζ − τ k | < (cid:15) } and define S (cid:15)/ analogously. By Lemma 3.7 and the definition of B α , we can find a smallarc A α ⊆ T centered at α such that both W ( x, ζ ) and B ( x, ζ ) := B x ( ζ ) are uniformlycontinuous on A α × ( T \ S (cid:15)/ ) . Choose ( α n ) ⊆ T such that each α n is generic and ( α n ) → α .Then by [12, Corollary 2.2], ( σ α n ) converges weak- (cid:63) to σ α .To exploit that fact, let Ψ (cid:15) be a continuous function on T such thatΨ (cid:15) ≡ T \ S (cid:15) , Ψ (cid:15) ≡ S (cid:15)/ , ≤ Ψ (cid:15) ≤ S (cid:15) \ S (cid:15)/ . Fix f ∈ C ( T ). By our assumptions and by Proposition 3.8, (cid:90) T f ( ζ )Ψ (cid:15) ( ζ ) dσ α ( ζ ) = lim n →∞ (cid:90) T f ( ζ )Ψ (cid:15) ( ζ ) dσ α n ( ζ )= lim n →∞ (cid:90) T f α n ( ζ ) Ψ (cid:15) ( ζ ) dν α n ( ζ ) = (cid:90) T f α ( ζ ) Ψ (cid:15) ( ζ ) dν α ( ζ ) , (15)where the last equality follows because (cid:90) T | f α n ( ζ ) W α n ( ζ ) − f α ( ζ ) W α ( ζ ) | Ψ (cid:15) ( ζ ) dm ( ζ ) ≤ sup ζ ∈ T \ S (cid:15)/ | ( f α n W α n − f α W α ) ( ζ ) | → , as n → ∞ because f x ( ζ ) W ( x, ζ ) is uniformly continuous on A α × ( T \ S (cid:15)/ ) . Furthermore,observe that since Ψ (cid:15) ( ζ ) ≡ L k , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T f ( ζ ) χ E α ( ζ ) dσ α ( ζ ) − (cid:90) T f ( ζ )Ψ (cid:15) ( ζ ) dσ α ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) T | f ( ζ ) | (1 − Ψ (cid:15) ( ζ )) χ E α ( ζ ) dσ α ( ζ ) ≤ (cid:107) f (cid:107) L ∞ ( T ) σ α (( S (cid:15) × T ) ∩ E α ) . Here, ( S (cid:15) × T ) ∩ E α is the intersection of the curve E α with thin strips in T , see Figure1. Letting (cid:15) (cid:38) (cid:15) (cid:38) σ α (( S (cid:15) × T ) ∩ E α ) = σ α (cid:16) ∪ (cid:96)k =1 ( τ k , B α ( τ k )) (cid:17) = 0 . This in turn implies that (cid:90) T f ( ζ ) χ E α ( ζ ) dσ α ( ζ ) = lim (cid:15) (cid:38) (cid:90) T f ( ζ )Ψ (cid:15) ( ζ ) dσ α ( ζ ) . As f α and W α are both bounded, we can also conclude thatlim (cid:15) (cid:38) (cid:90) T f α ( ζ )Ψ (cid:15) ( ζ ) dν α ( ζ ) = (cid:90) T f α ( ζ ) dν α ( ζ ) . Combining these last two equalities with (15) yields (13). LARK MEASURES FOR RIFS 15 - - - - - - Figure 1. This shows the set ( S (cid:15) × T ) ∩ E α graphed on [ − π, π ] . E α isthe red curve, τ , τ , τ are the black tics on the horizontal axis, and S (cid:15) × T is the union of thin gray strips. Part 2. Now we prove (14). We will show that for z ∈ D , (14) holds for P z . Sincelinear combinations of these are dense in C ( T ), the result will follow. To that end, fix0 < r < 1. Then the definition of σ α gives(16) (cid:90) T P ( rτ k ,z ) ( ζ ) dσ α ( ζ ) = (cid:60) (cid:18) α + φ ( rτ k , z ) α − φ ( rτ k , z ) (cid:19) . We will multiply both sides by (1 − r ) and let r (cid:37) 1. First, observe that for ζ ∈ T ,lim r (cid:37) (1 − r ) P ( rτ k ,z ) ( ζ ) = (cid:26) ζ (cid:54) = τ k , P z ( ζ ) if ζ = τ k . Then by the dominated convergence theorem,lim r (cid:37) (cid:90) T (1 − r ) P ( rτ k ,z ) ( ζ ) dσ α ( ζ ) = (cid:90) T P z ( ζ ) χ L k ( ζ ) dσ α ( ζ ) . Observe that L k ⊆ C α actually implies that φ ( τ k , z ) = α for all z ∈ D . Furthermore,since φ is analytic at each ( τ k , z ), we havelim z → τ k φ ( z , z ) = α and lim z → τ k φ ( z , z ) − αz − τ k = ∂φ∂z ( τ k , z ) := C (cid:54) = 0 , by Lemma 3.5. Then Carath´eodory’s theorem, see (VI-3) in [34], implieslim r (cid:37) − | φ ( rτ k , z ) | − r = Cτ k ¯ α = | C | and solim r (cid:37) (cid:60) (cid:18) (1 − r )( α + φ ( rτ k , z )) α − φ ( rτ k , z ) (cid:19) = lim r (cid:37) (1 − r ) 1 − | φ ( rτ k , z ) | | α − φ ( rτ k , z ) | = lim r (cid:37) (cid:12)(cid:12)(cid:12)(cid:12) τ k − rτ k α − φ ( rτ k , z ) (cid:12)(cid:12)(cid:12)(cid:12) − | φ ( rτ k , z ) | − r = 2 | C | . Now set(17) c αk = 1 | C | = 1 | ∂φ∂z ( τ k , z ) | > . Then (16) and our subsequent computations combine to give (cid:90) T P z ( ζ ) χ L k ( ζ ) dσ α ( ζ ) = c αk = c αk (cid:90) T P z ( ζ ) dm ( ζ ) . Multiplying both sides by P z ( τ k ) establishes (14) for f = P z and completes the proof. (cid:3) The following lemma, which was used in the above proof, is a consequence of standardmeasure-theory facts. We include its proof here for the ease of the reader. Lemma 3.10. Let σ be a Borel measure on T and let ζ = g ( ζ ) be a continuous curvein T . If W is a continuous function defined on T such that (18) (cid:90) T f ( ζ ) dσ ( ζ ) = (cid:90) T f ( ζ, g ( ζ )) W ( ζ ) dm ( ζ ) for all f ∈ C ( T ) , then (18) holds for all f ∈ L ( σ ) . Proof. For ease of notation, set dν = W dm . Then ν is a Borel measure on T . Furthermore,if E ⊆ T is a Borel set, then(19) E g := { ζ ∈ T : there exists ζ ∈ T with ( ζ , ζ ) ∈ E ∩ { ( ζ, g ( ζ )) : ζ ∈ T }} is the projection of a Borel set in T onto its first coordinate. This implies that E g is an an-alytic set and its characteristic function is Lesbesgue measurable and hence, ν -measurable,see for example Chapter 13 in [13]. We will use E g frequently because χ E ( ζ, g ( ζ )) = χ E g ( ζ )for ζ ∈ T . The following proof has three steps. Step 1: Establish (18) for f = χ U , where U is an