 

Marek Rychlik and Mark Torgerson
Algebraic NonIntegrability of the Cohen Map

Published: 
April 13, 1998 
Keywords: 
algebraic, integrability, multivalued map, complex dynamics 
Subject: 
58F23, 39A, 52A10 


Abstract
The map φ(x,y)=(\sqrt{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
nonconstant function F(x,y) such that:
 F∘φ=F.
 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.


Acknowledgements
This research has been supported in part by the National Science Foundation under grant no. DMS 9404419


Author information
Marek Rychlik:
Department of Mathematics, University of Arizona, AZ 85721, USA
rychlik@math.arizona.edu
http://alamos.math.arizona.edu/
Mark Torgerson:
Department of Mathematics, University of Arizona, AZ 85721, USA

