Badly approximable functions and interpolation by Blaschke products

We study geometrical properties of finite Blaschke products. For a Blaschke product B of degree d, let \(L_{\lambda }\) be the set of the lines tangent to the unit circle at the d preimages \( B^{-1}(\lambda ) \). We show that the trace of the intersection points of each pair of two elements in \( L_{\lambda } \) as \( \lambda \) ranges over the unit circle forms an algebraic curve of degree at most \( d-1 \). In case of low degree, we have more precise results. For instance, for \( d=3 \), the trace forms a conic section. For \( d=4 \), we provide a necessary and sufficient condition for Blaschke products whose trace include a conic section.

The result by Bourgain that every unimodular function ψ \psi on the unit circle can be factored as ψ = e i v ~ B 1 B ¯ 2 \psi = e^{i \tilde v} B_1 \overline B_2 with B 1 B_1 and B 2 B_2 Blaschke products can be improved. We show that the same result holds with B 1 B_1 and B 2 B_2 interpolating Blaschke products. This will at the same time be a refinement of Jones’s result that every unimodular function can be approximated in the H ∞ H^\infty -norm by a ratio of interpolating Blaschke products.

In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron–Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron–Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and -\infty with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi [4], that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron–Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron–Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on C^r function spaces.

The first part of the paper provides a self-contained introduction on conformally barycentric extensions for the continuous circle maps under which the pushforward measures of the Lebesgue measure have no atom. Then it studies properties of such extensions, for examples, it shows that such extensions remain to be conformally natural and the extensions of the restrictions to the circle of Blaschke products preserving the unit disk are equal to those Blaschke products. Furthermore, it shows that the conformally barycentric extension of a locally quasisymmetric orientation-preserving endomorphism is a quasiregular covering from an annulus domain onto an annulus, both taking the unit circle as outer boundary.

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

Wilson-Polyakov loops for critical strings and superstrings at finite temperature

We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green’s function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and nonzero winding numbers, and to calculate any dynamical correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.

Szegö via Jacobi

“Worm” algorithm in quantum Monte Carlo simulations

If the symbol of an infinite Toeplitz band matrix has nonzero winding number, then the distances of the n × n truncations of this matrix to the nearest singular matrices go to zero with exponential speed as n → ∞. The purpose of this note is to show that if we measure the distance to the nearest singular matrix within the matrices of the same Toeplitz band structure as the original matrix, then things are less dramatic: this distance stays away from zero if only the origin does not belong to the Schmidt-Spitzer set.

Recent symmetry considerations [Flach et al., Phys. Rev. Lett. 84, 2358 (2000)] have shown that dc currents may be generated in the stochastic layer of a system describing the motion of a particle in a one-dimensional potential in the presence of an ac time-periodic drive. In this paper we explain the dynamical origin of this current. We show that the dc current is induced by the presence and desymmetrization of ballistic channels inside the stochastic layer. The existence of these channels is due to resonance islands with nonzero winding numbers. The characterization of the flight dynamics inside ballistic channels is described by distribution functions. We obtain these distribution functions numerically and find very good agreement with simulation data.

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.

More on lattice chiral gauge theories with Wilson-Yukawa terms

On the Polynomials Orthogonal on Regular Polygons

We consider continuous functions f which map the open unit disk D into the Riemann sphere W. For a point ζ on the unit circle C, we say that χ is a chord at ζ if χ is a chord of C having one endpoint at ζ and that Δ is a Stolz angle at ζ if Δ is a Stolz angle with vertex ζ. Suppose S denotes either a chord at ζ, a Stolz angle at ζ, or D.