arbitrary open set in T . Let { K n } ∞ n =1 be a sequence of nested compact sets with U = (cid:83) n K n . Then, by Urysohn’s lemma, thereexists a sequence { f n } ∞ n =1 of continuous functions on T having 0 ≤ f n ≤ T and f n = 1 on K n and f n = 0 on U c . Then lim n →∞ f n ( ζ ) = χ U ( ζ ) for every ζ ∈ T , and since ν is a finite measure, we canapply the dominated convergence theorem to obtain σ ( U ) = lim n →∞ (cid:90) T f n ( ζ ) dσ ( ζ ) = lim n →∞ (cid:90) T f n ( ζ, g ( ζ )) dν ( ζ ) = (cid:90) T χ U ( ζ, g ( ζ )) dν ( ζ ) , as desired. Step 2: Establish (18) for f = χ E , where E is an arbitrary Borel set in T . Since σ isa finite Borel measure on T , and hence is Radon, for each n ∈ N there exists a compactset K n and an open set U n such that K n ⊆ E ⊆ U n and σ ( U n \ K n ) < /n . Urysohn’slemma again guarantees the existence of a sequence { f n } ∞ n =1 of continuous functions with0 ≤ f n ≤ f n = 1 on K n , and f n = 0 on U cn . Since (cid:90) T | ( f n − χ E )( ζ ) | dσ ( ζ ) ≤ (cid:90) T χ U n \ K n ( ζ ) dσ ( ζ ) = σ ( U n \ K n ) < n , LARK MEASURES FOR RIFS 17 we have (cid:107) f − f n (cid:107) L ( σ ) < /n and f n → f in L ( σ ) as n → ∞ . Since the f n are continuous,this implies (cid:90) T χ E ( ζ ) dσ ( ζ ) = lim n →∞ (cid:90) T f n ( ζ ) dσ ( ζ ) = lim n →∞ (cid:90) T f n ( ζ, g ( ζ )) dν ( ζ ) . Because χ K n ≤ f n ≤ χ U n on T , we then obtain(20) (cid:90) T χ K n ( ζ, g ( ζ )) dν ( ζ ) ≤ (cid:90) T f n ( ζ ) dσ ( ζ ) ≤ (cid:90) T χ U n ( ζ, g ( ζ )) dν ( ζ )and as χ K n ≤ χ E ≤ χ U n on T ,(21) (cid:90) T χ K n ( ζ, g ( ζ )) dν ( ζ ) ≤ (cid:90) T χ E ( ζ, g ( ζ )) dν ( ζ ) ≤ (cid:90) T χ U n ( ζ, g ( ζ )) dν ( ζ ) . Combining (20) and (21) gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T χ E ( ζ ) dσ ( ζ ) − (cid:90) T χ E ( ζ, g ( ζ )) dν ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) χ E − f n (cid:107) L ( σ ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T f n ( ζ ) dσ ( ζ ) − (cid:90) T χ E ( ζ, g ( ζ )) dν ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n + (cid:90) T ( χ U n − χ K n )( ζ, g ( ζ )) dν ( ζ )= n + σ ( U n \ K n ) < n for all n , where we used Step 1 applied to U n \ K n = U n ∩ K cn . Letting n → ∞ gives (18)for f = χ E . Step 3: Establish (18) for a general f ∈ L ( σ ). Pick a sequence { f n } ∞ n =1 of continuousfunctions on T such that f n → f in L ( σ ) and pointwise σ -almost everywhere on T .Then there exists some Borel set E ⊂ T with σ ( E ) = 0 such that if f n ( ζ ) (cid:57) f ( ζ ), n → ∞ , then ζ ∈ E . Then, if f n ( ζ, g ( ζ )) (cid:57) f ( ζ, g ( ζ )) then ζ ∈ E g , where E g is definedin (19). By Step 2, ν ( E g ) = σ ( E ) = 0. Hence(22) f n ( ζ, g ( ζ )) → f ( ζ, g ( ζ )) for ν − a . e ζ ∈ T . Since the f n are continuous, we have (cid:90) T | f n ( ζ ) − f m ( ζ ) | dσ ( ζ ) = (cid:90) T | f n ( ζ, g ( ζ )) − f m ( ζ, g ( ζ )) | dν ( ζ ) . This implies that { f n ( ζ, g ( ζ )) } ∞ n =1 is a Cauchy sequence and hence has a limit F in L ( ν ),and by (22) we must have F = f in L ( ν ). Then L -convergence gives (cid:90) T f ( ζ ) dσ ( ζ ) = lim n →∞ (cid:90) T f n ( ζ ) dσ ( ζ ) = lim n →∞ (cid:90) T f n ( ζ, g ( ζ )) dν ( ζ ) = (cid:90) T f ( ζ, g ( ζ )) dν ( ζ ) , which gives (18) for f . (cid:3) Properties of J α . Recall that the isometry J α : K φ → L ( σ α ) is obtained by firstdefining the operator on reproducing kernels k w as J α [ k w ]( ζ ) := (1 − αφ ( w )) C w ( ζ ) , for w ∈ D , ζ ∈ T , and then extending it to the rest of K φ . Theorem 1.3 details our main results about J α ,which are proved below in two propositions.First, unlike the one-variable case, these isometries J α need not be unitary. The exactsituation in our setting is encoded in the following result: Proposition 3.11. The isometric embedding J α : K φ → L ( σ α ) is unitary if and only if α is a generic value for φ .Proof. ( ⇒ ) Assume that α is generic. By Theorem 3.2 in [12], we need only show that A ( D ) is dense in L ( σ α ). Since σ α is a finite Radon measure, C ( T ) is dense in L ( σ α )and by the Stone-Weierstrass theorem, the set of two-variable trigonometric polynomialsis dense in C ( T ) and hence, in L ( σ α ). Thus, to show J α is unitary, we need only showthat each two-variable trigonometric polynomial agrees with some function in A ( D ) on E α , which contains the support of σ α . Let h ( ζ ) = ζ m ζ n be an arbitrary trigonometric monomial. To construct a function in A ( D ) that agrees with h on E α , first define t , t ∈ A ( D ) by t ( z ) = z and t ( z ) = z .Then, recall from Theorem 3.3 that deg B α = n . Write B α = γ (cid:81) nj =1 b a j , where γ ∈ T and each b a j = ( z − a j ) / (1 − ¯ a j z ) is the Blaschke factor with zero a j ∈ D . Let Let b − a denote the inverse function of b a and define s , s ∈ A ( D ) by s ( z ) = b − a (cid:104)(cid:16) γ n (cid:89) j =2 b a j ( z ) (cid:17) z (cid:105) , and s ( z ) = B α ( z ). Then, restricting to E α , we have s ( ζ, B α ( ζ )) = b − a (cid:104)(cid:16) γ n (cid:89) j =2 b a j ( ζ ) (cid:17)(cid:16) γ n (cid:89) j =1 b a j ( ζ ) (cid:17)(cid:105) = b − a (cid:2) b a ( ¯ ζ ) (cid:3) = ¯ ζ and s ( ζ, B α ( ζ )) = B α ( ζ ) . As h ( ζ, B α ( ζ )) = ζ m B α ( ζ ) n ,h agrees with one of t | m | t | n | , t | m | s | n | , s | m | t | n | , s | m | s | n | on E α . Taking linear combinations ofthese shows that every two-variable trigonometric polynomial agrees with some F ∈ A ( D )on E α and completes the proof of this forward direction.