ON A PARTICULAR FOLIATION ASSOCIATED WITH A POLYNOMIAL FAMILY OF NONINVERTIBLE MAPS OF THE PLANE

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z₀) each point having no real preimage, and (ii) (denoted by Z₂) each point having two real preimages. The regions Z₀, Z₂ are separated by the critical curve LC, locus of points having two coincident preimages. Z₂ is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z₂ is separated into two zones, Z'₂ and , inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z'₂ and is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Z_k (k=1,3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map.