184 research outputs found
On asymptotic Teichmüller space
In this article we prove that for any hyperbolic Riemann surface M of infinite analytic type, the little Bers space Q0(M) is isomorphic to c0. As a consequence of this result, if M is such a Riemann surface, then its asymptotic Teichm¨uller space AT(M) is bi-Lipschitz equivalent to a bounded open subset of the Banach space l∞/c0. Further, if M and N are two such Riemann surfaces, their asymptotic Teichm¨uller spaces, AT(M) and AT(N), are locally bi-Lipschitz equivalen
Local rigidity of infinite-dimensional Teichmüller spaces
This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space , for such a Riemann surface , is isomorphic to the Banach space of summable sequence, . This implies that whenever and are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then is isomorphic to . It is known from V. Markovic that if there is a linear isometry between and , for two Riemann surfaces and of non-exceptional type, then this isometry is induced by a conformal mapping between and . As a corollary to this rigidity theorem presented here, taking the Banach duals of and shows that the space of holomorphic quadratic differentials on , is isomorphic to the Banach space of bounded sequences, . As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent
Quasiregular mappings of polynomial type in R^2
Complex dynamics deals with the iteration of holomorphic functions. As is
well- known, the first functions to be studied which gave non-trivial dynamics
were quadratic polynomials, which produced beautiful computer generated
pictures of Julia sets and the Mandelbrot set. In the same spirit, this article
aims to study the dynamics of the simplest non-trivial quasiregular mappings.
These are mappings in R^2 which are a composition of a quadratic polynomial and
an affine stretch.Comment: 17 pages, 7 figure
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