A family of rational maps of the plane with prefocal set at infinity

The object of the present work is to study a one-dimensional nonautonomous equation (a Mann iteration) with geometric weights. Its study amounts to that of a two-dimensional autonomous rational map having a vanishing denominator in a non classical case. A map with a vanishing denominator possesses a set of points, called the prefocal set, with particular properties, that is, whose preimages are related to a single point called the focal point. A contact of a basin boundary with a prefocal set is a bifurcation giving rise to a qualitative change of this boundary by creation of loops at the focal point. With respect to classical cases, the particularity of the map family considered here consists in the fact that the prefocal set is at infinite distance, i.e. it is the set of points (x,±) x varying in , the related focal points being at finite distance. The main result is related to the formulation of a one-to-one correspondence between arcs passing through the focal points and the points at infinity of type (x,±). A change of variables facilitates the study by relocating focal points and prefocal sets at finite distance.