HERMAN RINGS OF BLASCHKE PRODUCTS OF DEGREE 3

Let Fa,λbe the Blaschke product of the form Fa,λ= λz2((z - a)/(1 - āz)) and α denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any α there exists a constant a0≧ 3 and a continuous function λ(a) such that Fa,λ(a)possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a0. It is remarked that the question whether a0= 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a0is strictly larger than 3.

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LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

Bukovinian Mathematical Journal ◽

10.31861/bmj2021.01.13 ◽

2021 ◽

Vol 9 (1) ◽

pp. 164-170

Author(s):

Y. Gal ◽

M. Zabolotskyi ◽

M. Mostova

Keyword(s):

Blaschke Product ◽

Meromorphic Functions ◽

Logarithmic Derivative ◽

Counting Function ◽

Finite System ◽

Blaschke Products ◽

Nevanlinna Characteristic ◽

Number Of Zeros ◽

Accumulation Points ◽

Theory Of Value

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

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Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-026-2 ◽

1962 ◽

Vol 14 ◽

pp. 334-348 ◽

Cited By ~ 16

Author(s):

G. T. Cargo

Keyword(s):

Blaschke Product ◽

Common Knowledge ◽

Holomorphic Functions ◽

Blaschke Products ◽

Radial Limit ◽

Radial Limits ◽

Almost Everywhere ◽

Image Position ◽

Bounded Holomorphic ◽

Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

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Approximation by Unimodular Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-025-4 ◽

1971 ◽

Vol 23 (2) ◽

pp. 257-269 ◽

Cited By ~ 3

Author(s):

Stephen Fisher

Keyword(s):

Blaschke Product ◽

Uniform Approximation ◽

Unit Disc ◽

Open Unit ◽

Blaschke Products ◽

Open Unit Disc ◽

Pointwise Approximation ◽

Finite Blaschke Product ◽

Image Position ◽

Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

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Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003412 ◽

1986 ◽

Vol 6 (2) ◽

pp. 205-239 ◽

Cited By ~ 31

Author(s):

Kevin Hockett ◽

Philip Holmes

Keyword(s):

Symbolic Dynamics ◽

Homoclinic Orbits ◽

Irrational Rotation ◽

Cantor Sets ◽

Rotation Numbers ◽

Twist Maps ◽

Irrational Rotation Number ◽

Rotation Set ◽

Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

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On partial isometries with circular numerical range

Concrete Operators ◽

10.1515/conop-2020-0121 ◽

2021 ◽

Vol 8 (1) ◽

pp. 176-186

Author(s):

Elias Wegert ◽

Ilya Spitkovsky

Keyword(s):

Blaschke Product ◽

Partial Isometry ◽

Shift Operator ◽

Numerical Range ◽

Elliptic Functions ◽

Blaschke Products ◽

Partial Isometries ◽

Unitary Similarity ◽

Boundary Interpolation ◽

Poncelet Polygons

Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

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Integral Means and Zero Distributions of Blaschke Products

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-072-x ◽

1972 ◽

Vol 24 (5) ◽

pp. 755-760 ◽

Cited By ~ 2

Author(s):

C. N. Linden

Keyword(s):

Half Plane ◽

Blaschke Product ◽

Blaschke Products ◽

Integral Means ◽

Elementary Theorem ◽

Image Position ◽

Analogous Theorem

A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only ifIf 0 appears m times in {zn} thenis the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such thatas r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

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Derivatives of Blaschke Products and Model Space Functions

Canadian Mathematical Bulletin ◽

10.4153/s0008439519000675 ◽

2019 ◽

Vol 63 (4) ◽

pp. 716-725

Author(s):

David Protas

Keyword(s):

Blaschke Product ◽

Bergman Spaces ◽

Model Space ◽

Weighted Bergman Spaces ◽

Blaschke Products ◽

Distribution Of Zeros ◽

The Relationship ◽

Derivatives Of

AbstractThe relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_{\unicode[STIX]{x1D6FC}}^{p}$ of the derivative of $B$ or the derivative of functions in its model space $H^{2}\ominus \mathit{BH}^{2}$ is investigated.

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Maximal subalgebra of Douglas algebra

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171288000894 ◽

1988 ◽

Vol 11 (4) ◽

pp. 735-741

Author(s):

Carroll J. Gullory

Keyword(s):

Blaschke Product ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Blaschke Products ◽

Maximal Subalgebra ◽

Interpolating Blaschke Product ◽

Douglas Algebra ◽

Necessary And Sufficient ◽

Interpolating Blaschke Products ◽

Douglas Algebras

Whenqis an interpolating Blaschke product, we find necessary and sufficient conditions for a subalgebraBofH∞[q¯]to be a maximal subalgebra in terms of the nonanalytic points of the noninvertible interpolating Blaschke products inB. If the setM(B)⋂Z(q)is not open inZ(q), we also find a condition that guarantees the existence of a factorq0ofqinH∞such thatBis maximal inH∞[q¯]. We also give conditions that show when two arbitrary Douglas algebrasAandB, withA⫅Bhave property thatAis maximal inB.

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