Algebraic non-integrability of the Cohen map

Marek Rychlik, Mark Torgerson

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

The map φ(x, y) = (√1 + x2 - y, x) of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability: The plane is a union of smooth, disjoint, invariant curves of the map φ. However, the integral has not explicitly been known. In the current paper we will show that the map φ does not have an algebraic integral, i.e., there is no non-constant function F(x, y) such that 1. F ○ φ = F; 2. There exists a polynomial G(x, y, z) of three variables with G(x, y, F(x, y)) = 0. Thus, the integral of φ, if it does exist, will have complicated singularities. We also argue that if there is an analytic integral F, then there would be a dense set of its level curves which are algebraic, and an uncountable and dense set of its level curves which are not algebraic.

Original languageEnglish (US)
Pages (from-to)57-74
Number of pages18
JournalNew York Journal of Mathematics
Volume4
StatePublished - Apr 13 1998

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Keywords

  • Algebraic
  • Complex dynamics
  • Integrability
  • Multi-valued map

ASJC Scopus subject areas

  • Mathematics(all)

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