Differentiating Blaschke products

Differentiating Blaschke products Oleg Ivrii April 14, 2017

Differentiating Blaschke Products Consider the following curious differentiation procedure: to a Blaschke product of degree d 1, F (z) = e iψ d i=1 z a i 1 a i z, a i D,

Differentiating Blaschke Products Consider the following curious differentiation procedure: to a Blaschke product F (z) = e iψ d i=1 z a i 1 a i z, a i D, of degree d 1, one can associate a Blaschke product B of degree d 1 whose zeros are located at the critical points of F.

Differentiating Blaschke Products Consider the following curious differentiation procedure: to a Blaschke product F (z) = e iψ d i=1 z a i 1 a i z, a i D, of degree d 1, one can associate a Blaschke product B of degree d 1 whose zeros are located at the critical points of F. Theorem. (M. Heins) The above map is a bijection: B d / Aut(D) B d 1 /S 1.

Inner functions An inner function is a holomorphic self-map of D such that for almost every θ [0, 2π), the radial limit lim F r 1 (reiθ ) exists and is unimodular (has absolute value 1). We will denote the space of all inner function by Inn.

BS decomposition An inner function can be represented as a (possibly infinite) Blaschke product singular inner function: B = e iψ a i a i z a i 1 a i z, a i D, (1 ai ) <. i ( ) ζ + z S = exp S 1 ζ z dσ ζ, σ m, σ 0. Here, the Blaschke factor takes out the zeros, while the singular inner factor is zero-free.

Nevanlinna class and B(S/S 1 )O decomposition The Nevanlinna class N consists of all holomorphic functions f on D which satisfy 1 sup log + f (z) dθ <. 0<r<1 2π z =r B = e iψ a i a i i ( S/S 1 = exp ( O = exp z a i 1 a i z, ζ + z S 1 ζ z dσ ζ ), σ m, ). ζ + z S 1 ζ z log f (ζ) dm ζ

Inner functions of finite entropy We will also be concerned with the subclass J of inner functions whose derivative lies in Nevanlinna class: lim r 1 1 2π 2π 0 log + F (re iθ ) dθ <.

Inner functions of finite entropy We will also be concerned with the subclass J of inner functions whose derivative lies in Nevanlinna class: lim r 1 1 2π 2π 0 log + F (re iθ ) dθ <. Jensen s formula: the set of critical points {c i } of F satisfies the Blaschke condition (1 c i ) <,

Inner functions of finite entropy We will also be concerned with the subclass J of inner functions whose derivative lies in Nevanlinna class: lim r 1 1 2π 2π 0 log + F (re iθ ) dθ <. Jensen s formula: the set of critical points {c i } of F satisfies the Blaschke condition (1 c i ) <, and is therefore the zero set of some Blaschke product.

Dyakonov s conjecture Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F.

Dyakonov s conjecture Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F. Remark. This mapping generalizes the construction outlined for finite Blaschke products,

Dyakonov s conjecture Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F. Remark. This mapping generalizes the construction outlined for finite Blaschke products, however, in addition to recording the critical set of F, Inn F may also contain a non-trivial singular factor.

Some progress Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F. Theorem. (K. Dyakonov, 2013) z 1.

Some progress Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F. Theorem. (K. Dyakonov, 2013) z 1. Theorem. (D. Kraus, 2007) MBP J BP.

Some progress Conjecture. (K. Dyakonov) Is the following map a bijection? J / Aut(D) Inn /S 1, F Inn F. Theorem. (K. Dyakonov, 2013) z 1. Theorem. (D. Kraus, 2007) MBP J BP. Theorem. (I, 2017) The map is injective. The image is closed under multiplication and taking divisors.

Background on conformal metrics The curvature of a conformal metric λ(z) dz is given by Examples. The hyperbolic metric has curvature 4, k λ = log λ λ 2. λ D = dz 1 z 2 while the Euclidean metric dz has curvature 0.

Background on conformal metrics Theorem. (Gauss) The curvature is a holomorphic invariant, that is k λ (f (z)) = k f λ(z).

