Computing Polynomial Conformal Models for Low-Degree Blaschke Products

Two-dimensional Blaschke Products: Degree Growth and Ergodic Consequences

We study the dynamics of Blaschke products in two dimensions, particularly the rates of growth of the degrees of iterates and the corresponding implications for the ergodic properties of the map.

Low-Degree Polynomial Roots

5 Cubic Polynomials 7 5.1 Real Roots of Multiplicity Larger Than One . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 One Simple Real Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3 Three Simple Real Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.4 A Mixed-Type Implementation . . . . . . . . . . . . ....

Computable Analysis and Blaschke Products

We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.

AN Lp DEFINITION OF INTERPOLATING BLASCHKE PRODUCTS

Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z1, z′ 1, z2, z ′ 2, . . . , zn, z ′ n (in this order) on the unit circle, and n points w1, . . . , wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′ j) = B(z ′ k) for all j and k. Modifying this example yields a Blaschke product of degree n− 1 that interpolates the zj ’s to the wj ’s. ...