Geodesic
On random choice of elliptic and hyperbolic rotations of the Lorentz spaces
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 955-971

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Elliptic and hyperbolic rotations of the $(n+1)$-dimensional Lorentz space can be represented as exponential of rank $2$ matrices of the real Lie algebra $\mathfrak{so}(1, n)$. We shown that the ratio of the volumes of the corresponding sets of matrices Euclidean norm $\leqslant r$ is equal to $(\sqrt2)^{n-1}-1$ for all $r > 0$. Consequently the portion of hyperbolic rotations near identity decreases exponentially with increasing $n$. Another corollary is that in case of Minkovski space of special relativity choose of elliptic and hyperbolic rotations near identity is equiprobable.
Mots-clés : elliptic rotation, random matrix.
Keywords: hyperbolic rotation 
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     author = {V. A. Churkin and A. I. Ilin},
     title = {On random choice of elliptic and hyperbolic rotations of the {Lorentz} spaces},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {955--971},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a28/}
}
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V. A. Churkin; A. I. Ilin. On random choice of elliptic and hyperbolic rotations of the Lorentz spaces. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 955-971. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a28/