Blaschke product generated covering surfaces

Bârză, Ilie; Ghişa, Dorin

doi:10.21136/mb.2009.140652

Cited by 3 publications

(8 citation statements)

References 2 publications

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“…The arcs (4) have been obtained by simultaneous continuation (see [2]) of paths in ( C, B a ) over the real non negative half-axis in the w-plane. We have also found (see [3]) that the cover transformations of ( C, B a ) were Möbius transformations of the form:…”

Section: Blaschke Productsmentioning

confidence: 79%

“…In [2] and [3] we gave a complete description of the domains Ω k of injectivity (fundamental domains) of the mappings when the Blaschke product is of the form:…”

Section: Blaschke Productsmentioning

confidence: 99%

“…We notice that this surface has two branch points: 0 and ∞, which correspond to a and 1/a , the branch points of ( C, B a ). This last surface is the Riemann surface of the multivalued function w → e iθ r − w 1/n 1 − rw 1/n obtained when trying to find an inverse of the function (3).…”

Section: Blaschke Productsmentioning

confidence: 99%

“…It is known (see [2] and [3]) that if the Blaschke product w = B(z) has the (generalized) Cantor set E as the set of cluster points of its zeros, then the equation B(z) = 1 has infinitely many distinct solutions in every "removed" open arc I n ⊂ ∂D \ E. The only cluster points of these solutions are the end points of I n . Therefore,…”

Section: Infinite Blaschke Productsmentioning

confidence: 99%

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Colour visualization of Blaschke product mappings

Ballantine

Ghişa

2010

Complex Variables and Elliptic Equations

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A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products, for which coloring pre-images of annuli centered at the origin allow us to describe the mappings with a high degree of fidelity. Additional graphics and animations are provided on the web site of the project [1].

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Section: Blaschke Productsmentioning

confidence: 79%

“…In [2] and [3] we gave a complete description of the domains Ω k of injectivity (fundamental domains) of the mappings when the Blaschke product is of the form:…”

Section: Blaschke Productsmentioning

confidence: 99%

Section: Blaschke Productsmentioning

confidence: 99%

Section: Infinite Blaschke Productsmentioning

confidence: 99%

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Colour visualization of Blaschke product mappings

Ballantine

Ghişa

2010

Complex Variables and Elliptic Equations

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“…The papers [2]- [4] deal with the study of global geometric properties of Blaschke products, while [5] and [6] refer to Blaschke self mappings of the real projective plan. In the case of infinite Blaschke products and quotients we restricted our study to the situation where E is a (generalized) Cantor subset of the unit circle.…”

Section: Blaschke Products and Blaschke Quotientsmentioning

confidence: 99%

Color Visualization of Blaschke Self-Mappings of the Real Projective Plan

Ballantine,

Ghisa

2009

Preprint

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The real projective plan P 2 can be endowed with a dianalytic structure making it into a non orientable Klein surface. Dianalytic self-mappings of that surface are projections of analytic selfmappings of the Riemann sphere C. It is known that the only analytic bijective self-mappings of C are the Möbius transformations. The Blaschke products are obtained by multiplying particular Möbius transformations. They are no longer one-to-one mappings. However, some of these products can be projected on P 2 and they become dianalytic self-mappings of P 2 . More exactly, they represent canonical projections of non orientable branched covering Klein surfaces over P 2 . This article is devoted to color visualization of such mappings. The working tool is the technique of simultaneous continuation we introduced in previous papers. Additional graphics and animations are provided on the web site of the project [1].

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