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Attractors with vanishing rotation number

Rafael Ortega, Francisco Ruiz del Portal (2011)

Journal of the European Mathematical Society

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carath´eodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

Boundary approach filters for analytic functions

J. L. Doob (1973)

Annales de l'institut Fourier

Let H ∞ be the class of bounded analytic functions on D : | z | < 1 , and let D ‾ be the set of maximal ideals of the algebra H ∞ , a compactification of D . The relations between functions in H ∞ and their cluster values on D ‾ - D are studied. Let D 1 be the subset of D ‾ over the point 1. A subset A of D 1 is a “Fatou set” if every f in H ∞ has a limit at e i θ A for almost every θ . The nontangential subset of D 1 is a Fatou set according to the Fatou theorem. There are many larger Fatou sets, for example the fine topology subset of D 1 but...

Boundary behavior and Cesàro means of universal Taylor series.

Frédéric Bayart (2006)

Revista Matemática Complutense

We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its Cesàro means are universal.

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

Let u be harmonic in the half-space R + n + 1 , n ≥ 2 . We show that u can have a fine limit at almost every point of the unit cubs in R n = ∂ R + n + 1 but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary.In R + 2 it is known that the Hardy classes H p , 0 < p < ∞ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability...

Cluster sets of analytic multivalued functions

S. Harbottle (1992)

Studia Mathematica

Classical theorems about the cluster sets of holomorphic functions on the unit disc are extended to the more general setting of analytic multivalued functions, and examples are given to show that these extensions cannot be improved.

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