( ⇐ ) Assume that α is exceptional and φ ∗ ( τ k , λ k ) = α . By way of contradiction, assumethat A ( D ) is dense in L ( σ α ). Let f ( ζ ) = ¯ ζ . By assumption, there is a sequence( f n ) ⊆ A ( D ) that converges to f in L ( σ α ). Then by Theorem 1.2, there is a c αk > (cid:90) T | f ( τ k , ζ ) − f n ( τ k , ζ ) | dm ( ζ ) ≤ c αk (cid:107) f − f n (cid:107) L ( σ α ) → , as n → ∞ . Since each f n ( τ k , · ) is in H ( D ), so is the limit function f ( τ k , · ). Since f ( τ k , ζ ) = ¯ ζ , it is clearly not in H ( D ) and so, we obtain the needed contradiction. (cid:3) We can also identify the exact form of the isometric operator J α . LARK MEASURES FOR RIFS 19 Proposition 3.12. For each each f ∈ K φ , the isometry J α : K φ → L ( σ α ) satisfies ( J α f )( ζ ) = f ∗ ( ζ ) for σ α -a.e. ζ ∈ T . Proof. Fix f ∈ K φ . By Lemma 3.13, f ∗ exists and equals a Borel-measurable function σ α -a.e. on T . We claim that f ∗ ∈ L ( σ α ) and(23) (cid:107) f ∗ (cid:107) L ( σ α ) (cid:46) (cid:107) f (cid:107) K φ , where the implied constant does not depend on f . To see this, use Theorem 3.2 to write f ( z ) = Qp ( z ) h ( z ) + n (cid:88) j =1 R j p ( z ) g j ( z ) for z ∈ D and g , . . . , g n , h ∈ H ( D ). By the proof of Lemma 3.13, this formula extends to T vianon-tangential limits both Lebesgue and σ α -a.e. By Proposition 3.6, there is a b j ∈ A ( D )such that (cid:12)(cid:12)(cid:12) ( R j p )( ζ, B α ( ζ )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( Qp )( ζ, B α ( ζ )) (cid:12)(cid:12)(cid:12) | b j ( ζ ) | , for all ζ ∈ T . Working through the definitions and applying Proposition 3.6 give | ( Qp )( ζ, B α ( ζ )) | W α ( ζ ) = | p ( ζ ) | − | p ( ζ ) | ( | p ( ζ ) | − | p ( ζ ) | ) | (˜ p − αp )( ζ ) | | p ( ζ ) | − | p ( ζ ) | | (˜ p − αp )( ζ ) | = 1for all ζ ∈ T \ { τ , . . . , τ m } . If B α is non-constant, this immediately implies that (cid:90) T | f ∗ ( ζ, B α ( ζ )) | dν α ( ζ ) (cid:46) (cid:90) T | ( Qp )( ζ, B α ( ζ )) | (cid:16) | h ∗ ( ζ ) | + n (cid:88) j =1 | b j ( ζ ) | | g ∗ j ( B α ( ζ )) | (cid:17) W α ( ζ ) dm ( ζ )= (cid:90) T | h ∗ ( ζ ) | + n (cid:88) j =1 | b j ( ζ ) | | ¯ g ∗ j ( B α ( ζ )) | dm ( ζ ) (cid:46) (cid:16) (cid:107) h (cid:107) H + n (cid:88) j =1 (cid:107) g j ◦ B α (cid:107) H (cid:17) (cid:46) (cid:107) f (cid:107) K φ , where ¯ g is the function in H ( D ) whose Taylor coefficients are the complex conjugatesof those of g . In this computation, we used Theorem 3.2 and the well-known fact thatcomposition by a non-constant finite Blaschke product B α induces a bounded operatoron H ( D ), see Theorem 5.1.5 in [24]. Here, the implied constant does not depend on f .If B α is constant, then the one-variable model space ˆ K B α = { } , so each b j ≡ (cid:90) T | f ∗ ( ζ, B α ( ζ ))) | W α ( ζ ) dm ( ζ ) = (cid:107) h (cid:107) H ≤ (cid:107) f (cid:107) K φ . Similarly, for each 1 ≤ j ≤ n and 1 ≤ k ≤ m , Proposition 3.6 gives constants M jk and d αjk such that for ζ (cid:54) = λ k , Qp ( τ k , ζ ) ≡ R j p ( τ k , ζ ) = M jk − B α ( τ k ) ζp ( τ k , ζ ) =: d αjk , since both the numerator and denominator are linear and by Lemma 3.5, vanish at λ k .This shows c αk (cid:90) T | f ∗ ( τ k , ζ ) | dm ( ζ ) (cid:46) n (cid:88) j =1 ( d αjk ) (cid:90) T | g j ( ζ ) | dm ( ζ ) (cid:46) (cid:107) f (cid:107) K φ . By Lemma 3.13, this shows f ∗ ∈ L ( σ α ). Furthermore, if we define a linear map T α : K φ → L ( σ α ) by ( T α f ) = f ∗ , then T α is bounded. Moreover, observe that for ζ ∈C α \ { ( τ , λ ) , . . . , ( τ m , λ m ) } , we have T α [ k w ]( ζ ) = (1 − αφ ( w )) C w ( ζ ) = J α [ k w ]( ζ ) . Thus, these functions are equal in L ( σ α ). Since T α and J α agree on a dense set offunctions in K φ , it follows that T α = J α , which completes the proof. (cid:3) The proof of Proposition 3.12 required the following lemma. Lemma 3.13. If f ∈ K φ , then f ∗ exists and agrees with a Borel measurable function σ α -a.e. on T and (cid:90) T | f ∗ ( ζ ) | dσ α ( ζ ) = (cid:90) T | f ∗ ( ζ, B α ( ζ )) | dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T | f ∗ ( τ k , ζ ) | dm ( ζ ) , where dν α = W α dm , the functions B α , W α are from Definition 1.1, and the c αk are fromTheorem 1.2.Proof. By Theorem 3.2, there exist g , . . . , g n , h ∈ H ( D ) such that f ( z ) = Qp ( z ) h ( z ) + n (cid:88) j =1 R j p ( z ) g j ( z ) , for z ∈ D . Let A ⊆ T be a Borel set with m ( A ) = 0 such that h, g , . . . , g n have non-tangentiallimits at all ζ ∈ T \ A . To finish the set-up, assume ( z n ) = ( z n , z n ) → ( τ k , ζ ) ∈ T non-tangentially, where ζ (cid:54) = λ k . Then since ( z − τ k ) is a factor of Q and Q/p is continuousnear ( τ k , ζ ),(24) lim