Asymptotics of Chebyshev rational functions with respect to subsets of the real line
aa r X i v : . [ m a t h . C A ] J a n ASYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS WITH RESPECTTO SUBSETS OF THE REAL LINE
BENJAMIN EICHINGER, MILIVOJE LUKIĆ, GIORGIO YOUNG
Abstract.
There is a vast theory of Chebyshev and residual polynomials and their asymptoticbehavior. The former ones maximize the leading coefficient and the latter ones maximize the pointevaluation with respect to an L ∞ norm. We study Chebyshev and residual extremal problemsfor rational functions with real poles with respect to subsets of R . We prove root asymptoticsunder fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widomasymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widomand DCT conditions. Introduction
Chebyshev polynomials are extremal polynomials with respect to the supremum norm on acompact set E . First discovered with explicit formulas for the set E = [ − , E , with important classical and modern developments[5, 7, 11, 27, 29]. Aspects of this theory have been extended to the setting of residual polynomials[7] (which are extremizers with respect to a point evaluation rather than leading coefficient) and tothe setting of Chebyshev rational functions with poles in R = R ∪ {∞} [17].To state the problems precisely, we make the following definitions. For c ∈ R we denote r ( z, c ) = ( c − z , c = ∞ ,z, c = ∞ . We fix a compact proper subset E ⊂ R containing infinitely many points. Connected componentsof R \ E are called gaps of E . We fix a sequence of poles C = ( c k ) ∞ k =1 with c k ∈ R \ E . The sequence C can have repetitions, which are used to designate multiplicity: we consider the spaces of rationalfunctions L n defined as L n = (cid:26) P ( z ) R n ( z ) : P ∈ P n (cid:27) , (1.1)where P n denotes the set of polynomials of degree at most n and R n ( z ) = Y ≤ k ≤ n c k = ∞ ( z − c k ) . (1.2)Of course, the spaces L n could also be defined iteratively, by L n = span (cid:8) r ( z, c n ) d n (cid:9) ⊕ L n − , L = { } , B.E. was supported by Austrian Science Fund FWF, project no: J 4138-N32.M.L. was supported in part by NSF grant DMS–1700179.G.Y. was supported in part by NSF grant DMS–1745670. where d n denotes the multiplicity of the pole c n up to that point, d n = X ≤ k ≤ n c k = c n . Let k · k E denote the supremum norm on E . We consider the two related extremal problems: Problem 1.1 (Chebyshev extremal problem) . m n ( c n ) := sup { Re λ n : ∃ F n ∈ L n such that k F n k E ≤ F n − λ n r ( · , c n ) d n ∈ L n − } . (1.3) Problem 1.2 (Residual extremal problem) . For x ∗ ∈ R \ ( E ∪ { c k : 1 ≤ k ≤ n } ), m n ( x ∗ ) := sup { Re F n ( x ∗ ) : F n ∈ L n , k F n k E ≤ } . (1.4)If c k = ∞ for all k , Problem 1.1 is the standard extremal problem for Chebyshev polynomials on E . For this reason we refer to λ n still as the leading coefficient. Whereas the Chebyshev extremalproblem maximizes the leading coefficient at the pole x ∗ = c n , the residual extremal problemmaximizes the value at a point x ∗ which is not a pole. We will use the notation x ∗ for bothproblems when convenient.For both problems, an extremal function exists (i.e., the supremum is a maximum) and is unique(see Section 2). The goal of this paper is to study the extremal functions F n and their asymptoticsas n → ∞ .Problems 1.1/1.2 have a conformal invariance with respect to the group PSL(2 , R ) of R -preserving,orientation-preserving Möbius transformations. This conformal invariance is obfuscated by the useof polynomials in the definitions (1.1) and (1.2), but can be made explicit in the language of divi-sors. Divisors on the Riemann sphere C = C ∪ {∞} are elements of the free Abelian group over C . They can be implemented as formal sums or as functions D : C → Z which take nonzero valuesonly at finitely many points; we will find the second interpretation notationally convenient. Thedegree of D is the integer deg D = P z D ( z ), and the divisor D is integral if D ( z ) ≥ z . Wealso write D ≤ D , if D − D is integral and denote by supp D = { z ∈ C : D ( z ) = 0 } the supportof D . In particular, for a meromorphic nonconstant function f : C → C , we denote its polar divisorby ( f ) ∞ ; the polar divisor assigns to each pole the multiplicity of that pole, and takes zero valueselsewhere. Similarly, for w ∈ C , we define ( f ) w = (1 / ( f − w )) ∞ . The value deg( f ) w is independentof w and corresponds to the degree of f . We also follow the convention to set ( f ) w = 0, if f is aconstant. For any n , we define the divisor D ∞ n by D ∞ n ( c ) = { k : c k = c , ≤ k ≤ n } . (1.5)In other words, in the functional interpretation, D ∞ n = P nk =1 χ { c k } . Note that by definitiondeg D ∞ n = n . Any integral divisor D with degree n generates a n + 1 dimensional vector space L ( D ) = { f : C → C | f is meromorphic and ( f ) ∞ ≤ D } , (1.6)and the definition (1.1) is equivalent to L n = L ( D ∞ n ) . (1.7)Now Problems 1.1, 1.2 can be unified as follows: Problem 1.3.
For a real integral divisor D ∞ n with deg D ∞ n = n containing only points in R \ E ,and a point x ∗ ∈ R \ E , denote d n = D ∞ n ( x ∗ ) and L n = L ( D ∞ n ) and find m n ( x ∗ ) := sup { Re lim x → x ∗ F n ( x ) r ( x, x ∗ ) d n : F n ∈ L n , k F n k E ≤ } . (1.8) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 3
The Chebyshev problem corresponds to d n > c , . . . , c n ) and theresidual problem corresponds to d n = 0. Throughout this paper, we work in the general setting ofProblem 1.3.In order to state our results in a conformally invariant form, we use the following language: Definition 1.4.
For a sequence ( t j ) mj =0 in R with m ≥
2, we say that the sequence is cyclicallyordered if it has no repetitions and there exists f ∈ PSL(2 , R ) such that f ( t ) = ∞ and f ( t ) For a real function F ∈ L n with k F k E ≤ 1, a set of distinct points x , . . . , x m ∈ E such that the sequence ( x ∗ , x , . . . , x m ) is cyclically ordered and satisfies the following alternationproperty F ( x j ) = ( − m − j − S n ( x j ) (1.10)for all j = 1 , . . . , m is called an alternation set. We say that F has a maximal alternation set if m = n + 1.It should be noted that the notion of alternation set depends on the function F , the class L n ,and the reference point x ∗ . Theorem 1.6 (Alternation theorem) . A real function F ∈ L n with k F k E ≤ is an extremalfunction if and only if it has a maximal alternation set. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 4 These results generalize standard results from the polynomial case: in the Chebyshev polynomialcase, S n ( x ) ≡ 0, and in the residual polynomial case, S n has one jump which may or may not affectthe alternation criterion, depending on degree. The case of Chebyshev rational functions was alsopreviously formulated in [17]. In all the cases previously considered in the literature, the extremizeris seen to be nonconstant. However, in the setting of residual rational functions, the extremizer canbe a constant function, and the alternation theorem lets us characterize when this happens: Theorem 1.7. The extremal function F n is constant if and only if the divisor D ∞ n is of the form (1.5) for points c , . . . , c n such that the points x ∗ , c , c , . . . , c n are in n + 1 distinct gaps of E . In particular, for the Chebyshev problem, x ∗ = c n so F n is always nonconstant.These results will be proved in Section 2, along with additional properties of F n and its zeros.Let us assume that F n is not constant and recall that ( F n ) ∞ ≤ D ∞ n ; we call a point x a “generalizedzero” of F n if either ( F n ) ( x ) > D ∞ n ( x ) − ( F n ) ∞ ( x ) > . Thus, this notion includes both actual zeros of F n and places where there is a reduction in the orderof the pole compared to the maximal allowed order. These generalized zeros are precisely countedby the divisor D n := ( F n ) + D ∞ n = ( F n ) + D ∞ n − ( F n ) ∞ . Since an alternation set is on E , note that changing x ∗ through a single gap only changes thealternation conditions up to an overall j -independent ± ± sign, theextremizer F n for Problem 1.3 is unchanged as x ∗ varies through a single gap of E . Thus, F n should be regarded as an extremal function of a gap, rather than of a single point. In particular,the Chebyshev extremizer for Problem 1.1 is the same (up to ± sign) as the residual extremizerfor Problem 1.2 for any x ∗ in the gap containing c n . Moreover, F n might even be extremal formore than one gap. This phenomenon is already known for the so-called Widom maximizer definedbelow, and is the content of the following corollary. Corollary 1.8. Let F n be an extremal function for x ∗ ∈ ( a , b ) . If ( a j , b j ) is a gap such that | F n ( a j ) | = | F n ( b j ) | = 1 and D n = 0 on ( a j , b j ) , then up to a ± factor, F n is an extremal functionfor any x j ∗ ∈ ( a j , b j ) . From deg( F n ) = deg( F n ) ∞ it follows thatdeg D n = deg D ∞ n = n (1.11)so we can define the normalized pole counting measure µ n := 1 n X c D ∞ n ( c ) δ c (1.12)and normalized generalized zero counting measure ν n := 1 n X c D n ( c ) δ c . (1.13)In Section 3, we consider the asymptotics of the extremal rational functions as n → ∞ , extendingresults about root asymptotics from the polynomial setting. For a sequence of divisors D ∞ n as inProblem 1.3 we define K C = [ n ≥ supp D ∞ n SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 5 and assume that K C ∩ E = ∅ (1.14)and that that the sequence has a limiting distribution, i.e., that the normalized pole counting mea-sures µ n have a weak limit µ in the topology dual to C ( R ). A similar combination of assumptions,but with poles away from the convex hull of E , is used in [28, Chapter 6] to study rational interpo-lation. Some of our current work mirrors our work for orthogonal rational functions [10], but thatwork required a periodic sequence of poles. In this sense, in addition to studying a different extremalproblem, our current setting is more general. To the best of our knowledge all previous works alsoassumed that the sequence of divisors D ∞ n is monotonic. Let further ( x ∗ n ) ∞ n =0 be a sequence in R \ E which does not accumulate on E .The behavior of log | F n | is governed by the zero and pole distributions. This corresponds to twoRiesz representations, with log | F n | superharmonic (respectively, subharmonic) away from the set ofzeros (respectively, poles). The limiting pole distribution µ directly determines the root asymptoticsof the functions F n and the limiting zero distribution.We assume that E is not a polar set, i.e., the domain Ω = C \ E is Greenian, and we denote by G ( z, w ) = G E ( z, w ) the Green function and by ω E ( dz, x ) harmonic measure for this domain. Theorem 1.9 (Root asymptotics) . Assume