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- W2905074547 abstract "Two ideal polygons, $(p_1,ldots,p_n)$ and $(q_1,ldots,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $alpha$-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = alpha$ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Mobius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures, and show that these relations, with different values of the constants $alpha$, commute, in an appropriate sense. We investigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many $alpha$-related polygons, and study the ideal polygons that are $alpha$-related to themselves (with a cyclic shift of the indices)." @default.
- W2905074547 created "2018-12-22" @default.
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- W2905074547 date "2018-12-13" @default.
- W2905074547 modified "2023-09-27" @default.
- W2905074547 title "Cross-ratio dynamics on ideal polygons" @default.
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