n →∞ (cid:12)(cid:12)(cid:12) Qp ( z n ) h ( z n ) (cid:12)(cid:12)(cid:12) (cid:46) lim n →∞ | z n − τ k |(cid:107) h (cid:107) H √ −| z n | (cid:46) lim n →∞ (cid:112) − | z n | = 0 , where we also used the reproducing property of H and the non-tangential property of( z n ). Similarly, if B α equals some constant γ ∈ T , then each λ k = ¯ γ and in Proposition3.6, each b j ≡ z − γ ) divides R j . Thus, arguments analogous to those in (24)imply that if ( z n ) = ( z n , z n ) → ( ζ , γ ) ∈ T non-tangentially with ζ ∈ T \ { τ , . . . , τ m } ,then lim n →∞ (cid:12)(cid:12)(cid:12) R j p ( z n ) g j ( z n ) (cid:12)(cid:12)(cid:12) = 0 , for j = 1 , . . . , n. This implies that f ∗ ( ζ ) exists for all ζ ∈ T \ ˆ A , whereˆ A := { ( τ k , λ k ) : k = 1 , . . . , m } ∪ (( A \ { τ , . . . , τ m } ) × T ) ∪ ( T × A \ { ¯ γ } ) , where we only include ¯ γ if B α is constant. By definition, ˆ A is a Borel set and we claim σ α ( ˆ A ) = 0. Since σ α has no point masses, it is immediate that σ α ( { ( τ k , λ k ) : k = 1 , . . . , m } ) = 0 . LARK MEASURES FOR RIFS 21 Set A = ( A \ { τ , . . . , τ m } ) × T . Then as A ∩ L k = ∅ for each k and Lemma 3.7 shows W α is bounded, we can use Theorem 1.2 to compute σ α ( A ) = (cid:90) T χ A ( ζ, B α ( ζ )) dν α ( ζ ) (cid:46) (cid:90) T χ A ( ζ ) dm ( ζ ) = 0 . If B α is non-constant, set A = T × A . Again by Theorem 1.2, there are constants c αk such that σ α ( A ) = (cid:90) T χ A ( ζ, B α ( ζ )) dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T χ A ( τ k , ζ ) dm ( ζ ) (cid:46) m ( { ζ ∈ T : B α ( ζ ) ∈ A } ) + m (cid:88) k =1 c αk m ( A ) = 0 , where the first set has Lebesgue measure 0 because non-constant finite Blaschke productsare smooth, have non-zero derivatives on T , and are locally invertible on T . Hence, thepreimage ¯ B − α ( A ) must have measure 0 because A does. If B α = γ is constant, set A = T × ( A \ { ¯ γ } ) . Then σ α ( A ) = (cid:90) T χ A ( ζ, ¯ γ ) dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T χ A ( τ k , ζ ) dm ( ζ ) = 0by the definition of A . Thus, f ∗ exists σ α -a.e. on T . Finally, observe that F ( ζ ) = lim sup r (cid:37) (cid:60) ( f ( rζ )) + i lim sup r (cid:37) (cid:61) ( f ( rζ ))is Borel measurable since each f r ( ζ ) := f ( rζ ) is continuous on T and F = f ∗ on T \ ˆ A and hence σ α -a.e. Our prior arguments also imply f ∗ ( ζ, B α ( ζ )) = F ( ζ, B α ( ζ )) for ν α -a.e. ζ ∈ T and f ∗ ( τ k , ζ ) = F ( τ k , ζ ) for m -a.e. ζ ∈ T . To finish the proof, for each n ∈ N ,define the Borel set D n = { ζ ∈ T : | F ( ζ ) | < n } . Then Theorem 1.2 combined with the monotone convergence theorem gives (cid:90) T | f ∗ ( ζ ) | dσ α ( ζ ) = lim n →∞ (cid:90) T | F ( ζ ) | χ D n ( ζ ) dσ α ( ζ )= lim n →∞ (cid:32)(cid:90) T | ( F χ D n ( ζ, B α ( ζ )) | dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T | ( F χ D n )( τ k , ζ ) | dm ( ζ ) (cid:33) = (cid:90) T | f ∗ ( ζ, B α ( ζ )) | dν α ( ζ ) + m (cid:88) k =1 c αk (cid:90) T | f ∗ ( τ k , ζ ) | dm ( ζ ) , which is what we needed to show. (cid:3) Applications The results from Section 3 have implications for the structure of Agler decompositionsand connections to the study of extreme measures from [22, 25, 28] and the referencestherein. In this section, we again assume φ = ˜ pp is a degree ( n, 1) rational inner functionand throughout, will use the notation denoted earlier in Definition 1.1 and Remark 3.1. Agler Decompositions. Recall that each such φ possesses an Agler decompositionfrom Theorem 3.2 arising from a particular orthonormal list in K φ . Moreover, the poly-nomial Q in that decomposition can be computed directly on T via Proposition 3.6. Inthe case of exceptional α , we can apply Theorem 1.2 to specify some of the remainingpolynomials R , . . . , R n from (7). Theorem 4.1. Let α ∈ T be exceptional for φ and (after reordering if necessary) assume φ ∗ ( τ k , λ k ) = α for k = 1 , . . . , (cid:96) . Using Theorem 3.3, write B α = b α /b α , where each deg b iα = n − (cid:96). Then in (7) , we can take (25) R j ( z ) = d αj (cid:16) b α ( z ) − z b α ( z ) (cid:17) (cid:89) ≤ k ≤ (cid:96)k (cid:54) = j ( z − τ k ) , for j = 1 , . . . , (cid:96), where each d αj > is chosen so c αj (cid:107) R j /p ( τ j , · ) (cid:107) H ( D ) = 1 and c αj is fromProposition 3.9.Proof. By the proof of Theorem 3.2, the R , . . . , R n from (7) are exactly obtained byspecifying that R /p, . . . , R n /p be an orthonormal basis for H ( K ). Thus, we need onlyshow that for the R j defined in (25), R /p, . . . , R (cid:96) /p are in H ( K ) and form an orthonormalset there.To that end, as in Theorem 3.2, let ˆ R /p, . . . , ˆ R n /p be some orthonormal basis for H ( K ). Recall that ˆ K B α := H ( D ) (cid:9) B α H ( D ) denotes the one variable model spaceassociated to B α . Then Proposition 3.6 implies that for each j , there is a unique polynomialˆ r j with deg ˆ r j ≤ n − b j ∈ ˆ K B α such thatˆ R j ( z ) = ˆ r j ( z ) (cid:16) − B α ( z ) z (cid:17) + z Q ( z ) b j ( z ) . Define a linear map T : Span { ˆ R , . . . , ˆ R n } → ˆ K B α by first