### Abstract

The map φ(x, y) = (√1 + x^{2} - 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 language | English (US) |
---|---|

Pages (from-to) | 57-74 |

Number of pages | 18 |

Journal | New York Journal of Mathematics |

Volume | 4 |

State | Published - Apr 13 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*New York Journal of Mathematics*,

*4*, 57-74.

**Algebraic non-integrability of the Cohen map.** / Rychlik, Marek R; Torgerson, Mark.

Research output: Contribution to journal › Article

*New York Journal of Mathematics*, vol. 4, pp. 57-74.

}

TY - JOUR

T1 - Algebraic non-integrability of the Cohen map

AU - Rychlik, Marek R

AU - Torgerson, Mark

PY - 1998/4/13

Y1 - 1998/4/13

N2 - 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.

AB - 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.

KW - Algebraic

KW - Complex dynamics

KW - Integrability

KW - Multi-valued map

UR - http://www.scopus.com/inward/record.url?scp=0011873622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011873622&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011873622

VL - 4

SP - 57

EP - 74

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -