Geodesic
The Longest Curves of Given Degree and the Quasicrystallic Harnack Theorem in Pseudoperiodic Topology
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 1-8

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Upper bounds for ergodic averages of topological characteristics of pseudoperiodic functions and manifolds are found in terms of the degrees of trigonometric polynomials defining these functions and manifolds. The bounds are based on finding the longest trigonometric and spherical curves of a fixed degree.
Keywords: Betti numbers, ergodic theory, characteristic numbers, perihelion, quasicrystalls, Sturm theory, Morse theory. 
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V. I. Arnol'd. The Longest Curves of Given Degree and the Quasicrystallic Harnack Theorem in Pseudoperiodic Topology. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 1-8. http://geodesic.mathdoc.fr/item/FAA_2002_36_3_a0/