specifying T ( ˆ R j ) = b j and thenextending by linearity. As dim ˆ K B α = n − (cid:96) , it follows that dim(ker T ) ≥ (cid:96) . If R ∈ ker( T ),then for some r with deg r < n ,(26) R ( z ) = r ( z )(1 − B α ( z ) z ) = r ( z ) b α ( z ) (cid:16) b α ( z ) − z b α ( z ) (cid:17) = q ( z ) (cid:16) b α ( z ) − z b α ( z ) (cid:17) , where q ∈ C [ z ] with deg q < (cid:96) . Note that the set of such R has dimension (cid:96) . By comparingdimensions, each R given in (26) must be in ker( T ) and hence, each R given in (26) satifies R/p ∈ H ( K ). In particular, this implies that each R j from (25) satisfies R j /p ∈ H ( K ).To show R /p, . . . , R (cid:96) /p are orthonormal in K φ , we use Proposition 3.9 and Theorem3.12. First, observe that those two results combine to imply that R j /p ( τ j , · ) ∈ H ( D ) \{ } ,so d αj is well defined. Then, one can use the fact that each R j vanishes on E α and each LARK MEASURES FOR RIFS 23 L k with 1 ≤ k ≤ (cid:96) and k (cid:54) = j to conclude: (cid:68) R i p , R j p (cid:69) K φ = (cid:68) J α (cid:16) R i p (cid:17) , J α (cid:16) R j p (cid:17)(cid:69) L ( σ α ) = (cid:90) T R i p ( ζ, B α ( ζ )) R j p ( ζ, B α ( ζ )) dν α ( ζ ) + (cid:96) (cid:88) k =1 c αk (cid:90) T R i p ( τ k , ζ ) R j p ( τ k , ζ ) dm ( ζ )= 0 + (cid:88) k = i or k = j c αk (cid:90) T R i p ( τ k , ζ ) R j p ( τ k , ζ ) dm ( ζ )= (cid:26) i = j i (cid:54) = j. Thus, { R /p, . . . , R (cid:96) /p } is an orthonormal set in K φ and hence in H ( K ), which completesthe proof. (cid:3) Extreme Points. Recall that P = { f ∈ Hol( D ) : (cid:60) f ( z ) > f (0 , 0) = 1 } and M : P → P ( T ) is the map that takes f ∈ P to the unique Borel probability measure µ f on T with f ( z ) = (cid:90) T P z ( ζ ) dµ f ( ζ ) for z ∈ D . for some f ∈ P and f is an extreme point of P if and only if µ f is an extreme point of M ( P ). As mentioned in the introduction, Forelli, McDonald, and Knese have proved anumber of interesting results related to such extreme points. For example, Knese provedthe following result in [22, Theorem 1.5]: Theorem 4.2. Let q be a polynomial with no zeros on D and let ˜ q be the reflectionof q with deg ˜ q = deg q . Assume that q is T -saturated, ˜ q, q share no common factors, ˜ q (0 , 0) = 0 , and q − ˜ q is irreducible. Then f := q +˜ qq − ˜ q is an extreme point of P . As mentioned in the introduction, our results in the ( n, 1) setting coupled with Theorem4.2 yield Corollary 1.4, which we restate here for convenience. Corollary. Assume ˜ p (0 , 0) = 0 and let α ∈ T . Then i. If α is an exceptional value for φ , then σ α is not an extreme point of M ( P ) . ii. If deg p = deg ˜ p , p is T -saturated, and α is generic for φ , then σ α is an extremepoint of M ( P ) .Proof. For (i), without loss of generality, assume φ ∗ ( τ k , λ k ) = α for k = 1 , . . . , (cid:96) . ByProposition 3.9, we can write σ α ( ζ ) = µ α ( ζ ) + c α ( δ τ ( ζ ) ⊗ m ( ζ )) , for a positive Borel measure µ α on T and c α > 0. As φ (0 , 0) = 0, we have1 = σ α ( T ) = µ α ( T ) + c α , and as µ α ( T ) > 0, we have c α < 1. Then ˆ µ α := − c α µ α is a probability measure and(27) σ α ( ζ ) = (1 − c α )ˆ µ α ( ζ ) + c α ( δ τ ( ζ ) ⊗ m ( ζ )) , so σ α is a convex combination of two probability measures on T . Clearly, the second oneis in M ( P ), as (cid:60) (cid:18) τ + z τ − z (cid:19) = 1 − | z | | z − τ | = (cid:90) T P z ( ζ ) d ( δ τ ( ζ ) ⊗ m ( ζ )) . For the first, observe that for each z ∈ D , − c α (cid:60) (cid:18) α + φ ( z ) α − φ ( z ) − c α τ + z τ − z (cid:19) = (cid:90) T P z ( ζ ) d ˆ µ α ( ζ ) > . This implies that ˆ µ α ∈ M ( P ) and by (27), σ α is not an extreme point in M ( P ).For (ii), choose λ ∈ T with λ = α , define q = λp , and set f := α + φα − φ = αp + ˜ pαp − ˜ p = q + ˜ qq − ˜ q . Note that ˜ q − q = ¯ λ (˜ p − αp ) must be irreducible by the characterization of C α fromTheorem 3.3. By Theorem 4.2, f is an extreme point of P and so σ α from Theorem 1.2is extreme in M ( P ) . (cid:3) Examples We illustrate our results by examining some specific RIFs and their associated Clarkmeasures in detail. For the first example, we can confirm our general findings at excep-tional values α via direct computation. Example . Let φ ( z ) = ˜ p ( z ) p ( z ) = 2 z z − z − z − z − z , essentially the example considered in [12]. We have the sums of squares decomposition | p ( z ) | − | ˜ p ( z ) | = (1 − | z | )2 | − z | + (1 − | z | )2 | − z | and for each α ∈ T , the associated B α is B α ( z ) = 2 z − α α − αz + z . Note that if α = − 1, then B − ≡ 1. If α (cid:54) = − 1, then 2 α − αz + z does not vanish on T .Thus if α (cid:54) = − 1, then by Proposition 3.8, for all f ∈ L ( σ α ), we have (cid:90) T f ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) 2 | − ζ | | ζ − α | dm ( ζ ) , LARK MEASURES FOR RIFS 25 and by Theorem 3.11, the isometric embedding J α : K φ → L ( σ α ) is unitary. Finally, if α = − 1, then | z − α | = 4 | z − | . By the given sums of squares decomposition,1 − | φ ( z ) | | α − φ ( z ) | = | p ( z ) | − | ˜ p ( z ) | | αp ( z ) − ˜ p ( z ) | = | p ( z ) | − | ˜ p ( z ) | | z − | | − p ( z ) − ˜ p ( z ) | = (1 − | z | )2 | − z | + (1 − | z | )2 | − z | | z − | · | z − | = 12 (cid:18) − | z | | z − | + 1 − | z | | z − | (cid:19) , which shows σ α = ( δ ( ζ ) ⊗ m ( ζ ) + m ( ζ ) ⊗ δ ( ζ )) . This was observed in [12], andconfirms the contents of Theorem 1.2. Note in particular that ∂φ∂z ( z , z ) = − ( z − (2 − z − z ) ,so that ∂φ∂z (1 , z ) = − z .See Figure 2(a) for a visual representation of the sets C α . (cid:7) Now let us consider a RIF that was not studied in [12], and again illustrate how theexceptional measure σ α can be identified using both our results and concrete Agler de-compositions. Example . Let φ = ˜ pp , where p ( z ) = 4 − z − z − z z + z and ˜ p ( z ) = 4 z z − z − z z − z + z . This example was introduced by Agler-McCarthy-Young in [2]. In [21, Section 15], Kneseprovides the following sums of squares decomposition: | p ( z ) | − | ˜ p ( z ) | = 4(1 − | z | ) | − z | + 4(1 − | z | ) (cid:0) | − z | | − z | + 2 | − z z | (cid:1) . The only singularity of φ occurs at (1 , α ∈ T , setting φ ( z ) = α and solvingfor z yields z = 1 /B α ( z ), where B α ( z ) = 4 z − z + 1 + α + αz α − z α + z α + z + z . As φ has only one singularity, by previous discussions, the denominator of B α can vanish ata point on T for at most one α . This occurs at α = − 1, where B − reduces to B − ( z ) = z and φ = − z = 1. Thus, we can apply Proposition 3.8 to α (cid:54) = − f ∈ L ( σ α ), (cid:90) T f ( ζ ) dσ α ( ζ ) = (cid:90) T f ( ζ, B α ( ζ )) 4 | ζ − | | ζ − ζ + 1 + α + αζ | dm ( ζ ) . By Theorem 3.11, the isometric embedding J α : K φ → L ( σ α ) is unitary for every α (cid:54) = − α = − φ ∗ (1 , p ( z ) + ˜ p ( z ) = 4(1 − z )(1 − z z ) , and hence, for α = − 1, we have1 − | φ ( z ) | | α − φ ( z ) | = | p ( z ) | − | ˜ p ( z ) | | − z | | − z z | . - - - - - - (a) Level curves for φ = (2 z z − z − z ) / (2 − z − z ) corresponding to α = 1(black), α = e iπ/ (gray), α = e iπ/ (or-ange), and α = e iπ/ (pink). Level setcorresponding to exceptional value α = − - - - - - - (b) Level curves for φ = (4 z z − z − z z − z + z ) / (4 − z − z − z z + z ) corresponding to α = 1 (black), α = e iπ/ (gray), α = e iπ/ (orange), and α = e iπ/ (pink). Level set corresponding toexceptional value α = − Figure 2. Supports of σ α for two different RIFs. By the sums of squares formula above, we then obtain1 − | φ ( z ) | | α − φ ( z ) | = 14 | − z | − | z | | − z z | + 14 | − z | − | z | | − z z | + 12 1 − | z | | − z | = 14 (1 − | z | ) | − z | + (1 − | z | ) | − z | | − z z | + 12 1 − | z | | − z | . The second term is evidently the Poisson integral of the measure σ (2) − = ( δ ( ζ ) ⊗ m ( ζ )), which matches what we get from computing ∂φ∂z (1 , z ) = − σ (1) − on T having (cid:90) T f ( ζ ) dσ (1) − ( ζ ) = (cid:90) T f ( ζ, ζ ) | − ζ | dm ( ζ ) , as can be seen by examining the Fourier coefficients (cid:100) σ (1) − ( k, l ) = , k = l − , k = l + 1 − , l = k + 10 otherwiseand computing the Poisson integral of σ (1) − explicitly. The specific form of σ (1) − of courseagrees with Theorem 1.2 once we set α = − W − ( ζ ) = | ζ − | | ζ − ζ | = | ζ − | . LARK MEASURES FOR RIFS 27 - - - - - - Figure 3. Generic level curves for (4 z z − z + z − z − / (4 − z + z z − z z − z z ) corresponding to several values of α (black, gray, orange,pink). Level sets corresponding to exceptional values α = − and α = 1 marked in green and red, respectively. Level curves C α for several values of α are displayed in Figure 2(b). (cid:7) Remark . The RIF φ = ˜ pp with p ( z ) = 2 − z z − z z and ˜ p ( z ) = 2 z z − z − , φ ∗ (1 , 1) = − α = − B − ( z ) = z so that σ − for this example is supportedon the same set as the exceptional Clark measure in Example 5.2. However, we have W α ( ζ ) = | ζ − | | (2 + α ) ζ + α | , which collapses to W − ( ζ ) = 1 at the exceptional value, meaning that the two Clarkmeasures do not coincide.Our next example is a degree (3 , 1) RIF with two different singularities on T . Here,we are able to observe qualitative differences in W α for the two corresponding exceptionalvalues of α that reflect the finer distinctions between the two singularities. Example . Let p ( z ) = 4 − z + z z − z z − z z and ˜ p ( z ) = 4 z z − z + z − z − φ = ˜ pp . This function has singularities at (1 , 1) and ( − , α -values are φ ∗ (1 , 1) = − φ ∗ ( − , 1) = 1. Level sets for this example aredisplayed in Figure 3; see also [9, Example 7.4].For α (cid:54) = 1 , − 1, we have B α ( z ) = α − αz + 3 αz + 4 z + αz α + 3 z − z + z . Note that for α = − 1, we get B − ( z ) = z − z − z + 13 + z = 3 z + 13 + z , a Blaschke product of degree 2, while for α = 1, B ( z ) = z + 1 z + 1 5 z − z + 1 z − z + 5 = 5 z − z + 1 z − z + 5 , another degree 2 Blaschke product. The graphs { ( ζ, B − ( ζ )) and { ( ζ, B ( ζ )) } togetherwith vertical lines at ζ = 1 and ζ = − σ − ) and supp( σ ), respectively.We further read off that p ( z ) = 4 and p ( z ) = − z − z − z so that, with W α as in Remark 3.1, W α ( ζ ) = 16 − | − ζ + 3 ζ + ζ | | ζ + αζ + 3 αζ − αζ + α | . After some