that E is not a polar set, (1.14) holds, the measures µ n converge weakly to µ in the topology dual to C ( R ) and ( x ∗ n ) ∞ n =0 be a sequence in R \ E notaccumulating on E . Then uniformly on compact subsets of C \ R , lim n →∞ n log | F n ( z ) | = Z G E ( z, x ) dµ ( x ) . Moreover, w-lim n →∞ ν n = Z ω E ( dz, x ) dµ ( x ) . Our proof of root asymptotics relies on an explicit representation of F n in terms of the so-called n -extension E n = F − n ([ − , E ⊂ E n and monotonicity of the Green function, we obtain a Bernstein-Walshtype upper bound for F n in terms of the Green functions G E ( z, c ). This is the major differencebetween the L and the L ∞ setting. In the L setting [10] an asymptotic upper bound is equivalentto Stahl–Totik regularity of the measure, whereas in the L ∞ setting this bound holds for any n .As in [25, Corollary 1.2], this can be used to describe the behavior of the leading coefficient.Theorem 1.9 generalizes known polynomial results, which correspond to the degenerate poledistribution µ = δ ∞ . Another notable case, related to [10], is of a p -periodically repeating sequenceof poles µ = p P pj =1 δ c j .In Section 4, we prove so-called Szegő-Widom asymptotics for F n . To the best of our knowledge,all previous results are only for polynomial extremal problems. Let Ω be a domain in C whichcontains ∞ and E = ∂Ω be an analytic Jordan curve, T n the associated Chebyshev polynomial and B E denote the Riemann map that maps Ω → D and B E ( ∞ ) = 0, normalized so that lim z →∞ zB E ( z ) > 0. Faber [13] showed that uniformly on compact subsets of Ω lim n →∞ T n B n E = 1 . (1.15)In his landmark paper [29], Widom generalized this notion to multiply connected domains. In thefollowing let Ω be a domain in C which contains ∞ so that E = ∂Ω is not polar. We will describethe type of results for multiply connected domains, but refer the reader for the precise definitions SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 6 and statements to Section 4. The correct analog for the Riemann map for multiply connecteddomains is the so-called complex Green function B E ( z, ∞ ) = e − G E ( z, ∞ ) − i ^ G E ( z, ∞ ) , (1.16)where ^ G E ( z, ∞ ) denotes the harmonic conjugate of G E ( z, ∞ ). To be more precise, since G E ( z, ∞ )is harmonic, B E ( z, ∞ ) is first defined locally and then using the monodromy theorem [26, Theorem11.2.1] extended to a global multivalued analytic function in Ω . Due to the multivaluedness of B E ,one cannot expect that B n E T n converges to a single analytic function as in (1.15). For this reason,Widom considered a related character automorphic extremal problem. Let z ∈ Ω and let π ( Ω, z )denote the fundamental group of Ω with basepoint fixed at z , and π ( Ω ) ∗ the group of unitarycharacters of π ( Ω, z ); that is, group homomorphisms from π ( Ω, z ) into T := R / Z . If F is ananalytic function on Ω , then we call F ( π ( Ω ) ∗ -) character-automorphic with character α , if F ◦ ˜ γ = e πiα (˜ γ ) F, ∀ ˜ γ ∈ π ( Ω, z ∗ ) . Let H ∞ Ω ( α ) denote the space of analytic character-automorphic functions, F , in Ω which are uni-formly bounded, i.e., k F k Ω := sup z ∈ Ω | F ( z ) | < ∞ . (1.17)In his ‘69 paper [29], Widom considered the extremal problemsup { Re F ( x ∗ ) : F ∈ H ∞ Ω ( α ) , k F k Ω ≤ } (1.18)under the assumption that E is a finite union of C Jordan curves and arcs and showed existence anduniqueness of the extremizer; let us call this the Widom maximizer. Let χ n denote the characterof B n E and W n the Widom maximizer with character χ n for the extremal point x ∗ = ∞ . If E is thefinite union of C Jordan curves, Widom showed that uniformly on compact subsets of ΩB n E T n − W n → . If such type of convergence holds, we say T n has Szegő-Widom asymptotics. The cases of arcsturned out to be essentially harder and for non-real problems only very simple cases such as onearc of the unit circle [8] are known. If E ⊂ R the situation is essentially better, since in this casethere are many symmetry properties, which manifests in the fact that the extremal function is realand allows for the explicit representation of the type we will derive in (2.11). If E is a finite unionof intervals Christiansen, Simon and Zinchenko [7] showed that T n has Szegő-Widom asymptotics.In 1971 Widom [30] also showed that (1.18) has a non-trivial solution as long as Ω is of Parreau–Widom type. We will define this notion in Section 4, but mention at this place that it also includesinfinitely connected domains. Recently Christiansen, Simon, Yuditskii and Zinchenko [5] provedSzegő-Widom asymptotics for T n if E ⊂ R such that Ω is a regular Parreau–Widom domain withDirect Cauchy theorem and this was later also proved under the same assumptions for residualpolynomials [7].We point out that ( T n ) ∞ = n ( B E ( · , ∞ )) , which makes B n E T n analytic and in fact a normal family. Since by definition( F n ) ∞ ≤ D ∞ n , SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 7 in our setting B n E should be substituted by the product of complex Green functions associated tothe divisor D ∞ n , i.e. B ( n ) E ( z ) = e iφ n Y c D ∞ n ( c ) B E ( z, c ) , (1.19)where B E ( z, c ) = e − G E ( z, c ) − i ^ G E ( z, c ) (1.20)and the phase will be specified in Section 4. With this modification we prove: Theorem 1.10. Let Ω = C \ E be a regular Parreau–Widom domain so that the Direct Cauchytheorem holds in Ω . Assume further that (1.14) holds, the measures µ n converge weakly to µ in thetopology dual to C ( R ) and ( x ∗ n ) ∞ n =0 be a sequence in R \ E without accumulation points in E . Then F n admits Szegő-Widom asymptotics. We want to highlight that this generalizes the known results in several ways. First of all, poly-nomials correspond to the case that D ∞ n = nχ {∞} and so the class of functions that we allow ismore general. Secondly, we allow for a sequence of extremal points x ∗ n , which in particular meansthat depending on n , F n might be a residual or a Chebyshev maximizer.2. Properties of the extremal rational functions In this section we study the extremal functions for fixed n . Let us begin by acknowledging thattheir existence follows by usual arguments. Namely, the leading coefficient λ n and the value F n ( x ∗ )are continuous functions of polynomial coefficients of F n R n . Since L n is finite-dimensional, the norm k·k E is mutually equivalent with a norm made from the polynomial coefficients, so Problem 1.3 isan extremal problem for continuous maps on the compact unit ball k·k E ≤ , R ) transformations. This willrequire the following claim from [27], for which we provide a short proof. Lemma 2.1. For every z ∈ C \ R and x ∈ R , there exists t ∈ R such that max z ∈ R (cid:12)(cid:12)(cid:12) ( z − t )( x − z )( z − z )( x − t ) (cid:12)(cid:12)(cid:12) = 1 and z = x is the unique maximum.Proof. Let f be a Möbius transformation mapping R to ∂ D with f ( z ) = 0. Since Möbius trans-formations preserve cross-ratios, (cid:12)(cid:12)(cid:12)(cid:12) ( x − z )( z − t )( x − t )( z − z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( x )( f ( z ) − f ( t ))( f ( x ) − f ( t )) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − f ( t ) f ( x ) − f ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . By choosing t so that f ( t ) = − f ( x ), we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − z )( z − t )( x − t )( z − z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − f ( t ) f ( x ) − f ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = | f ( z ) + f ( x ) | ≤ f ( z ) = f ( x ), i.e., z = x . (cid:3) In the next lemma, we consider the effect of a conformal transformation on the extremal problems,so we will emphasize dependencies on the poles, the point x ∗ and the set E where appropriate. Wedenote by F n ( z, E , D ∞ n ; x ∗ ) a maximizer for (1.8), and by L ( D ∞ n ) the space defined in (1.6). Fora divisor D and a a conformal map f ∈ PSL(2 , R ) we define the pushforward f ∗ D = D ◦ f − .Lemma 2.2 is an analog of [10, Lemma 2.1] adapted to the L ∞ extremal problem (1.8). SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 8 We would like to claim that the extremizers move by a conformal map f ∈ PSL(2 , R ) by F n ( f ( z ) , f ( E ) , f ∗ D ∞ n ; f ( x ∗ )) = F n ( z, E , D ∞ n ; x ∗ ) . However, this statement would be ambiguous until we prove uniqueness of extremizers, so we haveto formulate the claim more carefully: Lemma 2.2. Let f ∈ PSL(2 , R ) and let F n ( z, f ( E ) , f ∗ D ∞ n , f ( x ∗ )) be a maximizer of (1.8) for f ∗ D ∞ n , f ( E ) and f ( x ∗ ) . Then F n ( f ( z ) , f ( E ) , f ∗ D ∞ n , f ( x ∗ )) is a maximizer for (1.8) for D ∞ n , E and x ∗ .Proof. Möbius transformations preserve zeros and their multiplicity, i.e., for any rational function F and any w ∈ C , f − ∗ ( F ) w = ( F ◦ f ) w . Therefore, since pushforwards of integral divisors are integral, it follows from (1.6) that F ∈ L ( f ∗ D ∞ n ) = ⇒ F ◦ f ∈ L ( D ∞ n ) . (2.1)In particular, F n ( f ( z ) , f ( E ) , f ∗ D ∞ n , f ( x ∗ )) ∈ L ( D ∞ n ). Since F n ( z, f ( E ) , f ∗ D ∞ n , f ( x ∗ )) solves theextremal problem on f ( E ), we have k F n ( f ( · ) , f ( E ) , f ∗ D ∞ n , f ( x ∗ )) k E = k F n ( · , f ( E ) , f ∗ D ∞ n , f ( x ∗ )) k f ( E ) ≤ . It remains then to show F ( z ) := F n ( f ( z ) , f ( E ) , f ∗ D ∞ n , f ( x ∗ )) is an extremizer for n, E , D ∞ n and x ∗ . This will follow from showing that for d n > r ( f ( z ) , f ( x ∗ )) d n − c n r ( z, x ∗ ) d n ∈ L ( D ∞ n − x ∗ ) , (2.2) r ( f − ( z ) , x ∗ ) d n − c n r ( z, f ( x ∗ )) d n ∈ L ( f ( D ∞ n ) − f ( x ∗ )) , (2.3)for constants c n > 0. Indeed, given (2.2), (2.3), we suppose for the sake of contradiction there is a˜ F ∈ L ( D ∞ n ) with Re lim x → x ∗ ˜ F ( x ) r ( x,x ∗ ) dn > Re lim x → x ∗ F ( x ) r ( x,x ∗ ) dn . Then, since ˜ F ◦ f − ∈ L ( f ∗ D ∞ n ) by(2.1) and k ˜ F ◦ f − k f ( E ) ≤ , we contradict extremality of F ( z ).To show (2.2) and (2.3), we note that for the inversions z 7→ − z and the affine transformations z az + b , b ∈ R and a > 0, (2.2) and (2.3) follow by elementary computations. Since thesegenerate the group PSL(2 , R ), by writing f in this group as f = f ◦ f ◦ f , with f , f affine, f an inversion, and applying the argument immediately above three times, we have Lemma 2.2. (cid:3) Before we state one of the main theorems of the section, we recall that the set supp( f ) a is calledthe set of a -points of the function f . Polynomials or entire functions with real ± E = R \ [ i ( a i , b i ) , where ( a i , b i ) are the gaps of E , indexed by i from a countable indexing set. Theorem 2.3. Let F n be a maximizer for Problem 1.3. Let ( a , b ) be the gap containing x ∗ .(i) F n has only real generalized zeros.(ii) F n is real. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 9 (iii) For any distinct points x , x ∈ R such that D n ( x i ) ≥ , there is a point y ∈ E ∩ ( x , x ) with | F n ( y ) | = 1 .(iv) F n has only simple generalized zeros, i.e., D n ≤ .(v) F n has at most one generalized zero in each gap.(vi) F n has no generalized zeros in the gap ( a , b ) containing x ∗ .(vii) There is a unique extremizer F n .(viii) If F n is not constant, { z ∈ C : F n ( z ) ∈ [ − , } ⊂ R . In particular, all ± -points of F n lie on R .(ix) If F n is not constant, let m = deg F n and let the connected components of F − n (( − , becalled open bands of E n := F − n [ − , . Then, there are m open bands on E n , F n is strictlymonotonic on each of them and their endpoints account for all ± points.(x) F n ( a ) = ( − P c ∈ ( a ,x ∗ ) D ∞ n ( c ) and F n ( b ) = ( − P c ∈ [ x ∗ , b ) D ∞ n ( c ) , (xi) For any gap ( a i , b i ) containing a pole c i , either | F n ( b i ) | = 1 or | F n ( a i ) | = 1 . If D n ( c i ) = 1 , | F n ( b i ) | = | F n ( a i ) | = 1 . Remark. Note that (iii) is stronger than saying between two zeros of F n , we find an extremalpoint on the set; this statement provides extremal points between a zero and a pole c j at which F n has a reduction in order.Many of the statements in Theorem 2.3 will be proved by Markov correction arguments. We willcall a rational function M a Markov correction term if M F n ∈ L n and M ( x ∗ ) = 0. We will definethe rational function ˜ F n = (1 − ǫM ) F n , and note that˜ m n ( x ∗ ) = Re lim x → x ∗ ˜ F n ( x ) r ( x, x ∗ ) d n = Re lim z → x ∗ F n ( x ) r ( x, x ∗ ) d n = m n ( x ∗ ) . If there exists ε so that k ˜ F n k E < 1, then considering the rescaled function ˜ F n / k ˜ F n k E ∈ L n , we seethat ˜ m n ( x ∗ ) / k ˜ F n k E > m n ( x ∗ ), contradicting the extremality of F n . Proof of Theorem 2.3. All the conclusions are invariant under PSL(2 , R ) maps, so by Lemma 2.2,it suffices to consider the case x ∗ = ∞ . In this case, E is a compact subset of R .(i): Suppose for the sake of contradiction that there is a generalized zero z ∈ C \ R . Define˜ F n ( z ) = (cid:18) z − tz − z (cid:19) F n ( z )where t is selected so that max z ∈ R (cid:12)(cid:12)(cid:12) z − tz − z (cid:12)(cid:12)(cid:12) = 1, using Lemma 2.1 for x = ∞ . Since the maximum at ∞ is unique and E is compact, we have k ˜ F n k E < 1, and by the discussion above, this would be acontradiction.(ii): Since all poles and zeros of F n are real, we may write F n = A ˜ F n , where A ∈ C with | A | = 1and ˜ F n is real. It remains to show that A ∈ R . Note that ± ˜ F n are also admissible functions for theextremal problem. Since F n is extremal and ˜ F n is real, we haveRe lim x → x ∗ A ˜ F n ( x ) r ( x, x ∗ ) d n = Re lim x → x ∗ F n ( x ) r ( x, x ∗ ) d n ≥ ± Re lim x → x ∗ ˜ F n ( x ) r ( x, x ∗ ) d n = ± lim x → x ∗ ˜ F n ( x ) r ( x, x ∗ ) d n . Since Re lim x → x ∗ F n ( x ) r ( x,x ∗ ) dn = 0, we conclude that | Re( A ) | ≥ A ∈ { , − } .(iii): We have sup E ∩ ( x ,x ) | F n | = sup E ∩ [ x ,x ] | F n | = max E ∩ [ x ,x ] | F n | . Since F n is continuous on E we only have to explain the first equality. We only argue for x since x follows analogously. We SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 10 distinguish two cases. If F n ( x ) = 0, then clearly the sup is not changed by adding x . If instead D ∞ n ( x ) > 0, then x / ∈ E and ( x , x ) ∩ E = [ x , x ) ∩ E .Now, we assume for the sake of contradiction that max E ∩ [ x ,x ] | F n | < 1. Recalling that x ∗ = ∞ so that D n ( ∞ ) = 0, define the Markov correction term M ( z ; x , x ) = ( z − x )( z − x ) , x < x z − x )( x − z ) , x > x By distinguishing again the cases F n ( x i ) = 0 and D ∞ n ( x i ) > 0, we see that in either case M F n iscontinuous on E ∩ [ x , x ]. Thus, by our assumption we find ε > [ x ,x ] ∩ E | ˜ F n | < E , the norm is lowered, we may conclude by contradiction.(iv): Clearly, D n ( x ∗ ) = 0. With our convention x ∗ = ∞ and by (i), all generalized zeros are in R . Suppose x ∈ R with D n ( x ) ≥ 2. First, we take x E . We define the Markov correction term M ( z, x ) = z − x ) . If x / ∈ E , z → M ( z, x ) is continuous on E and so we may find an ǫ > k ˜ F n k E < 1. If instead, x ∈ E , then we conclude as in (iii) by continuity of M F n that we may finda small enough ǫ > k ˜ F n k E < E .(vi): Assume there is a zero in R \ [ b , a ]. We use the Markov correction term M ( z ; x ) = ( z − x , x < b x − z , x > a which is continuous and strictly positive on E . By continuity and compactness, for all small enough ǫ > k − ǫM k E < 1, so ˜ F n = (1 − ǫM ) F n once again contradicts extremality.(vii): Assume that there are two extremizers F n , F n . By convexity, T n = ( F n + F n ) is then alsoan extremizer. Let y i ∈ E be the points given by (iii) with | T n ( y i ) | = 1. We note that by (iv) thereare n such points. Then since | F n ( y i ) | , | F n ( y i ) | ≤ | T n ( y i ) | = 1, F n ( y i ) = F n ( y i ) = T n ( y i ) sothat F n ( y i ) − F n ( y i ) = 0. Define H n = F n − F n and let D n denote its divisor of generalized zeros.Then D n ( x ∗ ) ≥ D n ( y j ) ≥ D n ≥ n + 1. Since H n ∈ L n , thisimplies H n ≡ F n = F n .(viii): We write F n in reduced form as F n = PQ , with deg( P ) = m and note that deg Q ≤ m sothat deg( F n ) = m . If F n is nonconstant, we use a counting argument.Take two consecutive zeros of F n , x and x . If there is no pole between them, there must bea critical value y and by (iii), it must obey | F n ( y ) | ≥ 1. Separating cases by whether | F n ( y ) | = 1,we either obtain an (at least) double zero of F n − y , or zeros on intervals ( x , y ) and ( y, x ).Similarly, if there is a pole y ∈ ( x , x ), by continuity there are ± x , y ) and( y, x ).Thus, counted with multiplicity, there are at least two ± m simplezeros of P partition R into m such intervals, so we have at least 2 m total ± F n ) = m , this construction gives all the ± F n . In particular, this now also holds for the set of ± a -points for any a ∈ [ − , I kn be the connected components of the open set F − n (( − , a ∈ ( − , ± a -points are simple. Thus, if F n ( x ) = ± a , then F ′ n ( x ) = 0, so bycontinuity, the derivative has the same sign on each open band I kn . In particular, F n ( I kn ) = ( − , k and there are m connected components, F − n (( − , ∪ mk =1 I kn . That the endpoints of I kn account for all ± SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 11 (x): First we show the modulus is 1 at each point. If F n ≡ 1, this is clear. If deg( F n ) ≥ 1, wewill make use of the zeros of F n . Suppose for the sake of contradiction that | F n ( b ) | < 1. Then,define x := min { y : F n ( y ) = 0 } , with x ≥ b by (vi). We have sup z ∈ [ b ,x ] | F n ( z ) | < M ( z, x ) = x − z and note that M ≤ b , x ]. By the same argumentsas (iii) we derive a contradiction. The same argument at a shows | F n ( a ) | = 1.By (vi), the sign changes on ( a , ∞ ) can only occur at the poles contained in this interval, whichwe order as c n < · · · < c n m . By (vi), F n has no reduction of order at the poles at the c n i , so for a t ∈ ( c n m , ∞ ), sgn( F n ( a )) = ( − P c ∈ ( a , ∞ ) D ∞ n ( c ) sgn( F n ( t )). By our definition of r ( z, c ), F n > c n m , ∞ ). Since | F n ( a ) | = 1 by our work above, this proves the claim at a . Similar analysis at b ,with the modification that the parity of d n contributes to the sign, completes the proof.(xi): If F n is constant, F n ≡ F n nonconstant. By(x), it suffices to consider gaps ( a i , b i ) = ( a , b ). If | F n ( a i ) | = 1 there is nothing to prove. If | F n ( a i ) | < 1, it follows from monotonicity on the bands and lim x →∞ F n ( x ) > x < a i with F n (˜ x ) = 0. Similar considerations hold for b i . If max {| F n ( a i ) | , | F n ( b i ) |} < 1, let x := max { y : y < c i , D n ( y ) = 1 } and x := min { y : y > c i , D n ( y ) = 1 } . By (iii) there must be y ∈ ( x , x ) ∩ E , with | F n ( y ) | = 1. As in the proof of (x), we conclude from monotonicity on thebands that that either y = a i or y = b i . If D n ( c i ) = 1, we conclude in the same way that there is y ∈ ( x , c ) and y ∈ ( c , x ) with | F n ( y j ) | = 1 and finally that y = a i and y = b i . (cid:3) Theorem 2.4. Let F ∈ L n be real and D n its generalized zero divisor. Then, any set of alternationpoints has at most n + 1 − D n ( x ∗ ) points.Proof. Set m = D n ( x ∗ ) ≥ y j ∈ R \ { x ∗ } be the k points with D n ( y j ) > 0, where regardlessof its multiplicity each point appears only once. Since deg D n = n , we see that k ≤ n − m . Adding x ∗ to this list, we cyclically order the points as ( x ∗ , y , . . . , y k ). We note that these points cannotbe part of an alternating set, as they either are zeros of F , or coincide with some c i / ∈ E or x ∗ / ∈ E .We also write y = y k +1 = x ∗ .Fix 1 ≤ j ≤ k + 1. On the interval ( y j − , y j ), F has no generalized zeros, so the sign changesof F only occur at poles, according to the divisor D ∞ n : if ( x , x ) ⊂ ( y j − , y j ), and x , x are notpoles, then F ( x ) = ( − P c ∈ ( x ,x D ∞ n ( c ) F ( x ) = ( − S n ( x ) − S n ( x ) F ( x ) . (2.4)Thus, x , x cannot be two consecutive points of the same alternation set, because by the definitionof alternation set, this would imply F ( x ) = ( − S n ( x ) − S n ( x ) F ( x ) and lead to contradiction.Thus, any alternation set has at most one point in each interval ( y j − , y j ) for 1 ≤ j ≤ k + 1, so anyalternation set has at most k + 1 ≤ n − m + 1 alternation points. (cid:3) The above theorem justifies the following definition. Definition 2.5. We say that F n has a maximal set of alternation points if it has a set of alternationpoints of size n + 1. Theorem 2.6. If F n is the maximizer for (1.8) , then it has a maximal set of alternation points.Proof. Due to Theorem 2.3(vi), D n ( y ) = 0 for all y ∈ ( a , b ) and therefore, using (1.11) andTheorem 2.3(ii),(iv), there is a cyclically ordered sequence ( b , y , . . . , y n , a ), so that D n ( y i ) = 1.By Theorem 2.3(iii), for 2 ≤ j ≤ n , there is a point x j ∈ ( y j − , y j ) and x j ∈ E , so that | F n ( x j ) | = 1.We claim that together with x n +1 = a and x = b these points form a maximal set of alternationpoints. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 12 We start with x n +1 and x . Let ( a , b ) be the gap containing x ∗ . We have S n ( a ) = X c ∈ ( a ,x ∗ ) D ∞ n ( c ) . Thus, it follows directly from Theorem 2.3(x) that x n +1 = a is an alternation point. Similarly, wesee that S n ( b ) = X c ∈ ( b ,x ∗ ) D ∞ n ( c )and therefore since deg D n = P c D ∞ n ( c ) = n and D ∞ n ( b ) = 0, n + 1 − − S n ( b ) = X c D ∞ n ( c ) − X c ∈ ( b ,x ∗ ) D ∞ n ( c ) = X c ∈ [ x ∗ , b ) D ∞ n ( c ) . Thus, again by Theorem 2.3(x), also x = b is an alternation point in the above sense.Now for j ≥ x j , x j +1 and y j ∈ ( x j , x j +1 ) and assume that x j is an alternation point. Notethat all sign changes of F n correspond either to a pole of F n or to y j . Thus, F n ( x j ) = ( − P c ∈ ( xj,xj +1) D ∞ n ( c ) F n ( x j +1 ) . (2.5)This is easily seen if D ∞ n ( y j ) = 0. If D ∞ n ( y j ) > 0, then ( F n ) ∞ ( y j ) = D ∞ n ( y j ) − S n ( x j ) − S n ( x j +1 ) = X c ∈ ( x j ,x j +1 ) D ∞ n ( c ) . Therefore, x j +1 is also an alternating point. Thus, by induction we conclude that { x i } n +1 i =1 form amaximal set of alternation points for F n . (cid:3) We also have a form of converse to Theorem 2.6, which we prove as the following theorem. Theorem 2.7. If F ∈ L n is real and has a maximal alternation set, then F is the unique maximizerfor Problem 1.3.Proof. Let F ∈ L n be real and suppose that it has a maximal set of alternation points { x , . . . , x n +1 } .By relabeling, we assume the cyclic ordering ( x ∗ , x , . . . , x n +1 ). By Theorem 2.4, if F has an alterna-tion set with n +1 points, then ( F ) ∞ ( x ∗ ) = d n . Therefore, we can define lim x → x ∗ F ( x ) /r ( x, x ∗ ) d n =: α n ∈ R \ { } . It is convenient to rephrase our extremal problem: F n solves (1.3) if and only if˜ F n := λ n F n solves the dual probleminf {k ˜ F n k E : lim x → x ∗ ˜ F n ( x ) r ( x, x ∗ ) d n = 1 , ˜ F n ∈ L n } . (2.6)By this duality and Theorem 2.3(vii), it will suffice to show ˜ F := α n F is also an extremizer for (2.6); k ˜ F k E = k ˜ F n k E . Suppose that k ˜ F k E > k ˜ F n k E . We define ˜ H n = ˜ F − ˜ F n and denote its generalizedzero divisor by D n . Our normalization implies that D n ( x ∗ ) ≥ 1. Since sgn( H n ( x j )) = sgn( F ( x j )),we have sgn( H n ( x j )) = ( − n +1 − j − S n ( x j ) for 1 ≤ j ≤ n + 1. By the computation (2.4), we concludethat there must be y j ∈ ( x j , x j +1 ) with D n ( y j ) ≥ ≤ j ≤ n . Thus, deg( D n ) ≥ n + 1, whichcontradicts H n ∈ L n . (cid:3) In particular, the proof of Theorem 1.6 is now complete and we may prove Corollary 1.8. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 13 Proof of Corollary 1.8. We let { x , . . . , x n +1 } be an alternation set for F n and the point x ∗ , withcyclic ordering ( x ∗ , x , . . . , x n +1 ), where we recall that x = a and x n +1 = b . By definition of S n we see that for 1 ≤ ℓ ≤ n we have F n ( x ℓ ) F n ( x ℓ +1 ) = ( − P c ∈ ( xℓ,xℓ +1) D ∞ n ( c ) . (2.7)However, F n ( a ) F n ( b ) = ( − P c ∈ ( a , b ) D ∞ n ( c ) , (2.8)which is easier to see by using the expressions in Theorem 2.3(x). The difference between (2.7) and(2.8) is manifested in the fact that S n is anchored at x ∗ ∈ ( a , b ). Moreover, by Theorem 1.6, ifthere exists a set { x , . . . , x n +1 } which can be cyclically ordered so that F n satisfies (2.7) and (2.8),then for any ˜ x ∗ ∈ ( a , b ) up to a factor ± F n is the maximizer of (1.8).Denote by x ∗ j a point in the gap ( a j , b j ). Let S jn ( x ) := P c ∈ R \{ x j ∗ } D ∞ n ( c ) χ [ x j ∗ , c ) ( x ). There is1 ≤ k ≤ n so that x j ∗ ∈ ( x k , x k +1 ). Let us order the y i with D n ( y i ) = 1 cyclically as ( a , b , y , . . . , y n ).The assumption D n = 0 on ( a j , b j ) implies a j , b j ∈ ( y i , y i +1 ) for some 1 ≤ i ≤ n − 1. By (ix), a j and b j are the only points in ( y i , y i +1 ) with | F n | = 1, and since there is exactly one of the x i in each of the ( y i , y i +1 ), one and only one of a j and b j is in the alternation set { x , . . . , x n +1 } .Without a loss of generality we take x k = a j . We now claim the set { x , . . . , x k , b j , x k +1 , . . . x n } will form our alternation set. Since S jn is now anchored at x ∗ j ∈ ( a j , b j ), we need to check (2.8) forthe gap ( a j , b j ). From the assumption that D n = 0 on ( a j , b j ) it follows that F n ( a j ) F n ( b j ) = ( − P c ∈ ( a j, b j ) D ∞ n ( c ) . By the assumption that { x , . . . , x n +1 } form an alternation set (for S n ), (2.7) (for S jn ) is clearly sat-isfied for { x , . . . , x k } and for { x k +1 , . . . , x n } . Using again that { x , . . . , x n +1 } form an alternationset and that a j = x k , we have F n ( x k +1 ) = ( − P c ∈ ( xk,xk +1) D ∞ n ( c ) F n ( x k ) = ( − P c ∈ ( xk,xk +1) D ∞ n ( c ) ( − P c ∈ ( a j, b j ) D ∞ n ( c ) F n ( b j )= ( − P c ∈ ( b j,xk +1) D ∞ n ( c ) F n ( b j ) . Thus, (2.7) is also satisfied for x k +1 and b j . Similarly we can check (2.7) for x n and x and concludethat up to a factor of ± F n is also extremal for x ∗ j . (cid:3) Remark. In the above argument, one could have removed x and kept x n +1 to form an alternationset for x j ∗ .Next, we describe when the extremizer is constant: Proof of Theorem 1.7. Suppose E takes the above form. Without a loss of generality we assumethat ( x ∗ , c , . . . , c n ) are cyclically ordered. Then, ( b , a , x , . . . , x n − ), where x ℓ ∈ E ∩ ( c ℓ , c ℓ +1 )for 1 ≤ ℓ ≤ n − F n ≡ 1. By Theorem 2.7, F n is the maximizer for(1.8).Suppose now the set is not of the above form. If there is a c j with D ∞ n ( c j ) ≥ 2, by (iv), theextremizer F n is nonconstant. If there are two distinct poles c i and c j in a single gap, then F n cannot be constant by (v). In either case, F n is nonconstant. (cid:3) We record a final corollary of Theorem 2.6. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 14 Corollary 2.8. If the extremal function F n is not constant, then deg F n ≥ ⌈ n +12 ⌉ .Proof. By Theorem 2.6, F n has at least ⌈ n +12 ⌉ points with | F n | = 1 with the same sign. Thus, if F n is nonconstant, it has degree at least ⌈ n +12 ⌉ , and we can have at most ⌊ n − ⌋ cancellations. (cid:3) The set E n = F − n ([ − , n extension of E . Note that by definition it is anextension, i.e., E ⊂ E n . Theorem 2.3, particularly the locating of ± u i , v i ∈ R \ ˚ E with v i ∈ [ a i , b i ] and u i ∈ [ a i , v i ]. Then Theorem 2.9. For F n nonconstant, the n extension of E is of the form E n = E ∪ [ i ≥ [ u i , v i ] with [ u i , v i ] ⊆ [ a i , b i ] .The following cases are possible:(1) The gap remains unchanged, corresponding to u i = v i = a i .(2) E is extended on one edge, corresponding to a i = u i and v i = u i , v i = b i , or on the otherside, v i = b i and u i = a i , u i = v i .(3) An internal interval is added, corresponding to [ u i , v i ] ⊂ ( a i , b i ) , u i = v i .(4) The gap ( a i , b i ) may close, corresponding to a i = u i and b i = v i .Moreover, in the following cases there is not extension into a gap:(i) If x ∗ ∈ ( a i , b i ) , then this gap remains unchanged, i.e., u i = v i = a i .(ii) If there is a pole c i ∈ ( a i , b i ) and D n ( c i ) = 1 , then this gap remains unchanged, i.e., u i = v i = a i . Remark. (i) As we will see in the proof, for gaps ( a i , b i ) containing poles of F n , which isguaranteed for D ∞ n ( c i ) ≥ E n as in (3) above, then this is always related to a zero x i of F n and moreover | F n ( a i ) | = | F n ( b i ) | = 1. Clearly, if this zero approaches a pole, the intervalaround it becomes smaller. In this sense (ii) of the above theorem can be viewed as a limit ofsuch situations, where the additional interval degenerates to a point. Proof. Applying conformal invariance of the setting, we assume again that x ∗ = ∞ and E is acompact subset of R . Since we will prove (i) independently, we can assume that all extensionsoccur in bounded gaps. We first note that any internal interval cannot degenerate to a point, i.e.when u i = v i in (3), since due to 2.3(ix) there are m open bands and their endpoints accountfor all ± I k = ( y − k , y + k ), k = 1 , 2, so that y +1 , y − / ∈ E . Let x k denote the simple zero of F n on these open bands. Using that | F n | < I k and y +1 , y − / ∈ E , we see that max [ x ,x ] ∩ E | F n | < 1, contradicting Theorem 2.3(iii).Let us now prove (i): Due to (x) and (ix) of Theorem 2.3, an extension would contain an openband that lies entirely in ( a i , b i ). Therefore in particular, this would lead to a zero of F n on theextremal gap contradicting (vi) of Theorem 2.3.It remains to prove (ii): In this case again due to (xi) and (ix) of Theorem 2.3, this would leadto an open band that lies entirely in ( a i , b i ) forcing F n to have an additional zero in this gap. Butsince already D n ( c i ) = 1, this would contradict (v) of Theorem 2.3. (cid:3) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 15 In the following let us assume that F n is nonconstant so that C \ E n is Greenian. Note thatdue to Theorem 2.9(3), E n is a finite union of proper intervals and in particular is regular for theDirichlet problem. We define B n ( z ) = e iφ n Y c ( F n ) ∞ ( c ) B E n ( z, c ) , (2.9)and normalize the phase of B n by the conditionlim x → x ∗ B n ( x ) r ( x, x ∗ ) d n > . (2.10)Recall that in general B E n ( z, c ) define multivalued functions. However, we will show that theirproduct B n ( z ) is in fact single valued in C \ E n . Theorem 2.10. B n is a single-valued analytic function on C \ E n and F n ( z ) = 12 (cid:18) B n ( z ) + 1 B n ( z ) (cid:19) . (2.11) Proof. Recall that E n = { z ∈ C : F n ( z ) ∈ [ − , } . Therefore, since the Joukowsky map J ( ζ ) = (cid:16) ζ + ζ (cid:17) maps D conformally onto C \ [ − , Ψ n ( z ) = J − ( F n ( z )) , is well defined and single-valued in C \ E n . Moreover, for x ∈ E n , lim z → x | Ψ n ( z ) | = 1 and Ψ n ( z ) hasa zero of multiplicity ( F n ) ∞ ( c ) at each c . Thus, we conclude by the maximum principle that − log | Ψ n ( z ) | = X c ( F n ) ∞ ( c ) G E n ( z, c ) = − log | B n ( z ) | . Thus, by adding the complex conjugate, B n is defined up to a unimodular constant c . Finally,0 < lim x → x ∗ F n ( x ) r ( x, x ∗ ) d n = 12 lim x → x ∗ (cid:18) cB n ( x ) r ( x, x ∗ ) d n + 1 cB n ( x ) r ( x, x ∗ ) d n (cid:19) = 12 lim x → x ∗ cB n ( x ) r ( x, x ∗ ) d n . Using the normalization (2.10), we conclude c = 1 and obtain (2.11). (cid:3) This has the following consequence: Lemma 2.11. Let F n be represented as in (2.11) and let I n be an open band of E n . Then X c ( F n ) ∞ ( c ) ω E n ( I n , c ) . (2.12) Proof. Recall that F n ( z ) = J ( B n ( z )) (2.13)and that F n is strictly monotonic on I n . That is, either F n increases from − − I n = ( a, b ). Since J : ∂ D ∩ C ± → ( − , 1) bijectively, itfollows from the definition of I n and (2.13) that | arg B n ( b ) − arg B n ( a ) | = π. By using the Cauchy-Riemann equations, we getarg B n ( b ) − arg B n ( a ) = Z ba ∂G n ( x ) ∂n dx. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 16 On the other hand ω E n ( dx, c ) = 1 π ∂G E n ( x, c ) ∂n dx. Thus, we get arg B n ( b ) − arg B n ( a ) = π X c ( F n ) ∞ ( c ) ω E n ( I n , c )and the claim follows. (cid:3) We finish this section with a Bernstein-Walsh lemma for rational functions. Lemma 2.12. Let K ⊂ C be a compact, nonpolar set such that C \ K is connected. Let h be ameromorphic function on C Then, | h ( z ) |k h k K ≤ e P c ( h ) ∞ ( c ) G K ( z, c ) (2.14) If we assume in addition that K ⊂ R and that h is real, then | h ( z ) |k h k K ≤ (cid:16) e P c ( h ) ∞ ( c ) G K ( z, c ) + e − P c ( h ) ∞ ( c ) G K ( z, c ) (cid:17) . (2.15) Proof. For (2.14) we follow the standard proof of the Bernstein-Walsh lemma. Set H = h/ k h k K and consider F ( z ) = log | H ( z ) | − P c ( h ) ∞ ( c ) G K ( z, c ). Then, F is subharmonic in Ω = C \ K andfor q.e. ζ ∈ ∂Ω we have lim sup z → ζ F ( z ) ≤ 0. Moreover, if V c are vicinities of the points with( h ) ∞ ( c ) > V = ∪ c V c , then log | H ( z ) | is subharmonic on C \ V and thus bounded above by[19, Theorem 2.1.2]. Since the logarithmic pole on V c is canceled, F is also bounded above on V and we conclude from the maximum principle [14, Theorem 8.1] that F ( z ) ≤ Ω .Assume that H is real and that K is real. Define K H = { z ∈ C : H ( z ) ∈ [ − , } , but note that K is not necessarily a subset of R . However, using that H is real, we have that K ⊂ K H . Now, asin the proof of Theorem 2.10 we see that H ( z ) = 12 (cid:16) e G H ( z )+ i ^ G H ( z ) + e − ( G H ( z )+ i ^ G H ( z )) (cid:17) , G H ( z ) = X c ( h ) ∞ ( c ) G K H ( z, c ) . Let us also put G ( z ) = P c ( h ) ∞ ( c ) G K ( z, c ℓ ). Then it follows from the monotonicity of Greenfunctions with respect to the domain that for z ∈ C \ K H , we have | H ( z ) | = (cid:12)(cid:12)(cid:12) cosh (cid:16) G H ( z ) + i ^ G H ( z ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ cosh G H ( z ) ≤ cosh G ( z ) . Note that for z ∈ K H \ K , G ( z ) > z . This finishes theproof. (cid:3) We point out that (2.14) is an analog of the standard Bernstein-Walsh lemma, whereas (2.15)is a fairly recent improvement of Schiefermayr for real polynomial problems [23]. Note that thisalso implies that (2.15) holds for x ∗ ∈ R \ K , without the extra assumption on h n to be real. Thisfollows from Theorem 2.3, where we showed that the residual extremizer is always real. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 17 Root asymptotics We now turn to the study of the limiting behavior of F n as n → ∞ . We assume (1.14) andassume that the measures (1.12) have a limit µ = w-lim n →∞ µ n in the topology dual to C ( R ) and that ( x ∗ n ) is a sequence in R \ E without accumulation points in E . Note that (1.14) implies supp µ ∩ E = ∅ . Let further ν n be the normalized counting measure ofgeneralized zeros of F n , i.e, ν n = 1 n X x D n ( x ) δ x . Define the family of functions h n ( z ) = 1 n log | F n ( z ) | (3.1)and note that h n is subharmonic in C \ supp D ∞ n ; in particular, all functions h n are subharmonic in Ω C = C \ K C . We start with an upper estimate: Lemma 3.1. For any z ∈ C \ R we have lim sup h n ( z ) ≤ Z G E ( z, x ) dµ ( x ) . (3.2) Proof. Due to Lemma 2.12 and the definition of µ n we have h n ( z ) ≤ Z G E ( z, x ) dµ n ( x ) . On the other hand, since µ n → µ and by continuity of G E ( z, y ) on K C we havelim n →∞ Z G E ( z, x ) dµ n ( x ) = Z G E ( z, x ) dµ ( x ) . (cid:3) We continue with some facts about potentials. Lemma 3.2. Let E ( R be closed and not polar so that Ω = C \ E is Greenian and µ be a probabilitymeasure supported on R with supp µ ∩ E = ∅ . Then R G E ( z, x ) dµ ( x ) defines a positive superharmonicfunction in Ω and a harmonic function in Ω \ supp µ . Moreover, as a harmonic function, it has aunique subharmonic extension to C \ supp µ , which vanishes q.e. on E .Proof. If supp µ ⊂ R , it follows from [21, Theorem II.5.1] and the minimum principle for super-harmonic functions that R G E ( z, x ) dµ ( x ) defines a positive superharmonic function in Ω and aharmonic function in Ω \ supp µ that vanishes q.e. on E . In particular, locally in vicinities of E itis subharmonic and vanishes away from a polar set. Thus, by [1, Theorem 5.2.1.], for ζ ∈ E Z G E ( ζ, x ) dµ ( x ) = lim sup z → ζ Z G E ( z, x ) dµ ( x )defines the unique subharmonic extension to Ω \ supp µ ; since all claims are conformally invariant,the general case follows. (cid:3) Lemma 3.3. The set K C intersects only finitely many open gaps. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 18 Proof. K C is a closed subset of R , so it is compact. It is contained in R \ E , so its cover by theopen sets ( a j , b j ) has a finite subcover; in other words, K C only intersects finitely many gaps. (cid:3) We obtain immediately the following corollary: Corollary 3.4. For n sufficiently large, F n is non-constant.Proof. Because D n ≤ D ∞ n = n . (cid:3) Since we are interested in asymptotics of F n as n → ∞ , we assume from now on that F n isnon-constant. Lemma 3.5. Fix an open set O ⊂ R \ E so that µ ( O ) > . Then lim n →∞ X c ∈ O D ∞ n ( c ) = + ∞ . (3.3) Proof. By definition, µ n ( O ) = 1 n X c ∈ O D ∞ n ( c ) . By the Portmanteau theorem, lim inf n →∞ µ n ( O ) ≥ µ ( O ) > 0, solim inf n →∞ n X c ∈ O D ∞ n ( c ) > (cid:3) The following analog of Koosis’s formula for the Martin or Phragmén Lindelöf function [16,Theorem on page 407] will be very useful. It was already used in [5, Proposition 4.3]. Lemma 3.6. Let E ⊂ E ⊂ R so that E is not polar and let c ∈ R \ E . Then, G E ( z, c ) − G E ( z, c ) = Z E \ E G E ( z, x ) ω E ( dx, c ) . (3.4) Proof. Since (3.4) is conformally invariant, by applying a conformal map we can assume that ∞ ∈ E , i.e., Ω = C \ E ⊂ C . Define also Ω = C \ E . Since the logarithmic pole at c is canceled, G E ( z, c ) − G E ( z, c ) defines a superharmonic function on Ω which is bounded. Moreover, itsRiesz measure is given by ω E ( dx, c ) | Ω . Since E ⊂ E it follows by the maximum principle that G E ( z, c ) − G E ( z, c ) ≥ 0. Thus, in particular it has a nonnegative subharmonic minorant in Ω and it follows by the Riesz decomposition theorem that G E ( z, c ) − G E ( z, c ) = Z Ω G E ( z, x ) ω E ( dx, c ) + u ( z ) , where u is the greatest harmonic minorant of G E ( z, c ) − G E ( z, c ). We have already seen that u ≥ 0. On the other hand, since E is the boundary for Ω and Ω it follows that for q.e. x ∈ E we have lim sup z → x u ( z ) ≤ lim sup z → x ( G E ( z, c ) − G E ( z, c )) = 0 . Thus, u is a bounded harmonic function in Ω which vanishes q.e. on E . It follows by the maximumprinciple [14, Corollary 8.3] that u = 0 and we obtain (3.4). (cid:3) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 19 Compared to the standard Chebyshev problem, we encounter a technical difference for residualextremal functions. Let ( a i , b i ) be a gap so that (3.3) is satisfied for O = ( a i , b i ). We want toestimate G E n ( z, c ) for c ∈ ( a i , b i ). But since ( a i , b i ) is not necessarily the extremal gap, there canbe an extension ( u i , v i ) in this gap, which intuitively makes G E n ( z, c ) smaller if [ u i , v i ] is close to c . However, we have already encountered in Theorem 2.9(ii), that a cancellation of a pole can beregarded as a degenerated internal interval. Thus, we are led to expect that an additional intervalcan have no more “effect” than reducing the number of Green functions in the sum by one. This isthe content of the following lemma: Lemma 3.7. Let E n and u i , v i be defined as in Theorem 2.9. Fix a gap ( a i , b i ) and define E in = E n \ ( a i , b i ) . Let z ∈ C \ E n and ( F n ) ∞ ( z ) = 0 . Then there is a t ∈ [ a i , b i ] such that G E ( t, z ) =max x ∈ [ a i , b i ] G E ( x, z ) and we have X c ( F n ) ∞ ( c ) G E n ( z, c ) ≥ X c ( F n ) ∞ ( c ) G E in ( z, c ) − G E ( z, t ) . (3.5) In particular, if z / ∈ ( a i , b i ) , then lim n →∞ X c ( F n ) ∞ ( c ) G E in ( z, c ) = ∞ = ⇒ lim n →∞ X c ( F n ) ∞ ( c ) G E n ( z, c ) = ∞ . (3.6) Proof. If z ∈ ( a i , b i ), then t = z and (3.5) is trivial. Thus, let z / ∈ ( a i , b i ).Since E n is a finite union of intervals it is clearly not polar and putting E n \ E in = [ u i , v i ], weobtain from Lemma 3.6 that G E in ( z, c ) − G E n ( z, c ) = Z v i u i G E in ( z, x ) ω E n (d x, c ) . (3.7)By [19, Theorem 2.1.2] a subharmonic function attains its maximum on compacts and thus t is welldefined. Define ρ n ( dx ) = X c ( F n ) ∞ ( c ) ω E n ( dx, c ) , and note that it follows from Theorem 2.9 and Lemma 2.11 that ρ n ([ u i , v i ]) ≤ 1. Moreover, by themaximum principle G E in ( z, x ) ≤ G E ( z, x ) . Thus, X c ( F n ) ∞ ( c ) Z v i u i G E in ( z, x ) ω E n (d x, c ) = Z v i u i G E in ( z, x ) ρ n ( dx ) ≤ Z v i u i G E ( z, x ) ρ n ( dx ) ≤ G E ( z, t ) . Combining this with (3.7) yields (3.5). (cid:3) By the representation (2.11), we have h n ( z ) = − n log | B n ( z ) | − n log 2 + 1 n log | B n ( z ) | . (3.8)The next lemma shows that the the asymptotics of h n for n → ∞ are determined by the term − n log | B n ( z ) | . In fact, we even prove a stronger statement, which will be needed in Section 4. SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 20 Lemma 3.8. Uniformly on compact subsets of C \ R we have lim n →∞ log (cid:12)(cid:12) B n ( z ) (cid:12)(cid:12) = 0 . (3.9) If we pass to a subsequence such that lim ℓ →∞ x ∗ n ℓ = x ∗∞ and ( a , b ) denotes the gap containing x ∗∞ ,then also for z ∈ ( a , b ) lim ℓ →∞ log (cid:12)(cid:12) B n ℓ ( z ) (cid:12)(cid:12) = 0 . (3.10) Proof. Consider B n as an analytic single-valued function on C + or C − and note that 0 < | B n ( z ) | < 1. Thus, log (cid:12)(cid:12) B n ( z ) (cid:12)(cid:12) = Re log(1 + B n ( z ) ) defines a family of harmonic functions which isuniformly bounded from above. Thus, by the Harnack principle, the family is precompact in thespace of harmonic functions together with the function which is identically −∞ . Therefore, itsuffices to show that pointwise for fixed z every subsequence has a subsequence so that (3.9) holds.Let us pass to a subsequence so that lim ℓ →∞ x ∗ n ℓ = x ∗∞ and let ( a , b ) denote the gap containing x ∗∞ . If necessary, we pass to a further subsequence so that x n ℓ ∈ ( a , b ) for all ℓ > 0. Since for | z | < | log( | z | ) | = | Re(log(1 + z )) | ≤ | log(1 + z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z z zt d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ | z | − | z | , it suffices to show that lim ℓ →∞ | B n ℓ ( z ) | = 0 . By (2.9) and (1.20) this is equivalent tolim ℓ →∞ X c ( F n ℓ ) ∞ ( c ) G E nℓ ( z, c ) = + ∞ . (3.11)Since E ∩ K C = ∅ , we find a gap ( a i , b i ) and ε > µ (( a i + ε, b i − ε )) > 0. Thus, byLemma 3.5 we have lim ℓ →∞ X c ∈ ( a i + ε, b i − ε ) D ∞ n ℓ ( c ) = + ∞ . (3.12)Note that it could be that ( a i , b i ) = ( a , b ), which causes no problems in the following.Set E i = R \ (( a , b ) ∪ ( a i , b i )) and E in ℓ = E n ℓ \ ( a i , b i ). By Theorem 2.9(i), E in ℓ ⊂ E i , so themaximum principle yields G E i ( z, c ) ≤ G E inℓ ( z, c ) . (3.13)Fix z ∈ C + ∪ C − ∪ ( a , b ) and note that lower semicontinuity implies0 < δ = min c ∈ [ a i + ε, b i − ε ] G E i ( z, c ) . Then, by Theorem 2.3(v) X c ∈ ( a i + ε, b i − ε ) ( F n ℓ ) ∞ ( c ) G E i ( z, c ) ≥ δ (cid:18) − X c ∈ ( a i + ε, b i − ε ) D ∞ n ℓ ( c ) (cid:19) . Thus, by (3.12) we obtain lim ℓ →∞ X c ∈ ( a i + ε, b i − ε ) ( F n ℓ ) ∞ ( c ) G E i ( z, c ) = ∞ . (3.14) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 21 Since by positivity of the Green function X c ( F n ℓ ) ∞ ( c ) G E i ( z, c ) ≥ X c ∈ ( a i + ε, b i − ε ) ( F n ℓ ) ∞ ( c ) G E i ( z, c )we obtain together with (3.13) thatlim ℓ →∞ X c ( F n ℓ ) ∞ ( c ) G E inℓ ( z, c ) = ∞ . By an application of Lemma 3.7 we obtain (3.11) which concludes the proof. (cid:3) Lemma 3.9. For z ∈ C \ R , we have lim inf n →∞ n log | F n ( z ) | ≥ . (3.15) Proof. Fix z ∈ C \ R and recall (3.8). Noting that | B n ( z ) | ≤ 1, the claim follows from Lemma3.8. (cid:3) In contrast to the classical polynomial setting, our limits will be described by the difference oftwo potentials, one corresponding to the zeros of F n , leading to a subharmonic part and one corre-sponding to the poles leading to a superharmonic part. Since in the following considerations we willwork with the Riesz measures for both of them, there is no natural choice of a “coordinate system”and it will be convenient to apply conformal maps to logarithmic potentials. For a probabiltiymeasure ν with supp ν ( R and z ∗ ∈ R \ supp ν , let us introduce the notation Φ ν ( z, z ∗ ) = Z K ( x, z ; z ∗ ) dν ( x ) , where K ( x, z ; z ∗ ) = ( log (cid:12)(cid:12)(cid:12) − z − z ∗ x − z ∗ (cid:12)(cid:12)(cid:12) , z ∗ = ∞ , log | z − x | , z ∗ = ∞ . It is straightforward to see that if z , z ∈ R \ supp ν , then there is β ∈ R so that Φ ν ( z, z ) = β + Φ ν ( z, z ) . Lemma 3.10. Let ν be a probability measure on R , supp ν ⊂ E and f ∈ PSL(2 , R ) . If f ( ∞ ) = ∞ ,then Φ ν ( z, z ∗ ) = Φ f ∗ ν ( f ( z ) , f ( z ∗ )) . Otherwise, Φ ν ( z, z ∗ ) = Φ f ∗ ν ( f ( z ) , f ( z ∗ )) − Φ f ∗ δ ∞ ( f ( z ) , f ( z ∗ )) . Proof. Let us first assume that f ( ∞ ) = ∞ , i.e., f ( z ) = az + b with a = 0. Then we have1 − z − z ∗ x − z ∗ = 1 − f ( z ) − f ( z ∗ ) f ( x ) − f ( z ∗ ) . Thus, the claim follows by the transformation rule for pushforward measures.Let now f ( ∞ ) = ∞ . Since f preserves cross-ratios, we get1 − z − z ∗ x − z ∗ = x − zx − z ∗ = f ( x ) − f ( z ) f ( x ) − f ( z ∗ ) f ( z ∗ ) − f ( ∞ ) f ( z ) − f ( ∞ ) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 22 = (cid:18) − f ( z ) − f ( z ∗ ) f ( ∞ ) − f ( z ∗ ) (cid:19) − (cid:18) − f ( z ) − f ( z ∗ ) f ( x ) − f ( z ∗ ) (cid:19) . Noting that f ∗ δ ∞ = δ f ( ∞ ) , again the claim follows by applying the transformation rule for push-forward measures. (cid:3) Lemma 3.11. The measures ν n are a precompact family with respect to weak convergence on C ( R ) .Any accumulation point ν = lim ℓ →∞ ν n ℓ is a probability measure and supp ν ⊂ E .Proof. Since deg D n = n , precompactness follows by the Banach-Alaoglu theorem and any accu-mulation point is a probability measure on R . Let ( a , b ) be a connected component of R \ E . Let usprove that ν (( a , b )) = 0. By Möbius invariance, it suffices to assume that ( a , b ) is a bounded subsetof R . Due to Theorem 2.3 (vi), there is at most one generalized zero in ( a , b ), thus ν n ℓ (( a , b )) ≤ n ℓ and by the Portmanteau theorem ν (( a , b )) = 0 and supp ν ⊂ E . (cid:3) In the following we will need statements also for a subsequence ( h n ℓ ) ∞ ℓ =1 . Therefore, for a fixedsubsequence let us define K ′ = [ ℓ ≥ supp D ∞ n ℓ , and Ω K ′ = C \ K ′ , (3.16)so that h n ℓ is subharmonic on Ω K ′ for all ℓ . Since lim n →∞ µ n = µ , we have for any subsequence(and therefore any K ′ ), that supp µ ⊂ K ′ ⊂ K C and therefore Ω C ⊂ Ω K ′ ⊂ C \ supp µ .If D n ( z ∗ ) = D ∞ n ( z ∗ ) = 0, then by factoring F n we see that there is β n ∈ R so that h n ( z ) = β n + Φ ν n ( z, z ∗ ) − Φ µ n ( z, z ∗ ) . (3.17) Theorem 3.12. Let us pass to a subsequence so that lim ℓ ν n ℓ = ν , lim ℓ x ∗ n ℓ = x ∞ and lim ℓ β n ℓ = β ∈ R ∪ {−∞ , + ∞} . Then, in fact β ∈ R and for z ∗ / ∈ K C we have uniformly on compact subsetsof C \ R lim ℓ →∞ h n ℓ ( z ) = β + Φ ν ( z, x ∞ ) − Φ µ ( z, z ∗ ) =: h ( z ) . (3.18) In particular, h extends to a positive superharmonic function on C \ E and to a subharmonic functionon C \ supp µ . Moreover, for q.e. every z ∈ Ω K ′ lim sup ℓ →∞ h n ℓ ( z ) = β + Φ ν ( z, x ∗∞ ) − Φ µ ( z, z ∗ ) . Proof. Let ( a , b ) denote the gap containing x ∞ and let us assume that ℓ is big enough so that all x ∗ n ℓ are in ( a , b ). Due to Theorem 2.3(vi), ν n ℓ (( a , b )) = 0. Thus, we can write h n ℓ ( z ) = β n ℓ + Φ ν nℓ ( z, x ∗∞ ) − Φ µ nℓ ( z, z ∗ ) . (3.19)Since K ( · , z, x ∞ ) is continuous on supp ν n ℓ ⊂ R \ ( a , b ) and K ( · , z, z ∗ ) is continuous on R \ K ′ , weget lim ℓ →∞ Φ ν nℓ ( z, x ∗∞ ) = Φ ν ( z, x ∗∞ ) , lim ℓ →∞ Φ µ nℓ ( z, x ∗∞ ) = Φ µ ( z, x ∗∞ ) . Since, for z ∈ C + , Φ ν ( z , x ∗∞ ) , Φ µ ( z , z ∗ ) ∈ R the upper and lower estimates (3.2) and (3.15) implythat β ∈ R . In fact, convergence is uniform on compact subsets of C \ R : since supp( ν n ℓ ) , supp( µ n ℓ ) ⊂ R for all ℓ and all measures are normalized, the estimatelog (cid:12)(cid:12)(cid:12)(cid:12) x − z x − z (cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:18) | z − z | dist( z , R ) (cid:19) ≤ | z − z | dist( z , R ) , z , z ∈ C \ R SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 23 implies uniform equicontinuity of the potentials R log (cid:12)(cid:12)(cid:12) − z − x ∗∞ x − x ∗∞ (cid:12)(cid:12)(cid:12) dν n ℓ ( x ) and R log (cid:12)(cid:12)(cid:12) − z − z ∗ x − z ∗ (cid:12)(cid:12)(cid:12) dµ n ℓ ( x )on compact subsets of C \ R , and the Arzelà–Ascoli theorem implies uniform convergence on com-pacts.By applying a conformal map f ∈ PSL(2 , R ) and Lemma 2.2 we assume that ∞ ∈ ( a , b ) so that E and K ′ are compact subsets of R .We note that Φ ρ , for ρ = µ, ν , are subharmonic in C and harmonic in C \ supp ρ . Thus, we onlyneed to argue why h is harmonic at ∞ . Since supp µ and E are bounded and µ, ν are probabilitymeasures, we have Φ ρ ( z ) = log | z | + O (1) (3.20)as z → ∞ and therefore, h ( z ) = O (1) there and h has a harmonic extension to ∞ .Finally, for z ∈ Ω K ′ \ {∞} , K ( · , z, z ∗ ) is continuous on K ′ and thuslim ℓ →∞ Φ µ nℓ ( z, z ∗ ) = Φ µ ( z, z ∗ ) . (3.21)By the upper envelope theorem for q.e. z ∈ C lim sup ℓ →∞ Φ ν nℓ ( z, x ∗∞ ) = Φ ν ( z, x ∗∞ ) . (3.22)Combining (3.21) and (3.22), for q.e. z ∈ Ω K ′ we havelim sup ℓ →∞ h n ℓ ( z ) = lim ℓ →∞ β n ℓ + lim sup ℓ →∞ Φ ν nℓ ( z, x ∗∞ ) − lim ℓ →∞ Φ µ nℓ ( z, z ∗ ) = h ( z ) . (cid:3) Lemma 3.13. For z ∈ C \ R , we have lim inf n →∞ n log | F n ( z ) | ≥ Z G E ( z, x ) dµ ( x ) . Proof. By applying a conformal map f , we assume ∞ ∈ E so that Ω ⊂ C . Fix z ∈ C \ R and let n ℓ be such that lim ℓ →∞ h n ℓ ( z ) = lim inf n →∞ h n ( z )and lim ℓ →∞ h n ℓ ( z ) = h ( z ) = β + Φ ν ( z, x ∞ ) − Φ µ ( z, z ∗ )in the sense of Theorem 3.12. Thus, h defines a positive superharmonic function on Ω and − ∆h = ∆Φ µ ( z, z ∗ ) = 2 πµ. By the Riesz decomposition theorem [1, Theorem 4.4.1], we have h ( z ) = Z G E ( z, x ) dµ ( x ) + u ( z ) , where u ( z ) is the greatest harmonic minorant of h . Since h is positive, it follows that u ≥ 0. Thus,we obtain h ( z ) ≥ Z G E ( z, x ) dµ ( x )and the claim follows. (cid:3) We can now prove the root asymptotics of F n and convergence of generalized zero countingmeasures: SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 24 Proof of Theorem 1.9. Root asymptotics follow by combining Lemma 3.1 and Lemma 3.13.By conformal invariance, we assume that ∞ ∈ supp µ so that Ω µ := C \ supp µ ⊂ C and E iscompact in R . Due to Lemma 3.11 the family { ν n } is precompact and we can consider a weaklyconvergent subsequence ν = lim j →∞ ν n j . Moreover, by Lemma 3.2, R G E ( z, x ) dµ ( x ) defines asubharmonic function in Ω µ . Let us compute its Riesz measure. Take φ ∈ C ∞ c ( Ω µ ) and compute Z Z G E ( z, x ) dµ ( x ) ∆φ ( z ) dA ( z ) = Z Z G E ( z, x ) ∆φ ( z ) dA ( z ) dµ ( x ) = 2 π Z Z ω E ( dz, x ) dµ ( x ) φ ( z ) dA ( z ) , where Fubini’s theorem is justified since supp( φ ) ⊂ C \ supp( µ ), sup supp( φ ) × supp( µ ) | G E ( z, x ) | < ∞ .That is, 12 π ∆ (cid:18)Z G E ( z, x ) dµ ( x ) (cid:19) = Z ω E ( dz, x ) dµ ( x ) =: ρ. Root asymptotics and Theorem 3.12 imply that on C \ R Z G E ( z, x ) dµ ( x ) = β + Φ ν ( z, x ∞ ) − Φ µ ( z, z ∗ ) . Applying the weak identity principle for subharmonic functions [19, Theorem 2.7.5], this equalityalso holds on Ω µ . Thus, computing the distributional Laplacian on both sides yields ν = ρ andw-lim ν n = ρ . (cid:3) Lemma 3.14. Let us fix a gap ( a , b ) and let [ u n , v n ] = E n ∩ [ a , b ] . Let us pass to a subsequence,such that there are limits u ∞ , v ∞ ∈ [ a , b ] , i.e., lim ℓ →∞ v n ℓ = v ∞ , lim ℓ →∞ u n ℓ = u ∞ Then u ∞ = v ∞ . Proof. By conformal invariance we can assume that ∞ / ∈ ( a , b ) and consider again { h n } as a familyof subharmonic functions in Ω C . We have E n ∩ supp D ∞ n = ∅ , since either F n has a pole at c or if F n has a generalized zero at c then by (ii) of Theorem 2.9 there is no extension in this gap. Due toTheorem 1.9, lim n →∞ h n = R G E ( · , x ) dµ ( x ) . Assume that v ∞ − u ∞ = δ > 0. For any 0 < ε < δ/ ℓ such that for all ℓ > ℓ , we have A := [ u ∞ + ε, v ∞ − ε ] ⊂ [ u n ℓ , v n ℓ ] . (3.23)Therefore, defining K ′ as in (3.16), we have A ∩ K ′ = ∅ . Note that first we only have emptyintersection without taking the closure, but since ε above can be made smaller, we also concludethat it holds for K ′ .By Theorem 3.12 we have for q.e. z ∈ Ω K ′ lim sup ℓ →∞ h n ℓ ( z ) = Z G E ( z, x ) dµ ( x ) . Since supp µ ⊂ K ′ , it follows from Lemma 3.2 that R G E ( z, x ) dµ ( x ) > z ∈ Ω K ′ andtherefore in particular for z ∈ A . On the other hand, by definition of E n and (3.23), we have h n ℓ ( z ) ≤ A has positive capacity, this gives a contradiction. (cid:3) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 25 Szegő–Widom asysmptotics Asymptotics of log | F n | . In the following in addition to the assumptions made in Section 3,we assume that E is a regular Parreau–Widom set. Let us recall its definition. First we assume that E is regular for the Dirichlet problem. Let z ∈ R \ E and denote the gap containing z by ( a , b ).Due to regularity and concavity of the Green function, G E ( z, z ) has exactly one critical point ineach gap ( a j , b j ) except in the gap ( a , b ). Let us denote these critical points of G E ( z, z ) by ξ j .Then we call E a regular Parreau–Widom set, if PW E ( z ) = X j G E ( ξ j , z ) < ∞ . (4.1)It is well known that this does not depend on the choice of z ; see e.g. [15, Chapter V].Denote the