Background on conformal metrics Theorem. (Gauss) The curvature is a holomorphic invariant, that is k λ (f (z)) = k f λ(z). Theorem. (Ahlfors) The hyperbolic metric on D is (pointwise) the largest metric which has curvature 4.

Background on conformal metrics Theorem. (Gauss) The curvature is a holomorphic invariant, that is k λ (f (z)) = k f λ(z). Theorem. (Ahlfors) The hyperbolic metric on D is (pointwise) the largest metric which has curvature 4. Theorem. (Liouville) Any metric of curvature 4 (whose zeros have integer multiplicity) on a simply-connected domain Ω is of the form f λ D, for some holomorphic function f : Ω D.

Heins theory of conformal metrics An SK-metric is a conformal metric with continuous density and curvature 4.

Heins theory of conformal metrics An SK-metric is a conformal metric with continuous density and curvature 4. A Perron family Φ of SK-metrics on a domain Ω: (Taking maxima) λ, µ Φ = max(λ, µ) Φ. (Disk modification) λ Φ = M D λ Φ. Here, D Ω is a round disk and M D λ = λ on Ω \ D and has curvature 4 in D.

Heins theory of conformal metrics An SK-metric is a conformal metric with continuous density and curvature 4. A Perron family Φ of SK-metrics on a domain Ω: (Taking maxima) λ, µ Φ = max(λ, µ) Φ. (Disk modification) λ Φ = M D λ Φ. Here, D Ω is a round disk and M D λ = λ on Ω \ D and has curvature 4 in D. Theorem. (Heins, 1962) Provided Φ, the metric λ Φ := sup λ Φ λ has curvature 4.

Maximal Blaschke products Given a countable set C in D, counted with multiplicity, construct a Blaschke product with critical set C, if one exists.

Maximal Blaschke products Given a countable set C in D, counted with multiplicity, construct a Blaschke product with critical set C, if one exists. Procedure: Let Φ C be the collection of all SK-metrics that vanish on C. Set λ C := sup λ ΦC λ. Liouville s theorem gives a holomorphic map F C : D D.

Maximal Blaschke products Given a countable set C in D, counted with multiplicity, construct a Blaschke product with critical set C, if one exists. Procedure: Let Φ C be the collection of all SK-metrics that vanish on C. Set λ C := sup λ ΦC λ. Liouville s theorem gives a holomorphic map F C : D D. Theorem. (D. Kraus, O. Roth, 2007) If Φ C, then F C is an indestructible Blaschke product.

Maximal Blaschke products Theorem. (M. Heins, 1962) Finite Blaschke products are maximal.

Maximal Blaschke products Theorem. (M. Heins, 1962) Finite Blaschke products are maximal. Theorem. (D. Kraus, 2007) If C satisfies the Blaschke condition, then B C λ D is an SK-metric, where B C is the Blaschke product with zero set C.

Maximal Blaschke products Theorem. (M. Heins, 1962) Finite Blaschke products are maximal. Theorem. (D. Kraus, 2007) If C satisfies the Blaschke condition, then B C λ D is an SK-metric, where B C is the Blaschke product with zero set C. Theorem. (D. Kraus, O. Roth, 2007) The collection of all critical sets of Blaschke products coincides with zero sets of functions in the weighted Bergman space A 2 1. In particular, if C 1 and C 2 are critical sets, their union may not be a critical set.

The fundamental lemma (1) If F C MBP J, then λ FC B C λ D by maximality.

The fundamental lemma (1) If F C MBP J, then λ FC B C λ D by maximality. (2) In fact, for any F J, λ F = F λ D Inn F λ D.

The fundamental lemma (1) If F C MBP J, then λ FC B C λ D by maximality. (2) In fact, for any F J, λ F = F λ D Inn F λ D. I proved (2) by considering Nevanlinna-stable approximations F n F by finite Blaschke products, using (1) for the F n, and taking the limit as n.

