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Schwarz reflection geometry. I: Continuous iteration of reflection. (English) Zbl 1078.53044

Let \(\Gamma_1, \Gamma_2\) be two regular real-analytic curves in the complex space \(\mathbb C\). Reflecting in the Schwarz sense \(\Gamma_1\) with respect to \(\Gamma_2\) gives rise to a third curve \(\Gamma_3=\Gamma_2\cdot \Gamma_1\). Iterating this process one obtains a discrete dynamical system (in the space of curves). Since each Schwarz reflection is associated to a holomorphic function – the Schwarz function – one can interpret the previous dynamical system as a sequence of mappings. Replacing the discrete variables with real continuous variables and deriving, one obtains then a partial differential equation of the type \({d\over dt} S(t,z)=g(z){d\over dz}S(t,z)\) with \(t\in \mathbb R\), \(z\in \mathbb C\) and \(g\) a holomorphic function. Such an equation is conformally invariant and can be normalized in order to consider the initial curve to be the real axis. This allows to obtain a canonical parametrization for the curve associated to the Schwarz function \(S(t,z)\) solution of the normalized equation. Starting from this, the authors prove then that the equations of continuous reflections are reducible to quadrature.
In the second part of the paper, the continuous reflection equation is interpreted as the geodesic equation on the space of analytic curves with respect to a symmetric affine connection. This construction is quite ingenious and delicate and this reviewer is not able to better describe it in few lines.
However, the paper is pretty well readable, contains a lot of clear examples and a useful appendix about symmetric spaces formalism.

MSC:

53C35 Differential geometry of symmetric spaces
53A30 Conformal differential geometry (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
58B25 Group structures and generalizations on infinite-dimensional manifolds
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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