simplifications, we find that W α ( z ) = | ζ − | | ζ + 1 | | ζ + αζ + 3 αζ − αζ + α | . For the exceptional values α = ± 1, the weights in the point mass parts of σ ± can beobtained by computing ∂φ∂z (1 , z ) = − ∂φ∂z ( − , z ) = − , which imply c − = 1 | ∂φ∂z (1 , z ) | = 1 and c = 1 | ∂φ∂z ( − , z ) | = 12 . (Note that φ (0 , 0) = − so that the Clark measures σ ± are not probability measures inthis example.) Putting α = ± W α , we have cancellation in numerator and denomina-tor, and we obtain W − ( ζ ) = | ζ + 1 | | ζ + 1 | and W ( ζ ) = | ζ − | | ζ + 1 | | ζ − ζ + 1 | . This gives us a complete description of the exceptional Clark measures.Furthermore, observe that, W − (1) (cid:54) = 0 and so, W − does not vanish at the z -coordinateof the singularity with non-tangential value − . In contrast, W ( − 1) = 0, so W doesvanish at the z -coordinate of the singularity with non-tangential value 1 . This mirrorsthe singular behavior in Example 5.2, where function W − vanishes at ζ = 1, the z -coordinate of the singularity where φ ∗ (1 , 1) = − 1. This pattern suggests a connection withcontact order, which was studied in [8] and governs the integrability of partial derivativesof a RIF φ ; in that sense, higher contact order indicates a stronger singularity. In ourcomputations, the singularities at (1 , 1) in Example 5.2 and at ( − , 1) in this example(where the exceptional W α vanish at the z -coordinate of the associated singularity) areinstances of singularities where φ exhibits contact order 4; the singularities in Example 5.1 LARK MEASURES FOR RIFS 29 and at (1 , 1) in this example (where the exceptional W α do not vanish at the z -coordinateof the associated singularity) are singularities where φ exhibits contact order 2, the lowestpossible contact order. (cid:7) Remark . It would interesting to investigate how the exact nature of a singularity τ ∈ T (contact order, number of branches of p coming together at τ , etc) of a RIF isreflected in the associated exceptional Clark measure. For example, if φ = ˜ pp is a generaldegree ( m, n ) RIF having contact order at least 4, does the corresponding exceptionalClark measure have a density along C α that vanishes at τ ?Our final example is a rational inner function having bidegree (3 , m, n ) with m, n ≥ 2, a general α -level set is not necessarily parametrizedby a single function. Example . Let φ ( z ) = ˜ pp ( z ) where p ( z ) = 2 − z z − z z and ˜ p ( z ) = z z (2 z z − z − z ) . This example is obtained by applying the level line embedding construction described in[9, Section 6.1] to the essentially T -symmetric polynomial r ( z ) = (1 − z z )(1 − z z ) . As is guaranteed by the embedding construction, we have p (1 , 1) = 0 = ˜ p (1 , 1) and φ ∗ (1 , 1) = − 1, as well as ˜ p + p = 2(1 − z z )(1 − z z ) . These facts can also be checked directly. We also note that p and ˜ p , and hence φ , are in-variant under the simultaneous coordinatewise rotations z j (cid:55)→ e iπ/ z j and z j (cid:55)→ e − iπ/ z j .Some level sets of φ are displayed in Figure 4.Recall from Example 5.1 that | − x − y | − | xy − x − y | = (1 − | x | )2 | − y | +(1 − | y | )2 | − x | . Substituting x = z z and y = z z into this formula, we get thedecomposition | − z z − z z | − | z z − z z − z z | = (1 − | z z | )2 | − z z | + (1 − | z z | )2 | − z z | . It follows that 1 − | φ ( z ) | | φ ( z ) | = 12 1 − | z z | | − z z | + 12 1 − | z z | | − z z | , and by computing Fourier coefficients, one can show that the two expressions on the rightare the Poisson integrals of the measures having (cid:90) T f ( ζ ) dσ (1) − = (cid:90) T f ( ζ, ζ ) dm ( ζ ) and (cid:90) T f ( ζ ) dσ (2) − = (cid:90) T f ( ζ , ζ ) dm ( ζ )respectively. By Doubtsov’s Theorem 3 . J − isunitary if and only if the bidisk algebra is dense in L ( σ − ).Let us show that this is indeed the case. By definition, h ( z ) = z and h ( z ) = z areelements of A ( D ). Next consider the function g ( z ) = ¯ z and the function f ( z ) = z z + (1 − z z ) z ∈ A ( D ) . - - - - - - Figure 4. Level curves for (2 z z − z z − z z ) / (2 − z z − z z ) corre-sponding to α = 1 (black) and α = e πi/ (orange). Level set correspondingto exceptional value α = − marked in red. Since f ( ζ, ¯ ζ ) = ζ ¯ ζ + (1 − ¯ ζ ζ ) ¯ ζ = ¯ ζ = g ( ζ, ¯ ζ )and f ( ¯ ζ , ζ ) = ¯ ζ ζ + (1 − ζ ¯ ζ ) ζ = ζ = g ( ¯ ζ , ζ )we have g = f on the support of σ − . A similar computation shows that the bidiskalgebra function f ( z ) = z z + (1 − z z ) z coincides with g ( z ) = ¯ z on supp( σ − ). Thus, if g ( ζ ) = ζ m ζ n is any trigonometricpolynomial, then, on the support of σ − , g coincides with one of functions h | m | h | n | , h | m | f | n | , f | m | h | n | , and f | m | f | n | , which are all in A ( D ). Since the trigonometric polynomials aredense in C ( T ), which in turn is dense in L ( σ − ), A ( D ) is also dense. Thus, J − isunitary even though α = − φ at a singularity. (cid:7) References [1] J. Agler, J.E. Mc Carthy, and M. Stankus, Toral algebraic sets and function theory on polydisks,J. Geom. Anal. (2006), no. 4, 551–562. 