topological circle T j = [ a j , b j ] / a j ∼ b j . Since a j , b j are Dirichlet regular points,lim x ↓ a j G E ( z, x ) = lim x ↑ b j G E ( z, x ) = 0so with the usual convention G E ( z, a j ) = G E ( z, b j ) = 0 , (4.2)the Green function G E ( z, t j ) depends continuously on t j ∈ T j . We also consider the compact space D ( E ) = ∞ Y j =0 T j , (4.3)equipped with the product topology. As for divisors, a functional interpretation will be convenient.Thus, for an element D ∈ D ( E ), D = ( t j ) ∞ j =0 , we also use the functional interpretation D ( x ) = ∞ X j =0 χ { t j } ( x ) . We want to associate to the divisor D n an element D n ∈ D ( E ). In principle we want to define D n as the restriction of D n to R \ E . Recall that due to Theorem 2.3(v) and (vi), there is at most onegeneralized zero in each gap and no generalized zero in the gap containing x ∗ . Since deg D n = n ,almost all gaps do not contain a generalized zero. To overcome this, we define D n = ( t jn ) ∞ j =0 , (4.4)where t jn = t , if there is t ∈ [ a j , b j ] such that D n ( t ) = 1 and otherwise we define t jn to be the cosetof a j ∼ b j in T j . Due to (4.2), these choices formally complete the definition of D n ∈ D ( E ) withoutaffecting certain sums below.In the previous section we have described root asymptotics, i.e., asymptotics of n log | F n ( z ) | .The following theorem describes asymptotics of log | F n ( z ) | and is the key to prove Szegő-Widomasymptotics in Theorem 4.5. Theorem 4.1. Let n ℓ be such that lim ℓ →∞ x ∗ n ℓ = x ∗∞ ∈ ( a , b ) and lim ℓ →∞ D n ℓ = D . Then for z ∈ C \ ( R \ ( a , b )) , we have lim ℓ →∞ log | F n ℓ ( z ) | − X c D ∞ n ℓ ( c ) G E ( z, c ) ! = − log 2 − X t D ( t ) G E ( z, t ) . (4.5) Moreover, D ( a ) = 1 . SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 26 Proof. Define H n ℓ ( z ) = X c (cid:16) ( F n ℓ ) ∞ ( c ) G E nℓ ( z, c ) − D ∞ n ℓ ( c ) G E ( z, c ) (cid:17) . Due to (3.8) and Lemma 3.8 it remains to show thatlim ℓ →∞ H n ℓ ( z ) = − X t D ( t ) G E ( z, t ) . (4.6)Let us assume without loss of generality that all x ∗ n ℓ lie in ( a , b ). Recall that by Theorem 2.9(i) D n ℓ ( a ) = 1, showing that D ( a ) = 1. Moreover, Theorem 2.9(i) implies E n ℓ ∩ ( a , b ) = ∅ and since G E ( z, c ) − G E n ( z, c ) ≥ F n ) ∞ = D ∞ n on ( a , b ), we conclude that ( − H n ℓ ) ℓ defines a familyof positive harmonic functions in C \ ( R \ ( a , b )) and is thus by the Harnack principle precompactin the space of positive harmonic functions together with the function which is identically + ∞ equipped with uniform convergence on compact subsets.Let us now turn to the other gaps. Let [ u jn , v jn ] denote the extension in the j th gap of the set E n as in Theorem 2.9 and consider G E nℓ ( z, c ) − G E ( z, c )as a subharmonic function in Ω = C \ E , which vanishes on E . Thus, by Lemma 3.6 G E nℓ ( z, c ) − G E ( z, c ) = − X j Z v jnℓ u jnℓ G E ( z, x ) ω E nℓ (d x, c ) , Let us define ω n ℓ ( dx ) = X c ( F n ℓ ) ∞ ( c ) ω E nℓ (d x, c ) , and recall that this is just a finite sum. We conclude that H n ℓ ( z ) = X c ( F n ℓ ) ∞ ( c ) (cid:16) G E nℓ ( z, c ) − G E ( z, c ) (cid:17) − X c (cid:0) ( D ∞ n ℓ ( c ) − ( F n ℓ ) ∞ ( c )) G E ( z, c ) (cid:1) = − ∞ X j =0 Z v jnℓ u jnℓ G E ( z, x ) ω n ℓ ( dx ) − X c (cid:0) ( D ∞ n ℓ ( c ) − ( F n ℓ ) ∞ ( c )) G E ( z, c ) (cid:1) . (4.7)Due to Lemma 3.3 there are finitely many gaps containing poles. So by partitioning into finitelymany subsequences, we can assume that for each j , for all ℓ > D ∞ n ℓ ( t jn ℓ ) > D ∞ n ℓ ( t jn ℓ ) = 0,i.e., in the first case t jn ℓ corresponds to a pole reduction of F n ℓ . We will show that both cases leadto the same limit.Let us first consider a gap ( a j , b j ) so that D ∞ n ℓ ( t jn ℓ ) = 0 and let us assume that t jn ℓ → t j ∞ ∈ ( a j , b j ). Due to Lemma 3.14, lim ℓ u jn ℓ = lim ℓ v jn ℓ = t j ∞ . (4.8)In particular for ℓ big enough we have [ u jn ℓ , v jn ℓ ] ⊂ ( a j , b j ) and it follows then from Lemma 2.11that ω n ℓ ([ u jn ℓ , v jn ℓ ]) = 1 . Hence, ω n ℓ | [ u jn ℓ , v jn ℓ ] → δ t j ∞ SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 27 and therefore Z v jnℓ u jnℓ G E ( z, x ) ω n ℓ ( dx ) → G E ( z, t j ∞ ) . If t j ∞ = a j , using that G E ( z, · ) vanishes at a j we conclude as above that Z v jnℓ u jnℓ G E ( z, x ) ω n ℓ ( dx ) → G E ( z, t j ∞ ) . It remains to discuss the gaps where D ∞ n ℓ ( t jn ℓ ) = 1. Due to Theorem 2.9(ii), u jn ℓ = v jn ℓ = a j , butin this case D ∞ n ℓ ( t jn ℓ ) − ( F n ℓ ) ∞ ( t jn ℓ ) = 1 . Thus, these are exactly the terms that contribute in the second sum in (4.7). Since G E is continuouswe conclude that G E ( z, t jn ℓ ) → G E ( z, t j ∞ ). Hence, if we are allowed to interchange the limit andsummation in (4.7), we have proved (4.6). As in [15, Chapter V], by a Harnack-type argument, theParreau–Widom condition implies P j sup x ∈ ( a j , b j ) G E ( z, x ) < ∞ and since moreover ω n ℓ (( a j , b j )) ≤ (cid:3) Blaschke products, character-automorphic Hardy spaces and a related H ∞ ex-tremal problem. We will now pass from asymptotics of the superharmonic function log | F n | toasymptotics of the rational function F n . Thus, essentially in (4.5) we need to add harmonic conju-gates and apply exp. Thus, the left-hand side in (4.5) will lead to complex Green functions B E ( z, c ) = e − ( G E ( z, c )+ i ^ G E ( z, c )) , as defined in (1.20). We have already mentioned that in general B E ( z, c ) is a multi-valued functionin Ω . Let us fix a normalization gap ( a , b ) and z ∈ ( a , b ) and define E j = [ z , a j ] ∩ E . Let ˜ γ j bethe generator of the fundamental group π ( Ω, z ), which starts at z and passes through the gap( a j , b j ), encircling the set E j once. If we extend B E ( z, c ) analytically along γ j , we get B E (˜ γ j ( z ) , c ) = e πiω E ( E j , c ) B E ( z, c ) . (4.9)When working with multi-valued functions, it is convenient to consider them as single-valuedfunctions on the universal cover of Ω = C \ E . By means of the Koebe–Poincaré uniformizationtheorem, Ω is uniformized by the disk D ; that is, there exists a Fuchsian group Γ and a meromorphicfunction z : D → Ω with the following properties:1 . ∀ z ∈ Ω ∃ ζ ∈ D : z ( ζ ) = z, . z ( ζ ) = z ( ζ ) ⇐⇒ ∃ ˜ γ ∈ Γ : ζ = ˜ γ ( ζ ) . We fix it by the normalization z (0) = z , z ′ (0) > 0. For Denjoy domains the covering map canbe explicitly constructed [20, Section 4]. Moreover, there exists a Ford fundamental domain F , sothat z : F → Ω is bijective. We denote by Γ ∗ the group of unitary characters of Γ ; that is, grouphomomorphisms from Γ into T := R / Z . By the covering space formalism, Γ is group isomorphicto the fundamental group π ( Ω, z ). For a fixed ζ ∈ D we denote by b ( ζ, ζ ) := Y γ ∈ Γ γ ( ζ ) | γ ( ζ ) | γ ( ζ ) − ζ − γ ( ζ ) ζ , (4.10) SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 28 the standard Blaschke product. Since Cap E > Γ is of convergent type and thus the productis indeed convergent. The functions b ( ζ, ζ ) are character-automorphic, i.e., there exists χ z ∈ Γ ∗ such that b ( γ ( ζ ) , ζ ) = e πiχ z ( γ ) b ( ζ, ζ ) , ∀ γ ∈ Γ. If z = z ( ζ ), then these Blaschke product are related to the Green function of Ω , by − log | b ( ζ, ζ ) | = G E ( z ( ζ ) , z ) . Thus, we can regard the multi-valued functions B E ( z, z ) as single-valued character-automorphicfunction on the universal cover. Definition 4.2. Let f be analytic in D . We call f ( Γ ∗ -) character-automorphic with character α ∈ Γ ∗ if f ◦ γ = e πiα ( γ ) f, ∀ γ ∈ Γ. Similarly, if F is an analytic function on Ω , then we call F ( π ( Ω ) ∗ -) character-automorphic withcharacter α ∈ π ( Ω ) ∗ , if F ◦ ˜ γ = e πiα (˜ γ ) F, ∀ ˜ γ ∈ π ( Ω ) . Via the covering map z , Γ ∗ - and π ( Ω ) ∗ -character-automorphic functions are in one-to-onecorrespondence. The advantage is that Γ ∗ - character-automorphic functions on the universal cover D are single-valued. Therefore, we will formulate all convergence results for the correspondingsingle-valued lifts on D .Recall that H ∞ Ω ( α ) denotes the space of bounded analytic character-automorphic functions, F ,in Ω ; see (1.17). It is a fundamental result of Widom [30] that if E is a Parreau–Widom set, then H ∞ Ω ( α ) = { } for every α ∈ π ( Ω ) ∗ . The Widom maximizer for x ∗ and character α is the uniquefunction W ( z ; α, x ∗ ) in the unit ball of H ∞ Ω ( α ) such that W ( x ∗ ; α, x ∗ ) = max { Re F ( x ∗ ) : F ∈ H ∞ Ω ( α ) , k F k Ω ≤ } . (4.11)We are now ready to state a definition of Direct Cauchy theorem. It is usually stated as apoint evaluation property for certain H functions in Ω [15], and hence the name, but it can beequivalently defined by the following: Definition 4.3. We say that the Direct Cauchy Theorem (DCT) holds in Ω , if for one and hencefor all x ∗ ∈ Ω , the map α W ( x ∗ ; α, x ∗ ) is continuous on π ( Ω ) ∗ equipped with the topology dualto the discrete topology on Γ .Let us for notational convenience also define B E ( z, z ) ≡ 1, if z ∈ E . Note that generally theharmonic conjugate is fixed up to an additive constant. So an additional normalization is requiredin (1.20). Since we will have varying normalizations, we will not fix it for a single function, butassume instead that for products of complex Green functions all of them are normalized to bepositive at the same point. In this way, we can associate to any divisor D ∈ D ( E ) a product ofcomplex Green functions, in other words a Blaschke product, by B E ( z, D ) = B E ( z, D, φ ) = e iφ Y t D ( t ) B E ( z, t ) . (4.12)Note that − log | B E ( z, D ) | = X t D ( t ) G E ( z, t ) , SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 29 that is, these are exactly expression of the type appearing in (4.5). Moreover, the Widom conditionguarantees that B E ( z, D ) converges to a non-trivial function for any D ∈ D ( E ). Let us define therestriction D k ( E ) = { D ∈ D ( E ) : D ( a k ) = 1 } . For D ∈ D k ( E ) it is natural to normalize B E ( z, D, φ ) such that B E ( z, D, φ ) > a k , b k ) whichwe fixes φ . To be more precise, since complex Green functions are defined locally and then extendedanalytically, this normalization holds only for one branch. Let us always assume that this branchcorresponds to the values of the lift to D in the fundamental domain F .The Abel map is an important object in the spectral theory of self adjoint difference and differ-ential operators. It is a map π from Divisors D k ( E ) to the characters π ( Ω ) ∗ . However, there is asubtle difference between this Abel map and the Abel map which we will implicitly use for Problem1.3. It can be seen from the definition of D ( E ). In spectral theory one would usually take a two-foldcover of the interval [ a j , b j ] and identify the endpoints of the two copies of the interval, whereasin our