This article is cited in 4 papers

Morse–Smale circle diffeomorphisms and moduli of elliptic curves

Yu. S. Ilyashenko^ab, V. S. Moldavskii<sup>b

a</sup> Steklov Mathematical Institute, Russian Academy of Sciences
^b Cornell University

Abstract: To any circle diffeomorphism there corresponds, by a classical construction of V. I. Arnold, a one-parameter family of elliptic curves. Arnold conjectured that, as the parameter approaches zero, the modulus of the corresponding elliptic curve tends to the (Diophantine) rotation number of the original diffeomorphism. In this paper, we disprove the generalization of this conjecture to the case when the diffeomorphism in question is Morse–Smale. The proof relies on the theory of quasiconformal mappings.

Key words and phrases: Circle diffeomorphism, rotation number, moduli of elliptic curves, quasiconformal mappings.

MSC: 37E10, 37F30

Received: January 10, 2003

Language: English

DOI: 10.17323/1609-4514-2003-3-2-531-540

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