3[2] J. Agler, J.E. McCarthy, and N.J. Young, A Carath´eodory theorem for the bidisk via Hilbertspace methods, Math. Ann. (2012), 581-624. 25[3] A.B. Aleksandrov and E. Doubtsov, Clark measures on the complex sphere, J. Funct. Anal. (2020), 108314. 1[4] J.A. Ball, C. Sadosky, and V. Vinnikov, Scattering systems with several evolutions and multi-dimensional input/state/output systems. Integral Equations Operator Theory 52 (2005), no. 3,323–393. 8, 9[5] K. Bickel, Fundamental Agler decompositions. Integral Equations Operator Theory 74 (2012),no. 2, 233–257. 8[6] Kelly Bickel and Greg Knese, Inner functions on the bidisk and associated Hilbert spaces, J.Funct. Anal. 265 (2013), no. 11, 2753–2790. 9[7] K. Bickel and P. Gorkin, Compressions of the shift on the bidisk and their numerical ranges. J.Operator Theory 79 (2018), no. 1, 225–265. 9 LARK MEASURES FOR RIFS 31 [8] K. Bickel, J.E. Pascoe, and A. Sola, Derivatives of rational inner functions: geometry of singu-larities and integrability at the boundary, Proc. London Math. Soc. (2018), 281-329. 4, 7,28[9] K. Bickel, J.E. Pascoe, and A. Sola, Level curve portraits of rational inner functions, Ann. Sc.Norm. Sup. Pisa Cl. Sc. XXI (2020), 451-494. 4, 7, 27, 29[10] K. Bickel, J.E. Pascoe, and A. Sola, Singularities of rational inner functions in higher dimensions.American J. Math, to appear. Preprint available at https://arxiv.org/pdf/1906.10913.pdf. 3[11] J.A. Cima, A.L. Matheson, and W.T. Ross, The Cauchy transform, Math. Surveys and Mono-graphs , Amer. Math. Soc., Providence, RI, 2006. 1, 2, 5, 6[12] E. Doubtsov, Clark measures on the torus. Proc. Amer. Math. Soc. 148 (2020), no. 5, 2009–2017.1, 2, 5, 6, 14, 18, 24, 25, 29[13] R.M. Dudley, Real Analysis and Probability. Revised reprint of the 1989 original. CambridgeStudies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. 16[14] G. Folland, Real analysis. Modern techniques and their applications. Second edition. Pure andApplied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc.,New York, 1999. 2[15] F. Forelli, A necessary condition on the extreme points of a class of holomorphic functions, II,Pacific J. Math. 92 (1981), no. 2, 277–281. 5[16] S.R. Garcia, J. Mashreghi, and W.T. Ross, Introduction to model spaces and their operators, Cambridge studies in advanced mathematics Harmonicanalysis, function theory, operator theory, and their applications , Theta Ser. Adv. Math. 19,Theta, Bucharest, 2017, 133-158. 7, 10[18] M.T. Jury, Clark theory in the Drury-Arveson space, J. Funct. Anal. (2014), 3855-3893. 1[19] G. Knese, Polynomials with no zeros on the bidisk, Anal. PDE 3 (2010), no. 2, 109–149. 9[20] G. Knese, Rational inner functions in the Schur-Agler class of the polydisk. Publ. Mat. 55 (2011),no. 2, 343-357. 3[21] G. Knese, Integrability and regularity of rational functions, Proc. London. Math. Soc. (2015),1261-1306. 3, 7, 10, 25[22] G. Knese, Extreme points and saturated polynomials. Illinois J. Math. 63 (2019), no. 1, 47–74.5, 21, 23[23] A. Luger and M. Nedic, Geometric properties of measures related to holomorphic functions havingpositive imaginary or real part, 2018, to appear in J. Geom. Anal. 6[24] R.A. Mart´ınez-Avenda˜no and P. Rosenthal, An introduction to operators on the Hardy-Hilbertspace. Graduate Texts in Mathematics, 237. Springer, New York, 2007. 19[25] J.N. McDonald, Measures on the torus which are real parts of holomorphic functions, MichiganMath. J. 29 (1982), 259–265. 5, 6, 21[26] J.N. McDonald, Examples of RP-measures, Rocky Mountain J. Math. 16 (1986), 191–201. 5[27] J.N. McDonald, Holomorphic functions on the polydisc having positive real part, Michigan Math.J. 34 (1987), no. 1, 77–84. 5[28] J.N. McDonald, An extreme absolutely continuous RP -measure, Proc. Amer. Math. Soc. (1990), 731–738. 3, 5, 6, 7, 21[29] J.E. Pascoe, A wedge-of-the-edge theorem: analytic continuation of multivariable Pick functionsin and around the boundary, Bull. London Math. Soc. , 916-925. 3[30] A.G. Poltoratski, Boundary behavior of pseudocontinuable functions. (Russian) Algebra i Analiz5 (1993), no. 2, 189–210; translation in St. Petersburg Math. J. 5 (1994), no. 2, 389–40. 5[31] W. Rudin, Function Theory in polydisks , W. A. Benjamin, Inc., New York-Amsterdam, 1969. 3[32] W. Rudin, “Harmonic analysis in polydiscs” in Actes du Congr`es International desMath´ematiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, 489–493. 5[33] W. Rudin and E. L. Stout, Bounday properties of functions of several complex variables, J. Math.Mech. 14 (1965), 991–1005. 3[34] D. Sarason, Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes inthe Mathematical Sciences, 10. A Wiley-Interscience Publication. John Wiley & Sons, Inc., NewYork, 1994. 15 [35] A. Zygmund, Trigonometric series. Vol. I, II . Third edition. With a foreword by Robert A.Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. 2 Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA. Email address : [email protected] Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599USA. 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