case we only took one copy and identified a j ∼ b j . This map π is also the reason why theDCT property is needed, because this assumption makes π a bijection which is used in the proofof the following theorem. The proof relies on the fundamental construction of the generalized Abelmap from Sodin and Yuditskii [22]. Theorem 4.4 ([5, Theorem 5.1],[11, Proposition 2.3]) . Let Ω be a regular Parreau–Widom domainsuch that DCT holds. Let D ∈ D k ( E ) and let α be the character of B E ( z, D ) defined by (4.12) .Then, for x ∗ ∈ ( a k , b k ) we have W ( z ; α, x ∗ ) = B E ( z, D ) . We see again that as for F n , the extremal function only depends on the chosen gap ( a k , b k )and not the particular extremal point in the gap. Since the above theorem holds for any gapand arbitrary Blaschke products associated to divisors in D k ( E ), we conclude that if D ∈ D j ( E )for j = k , then up to a unimodular constant B E ( z, D ) is also the Widom maximizer for the gap( a j , b j ). This is in line with the Corollary 1.8 for F n .Let D ∞ n , x ∗ n and d n be as in Problem 1.3 and define B ( n ) E ( z ) = e iφ n Y c D ∞ n ( c ) B E ( z, c ) , where e iφ n is chosen such that lim x → x ∗ n B ( n ) E ( x ) r ( x, x ∗ n ) d n > . (4.13)Let χ n denote the character of B ( n ) E . Let further W n ( z ) = W ( z ; χ n , x ∗ n ) , denote the Widom maximizer for the point x ∗ n and character χ n .For the following we follow the spirit of [5] and state convergence results on the universal cover D without introducing the corresponding lift of multi-valued functions on Ω . To give an example: if Q n are π ( Ω ) ∗ -character-automorphic function on Ω , we will write Q n → Q uniformly on compactsubsets of D , meaning that there are lifts q n of the Q n which are Γ ∗ -character-automorphic functionssuch that q n → q uniformly on compact subsets of D and Q is the projection of q . SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 30 Theorem 4.5. Let E be a regular Parreau–Widom set, such that DCT holds in Ω and F n be theextremizer of (1.3) . Then uniformly on compact subsets of D , we have lim n →∞ (cid:18) B ( n ) E ( z ) F n ( z ) − W n ( z ) (cid:19) = 0 . We will use the following simple criterion based on normality; note that it is simpler than thecorresponding criterion used in the polynomial case [5, Proposition 4.2], since our approach avoidsworking on multivalued functions on varying domains: Proposition 4.6. Let { q n } ∞ n =1 be a normal family on D . Let q ∞ be analytic on D so that for some ζ ∈ D and some neighborhood, V , of ζ we have that lim n →∞ | q n ( ζ ) | = | q ∞ ( ζ ) | for all ζ ∈ V ; (4.14) q n ( ζ ) > , q ∞ ( ζ ) > . (4.15) Then q n → q ∞ uniformly on compact subsets of D .Proof. By normality, it suffices to prove that any subsequence ( q n ℓ ) ∞ ℓ =1 which converges uniformlyon compacts has the limit q ∞ . Denote by f the limit of such a sequence. By (4.14), | f ( ζ ) | = | q ∞ ( ζ ) | for all ζ ∈ V . By (4.15), by possibly decreasing V , we can assume q ∞ ( ζ ) = 0 for ζ ∈ V , so by themaximum principle applied to f /q ∞ , we conclude f = e iφ q ∞ for some unimodular constant e iφ .By (4.15), f ( ζ ) ≥ q ∞ ( ζ ) > 0, so e iφ = 1 and f = q ∞ . (cid:3) Defining Q n ( z ) = F n ( z ) B ( n ) E ( z ) , the strategy is now clear: First we need to check that Q n ( z ) defines a normal family. Realizingthat log | Q n ( z ) | is exactly the left hand-side in (4.5), Theorem 4.1 and Proposition 4.6 imply thatall accumulation points are Blaschke products. Combining this with Theorem 4.4 finishes the proofof Theorem 4.5. Lemma 4.7. The sequence { Q n } ∞ n =1 forms a normal family in D .Proof. Since on Ω ( F n ) ∞ ≤ D ∞ n = ( B ( n ) E ) ,Q n are analytic π ( Ω ) ∗ -character automorphic functions in Ω . They have therefore Γ ∗ -characterautomorphic lifts to D . By Montel’s theorem [26, Chapter 6], it suffices to show that | F n B ( n ) E | ≤ Ω . The functions log | Q n | are subharmonic in Ω . Moreover, since E is regular and | F n | ≤ E , for every ζ ∈ E lim sup z → ζ log | Q n ( z ) | = lim sup z → ζ log | F n ( z ) | − lim z → ζ X t D ∞ n ( t ) lim z → ζ G E ( ζ, t ) ≤ , where we used Dirichlet regularity and the fact that the sum is only finite. The claim follows bythe maximum principle for subharmonic functions [19, Theorem 2.3.1]. (cid:3) Lemma 4.8. Let n ℓ be a subsequence such that lim ℓ →∞ D n ℓ = D and lim ℓ →∞ x ∗ n ℓ = x ∗∞ ∈ ( a j , b j ) for some j ≥ . Then, uniformly on compact subsets of D we have lim ℓ →∞ Q n ℓ ( z ) = 12 B E ( z, D ) , where D ∈ D j ( E ) and B E ( x ∗∞ , D ) > . SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 31 Proof. Let us assume without loss of generality that all x ∗ n ℓ lie in ( a j , b j ). By (4.13), we have Q n ℓ ( x ∗ n ℓ ) > . Moreover, Q n ℓ are real, i.e., Q n ℓ ( z ) = Q n ℓ ( z ). Since D n ( t ) = 0 for every t ∈ ( a j , b j ), it follows that Q n ℓ ( t ) > 0. Thus, in particular at x ∗∞ . Thus we can apply Proposition 4.6 in a vicinity of x ∗∞ andthen the claim follows from Theorem 4.1. (cid:3) Proof of Theorem 4.5. Since Q n , W n form normal families, by precompactness it suffices to provethat every subsequence has a subsubsequence so that lim ℓ Q n ℓ − W n ℓ = 0. Let us pass to asubsequence such that lim ℓ →∞ D n ℓ = D and lim ℓ →∞ x ∗ n ℓ = x ∗∞ ∈ ( a j , b j ) as in Lemma 4.8. Thenby Lemma 4.8 lim ℓ →∞ Q n ℓ ( z ) = 12 B E ( z, D ) . If α is the character of B E ( z, D ) this implies that χ n ℓ → α . By Theorem 4.4, B E ( z, D ) = W ( z, α, x ∗∞ ). On the other hand, it is proven in [5, Theorem 3.1] that DCT implies that W ( z ; χ n ℓ , x ∗ n ℓ ) → W ( z, α, x ∗∞ ) uniformly on compact subsets of D . In this reference there is no sequence of extremalpoints, but since the Widom maximizer only depends on the given gap and not on the particularpoint, the sequence W ( z ; χ n ℓ , x ∗ n ℓ ) eventually only depends on the character. This concludes theproof. (cid:3) References [1] D. H. Armitage and S. J. Gardiner, Classical potential theory , Springer Monographs in Math-ematics, Springer Verlag, London, 2001. MR 1801253[2] V. Azarin, Growth theory of subharmonic functions , Birkhäuser Advanced Texts: BaslerLehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009.MR 2463743[3] P. Chebyshev, Théorie des mécanismes connus sous le nom de parallélogrammes , Mémoiresprésentés à l’Académie Impériale des Sciences de Saint-Pétersbourg (1854), 539–586.[4] , Sur les questions de minima qui se rattachent à la représentation approximative desfonctions , Mémoires présentés à l’Académie Impériale des Sciences de Saint-Pétersbourg, Six-iéme serie (1859), 199–291.[5] J. S. Christiansen, B. Simon, P. Yuditskii, and M. Zinchenko, Asymptotics of Chebyshev poly-nomials, II: DCT subsets of R , Duke Math. J. (2019), no. 2, 325–349. MR 3909898[6] J. S. Christiansen, B. Simon, and M. Zinchenko, Asymptotics of Chebyshev polynomials, I:subsets of R , Invent. Math. (2017), no. 1, 217–245. MR 3621835[7] , Asymptotics of Chebyshev Polynomials, V. Residual Polynomials , arXiv:2008.09669(2020).[8] B. Eichinger, Szegő-Widom asymptotics of Chebyshev polynomials on circular arcs , J. Approx.Theory (2017), 15–25. MR 3628947[9] B. Eichinger and M. Lukić, Stahl–Totik regularity for continuum Schrödinger operators ,arXiv:2001.00875 (2020).[10] B. Eichinger, M. Lukić, and G. Young, Orthogonal rational functions with real poles, rootasymptotics, and GMP matrices , arXiv:2008.11884 (2020).[11] B. Eichinger and P. Yuditskii, The Ahlfors problem for polynomials , Mat. Sb. (2018), no. 3,34–66. MR 3769214 SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 32 [12] A. Eremenko and P. Yuditskii, Comb functions , Recent advances in orthogonal polynomials,special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Provi-dence, RI, 2012, pp. 99–118. MR 2964141[13] G. Faber, Über Tschebyscheffsche Polynome , J. Reine Angew. Math. (1920), 79–106.MR 1580974[14] J. B. Garnett and D. E. Marshall, Harmonic measure , New Mathematical Monographs, vol. 2,Cambridge University Press, Cambridge, 2005. MR 2150803[15] M. Hasumi, Hardy classes on infinitely connected Riemann surfaces , Lecture Notes in Mathe-matics, vol. 1027, Springer-Verlag, Berlin, 1983. MR 723502[16] P. Koosis, The logarithmic integral. I , Cambridge Studies in Advanced Mathematics, vol. 12,Cambridge University Press, Cambridge, 1998. MR 1670244[17] A. L. Lukashov, On Chebyshev-Markov rational functions over several intervals , J. Approx.Theory (1998), no. 3, 233–352. MR 1657679[18] V. A. Marčenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator ,Mat. Sb. (N.S.) (1975), no. 4(8), 540–606, 633–634. MR 0409965[19] T. Ransford, Potential theory in the complex plane , London Mathematical Society StudentTexts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766[20] L. A. Rubel and J. V. Ryff, The bounded weak-star topology and the bounded analytic functions ,J. Functional Analysis (1970), 167–183. MR 0254580[21] E. B. Saff and V. Totik, Logarithmic potentials with external fields , Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom. MR 1485778[22] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum,infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions ,J. Geom. Anal. (1997), no. 3, 387–435. MR 1674798[23] K. Schiefermayr, The growth of polynomials outside of a compact set—the Bernstein-Walshinequality revisited , J. Approx. Theory (2017), 9–18. MR 3707135[24] W. Schlag, A course in complex analysis and Riemann surfaces , Graduate Studies in Mathe-matics, vol. 154, American Mathematical Society, Providence, RI, 2014. MR 3186310[25] B. Simon, Equilibrium measures and capacities in spectral theory , Inverse Probl. Imaging (2007), no. 4, 713–772. MR 2350223[26] , Basic complex analysis , A Comprehensive Course in Analysis, Part 2A, AmericanMathematical Society, Providence, RI, 2015. MR 3443339[27] M. L. Sodin and P. M. Yuditskii, Functions deviating least from zero on closed subsets of thereal axis , St. Petersburg Math. J (1993), 201–249.[28] H. Stahl and V. Totik, General orthogonal polynomials , Encyclopedia of Mathematics and itsApplications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828[29] H. Widom, Extremal polynomials associated with a system of curves in the complex plane ,Advances in Math. (1969), 127–232. MR 239059[30] , H p sections of vector bundles over Riemann surfaces , Ann. of Math. (2) (1971),304–324. MR 288780 Institute of Analysis, Johannes Kepler University Linz, 4040 Linz, Austria. Email address : [email protected] Department of Mathematics, Rice University MS-136, Box 1892, Houston, TX 77251-1892, USA. Email address : [email protected] SYMPTOTICS OF CHEBYSHEV RATIONAL FUNCTIONS 33 Department of Mathematics, Rice University MS-136, Box 1892, Houston, TX 77251-1892, USA. Email address ::