Symplectic reversible maps, tiles and chaos
This nice paper extends studies by G. M. Zaslavskij and coworkers [\textit{A. A. Chernikov, R. Z. Sagdeev}, and \textit{G. M. Zaslavskij}, Physica D 33, No. 1-3, 65-76 (1988; Zbl 0696.58034)] on stochastic webs. The author first establishes the reversibility properties of the symplectic map \(F(z)= R_ \alpha z+\varepsilon (x)(-\sin \alpha,\cos\alpha)\). For periodic \(f\), some values of \(\alpha\) produce a kind of tiling of the plane which resembles some of Escher's pictures. Those correspond to the famous ``crystallographic condition''. However, for \(k\neq 1,2,3,4,6\), \(\alpha = {2\pi\over k}\) yields a quasicrystal (or an approximate) in the sense of \textit{V. I. Arnol'd} [Physica D 33, No. 1-3, 21-25 (1988; Zbl 0684.52008)].
scientific article; zbMATH DE number 64949