Celebratio Mathematica — Fuchs — Complete Bibliography

Complete Bibliography

Works connected to Maxim D. Arnold

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M. Arnold, D. Fuchs, I. Izmestiev, S. Tabachnikov, and E. Tsukerman: “Iterating evolutes and involutes,” Discrete Comput. Geom. 58 : 1 (2017), pp. 80–143. MR 3658331 Zbl 1381.51007 ArXiv 1510.07742 article

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (\( \mathcal{P} \)-evolutes), or by the incenters of the triples of consecutive sides (\( \mathcal{A} \)-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of \( \mathcal{P} \)-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study \( \mathcal{P} \)- and \( \mathcal{A} \)-involutes and prove that the side directions of iterated \( \mathcal{P} \)-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Maxim D. Arnold	Related
Emmanuel Tsukerman	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related

M. Arnold, D. Fuchs, I. Izmestiev, and S. Tabachnikov: Cross-ratio dynamics on ideal polygons. Preprint, December 2018. ArXiv 1812.05337 techreport

Two ideal polygons, \( (p_1,\dots,p_n) \) and \( (q_1,\dots,q_n) \), 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 Möbius-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).

Maxim D. Arnold	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related