1,721,143 research outputs found

    Moebius Transformations and the Poincarè distance in the quaternionic setting.

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    In the space H\mathbb{H} of quaternions, we investigate the natural, invariant geometry of the open, unit disc ΔH\Delta_{\mathbb{H}} and of the open half-space H+.\mathbb{H}^+. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of ΔH\Delta_{\mathbb{H}} and H+\mathbb{H}^+ and identify original ways of representing them in terms of two isomorphic groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M\"obius transformations, and use it to define the analog of the Poincare' distances and differential metrics on ΔH\Delta_{\mathbb{H}} and $\mathbb{H}^+.

    Schr¿der equation in several variables and composition operators.

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    Let ϕ\phi be a holomorphic self-map of the open unit ball BnB^n of CnC^n such that ϕ(0)=0\phi(0)=0 and that the differential dϕ0d\phi_0 of ϕ\phi at 0 is non singular. The study of the Schroder equation in several complex variables σϕ=dϕ0σ\sigma \circ \phi=d\phi_0 \circ \sigma is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on BnB^n and to the theory of the discrete, complex dynamical systems. An extensive use of the infinite matrix which represents the composition operator associated to the map ϕ\phi leads to a simpler approach, and provides new proofs, to results of existence of solutions for the Schroder equation
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