The fundamental lemma (1) If F C MBP J, then λ FC B C λ D by maximality. (2) In fact, for any F J, λ F = F λ D Inn F λ D. I proved (2) by considering Nevanlinna-stable approximations F n F by finite Blaschke products, using (1) for the F n, and taking the limit as n. K. Dyakonov proved this in 1992 using Julia s lemma. A little later, H. Hedenmalm came up with a different proof using yet another approach.

The fundamental lemma Lemma. (I, 2017) λ F is the minimal metric of curvature 4 satisfying λ F Inn F λ D.

The fundamental lemma Lemma. (I, 2017) λ F is the minimal metric of curvature 4 satisfying λ F Inn F λ D. For the proof, suppose λ is the minimal metric with the above property. Consider the function u(z) = log(λ F /λ), z D. It is non-negative and subharmonic ( u 0), yet 1 lim u(re iθ )dθ = 0. r 1 2π z =r

Ahern-Clark theory Lemma. If f N is a Nevanlinna class function, then gap(f ) := 1 log f (z) dθ 2π z =1 { } 1 lim log f (z) dθ r 1 2π z =r = σ(s 1 ).

Ahern-Clark theory Lemma. If f N is a Nevanlinna class function, then gap(f ) := 1 log f (z) dθ 2π Corollary. If F J then z =1 { } 1 lim log f (z) dθ r 1 2π z =r gap(f 1 ) = lim r 1 2π z =r log λ D λ F dθ. = σ(s 1 ).

Computing the entropy of inner functions Lemma. Suppose F J is a maximal Blaschke product with F (0) = 0 and F (0) 0. We have 1 log F (z) dθ = log 1 2π z =1 c cp i log 1 z zeros i, where we omit the trivial zero at the origin. Remark. This property characterizes maximal Blaschke products.

Computing the entropy of inner functions Lemma (Ch. Pommerenke, 1976). Suppose U P J is the universal covering map D \ {p 1, p 2, p 3,..., p k } with the normalization U P (0) = 0. Then, 1 log U P 2π (z) dθ = z =1 k i=1 where we omit the trivial zero at the origin. log 1 p i log 1 z zeros i. See his fantastic paper On Green s functions of Fuchsian groups.

Nevanlinna stability Suppose F J. A sequence F k F is Nevanlinna stable if Inn F k Inn F and Out F k Out F.

Nevanlinna stability Suppose F J. A sequence F k F is Nevanlinna stable if Inn F k Inn F and Out F k Out F. Not every sequence is Nevanlinna stable,

Nevanlinna stability Suppose F J. A sequence F k F is Nevanlinna stable if Inn F k Inn F and Out F k Out F. Not every sequence is Nevanlinna stable, but given any F J, a stable approximation exists. (M. Craizer, 1991)

Nevanlinna stability Suppose F J. A sequence F k F is Nevanlinna stable if Inn F k Inn F and Out F k Out F. Not every sequence is Nevanlinna stable, but given any F J, a stable approximation exists. (M. Craizer, 1991) Theorem. (I, 2017) Suppose F Ck F is a stable sequence and C 1,k C k such that F C1,k G. Then, F C1,k G is stable.

Divisor law Theorem. (I, 2017) Suppose F Ck F is a stable sequence and C 1,k C k such that F C1,k G. Then, F C1,k G is stable. Call I Inn constructible if its in the image of the map J / Aut(D) Inn /S 1, F Inn F. Corollary. If I is constructible and J I, then J is constructible.

Divisor law Theorem. (I, 2017) Suppose F Ck F is a stable sequence and C 1,k C k such that F C1,k G. Then, F C1,k G is stable. Call I Inn constructible if its in the image of the map J / Aut(D) Inn /S 1, F Inn F. Corollary. If I is constructible and J I, then J is constructible. Corollary. If S µ N then S µ is constructible (by the work of M. Cullen from 1971, this holds if supp µ is a Carleson set).

Product and decomposition laws Lemma. If I and J are constructible, then so is I J. The proof uses the divisor law and the Solynin-type estimate λ FC1 λ FC2 λ FC1 C 2 λ FC1 C 2. Corollary. The inner function B C S µ is constructible if and only if B C and S µ are.

Thank you for your attention!