Geodesic
Involutive birational transformations of arbitrary complexity in Euclidean spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 1, pp. 111-117.

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A broad family of involutive birational transformations of an open dense subset of $\mathbb R^n$ onto itself is constructed explicitly. Examples with arbitrarily high complexity are presented. Construction of birational transformations such that $\phi^k= \mathrm{Id}$ for a fixed integer $k>2$ is also presented.
Classification : 14E05
Keywords: rational mapping; birational transformation; involutive transformation 
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     title = {Involutive birational transformations of arbitrary complexity in {Euclidean} spaces},
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Dušek, Zdeněk; Kowalski, Oldřich. Involutive birational transformations of arbitrary complexity in Euclidean spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 1, pp. 111-117. http://geodesic.mathdoc.fr/item/CMUC_2013__54_1_a9/