From Lattès’ results to the notion of focal point

This contribution is a development of a Note [1] based on Samuel Lattes’ paper [2] Published in French, these two papers remained quasi unknown in the contemporary literature. Lattes’ contribution describes a method for generating particular families of Dimp nonlinear invertible maps T with vanishing denominators, Such a map

have an exceptional property : their inverse T^-1 is easily obtained and, with an initial condition (x₀,x₁,..., x_p-1), the solution x_n = g(x₀, x₁,...,x_p-1) can be expressed from the classical elementary functions of nontranscendental type. Considering $n \equiv t$ as a continuous parameter, the trajectories in the phase space are analytically defined in a parametric representation. For p = 2, Mira’s 1981 Note presents an example from Lattes’ paper [2], and another one constructed according to the same method. So, the solution x_n = g(x₀, x₁, ..., x_n) defines the trajectories in the phase plane, the basin of attractor of the attractor being well bounded by asymptotic arcs. Moreover, among the singular sets, the point Q where denominator and numerator of the map are both cancelling, plays a fundamental role. In French, it has been called point de focalisation (in English focal point).

Mira CRAS 1981

From a 2D example, the last page of the 1981 Note mentions another result (most likely unknown at that time) : a 2D polynomial map T, containing a product of two different variables (so its inverse T-1 being a map with a vanishing denominator may generate a focal point Q. In this contributino, it will be shown that two types of attractor appear : either a complex attracting set resembling to a closed curve with a loop at Q and its iterates, or to a chaotic attractor organized around Q and its iiterates, each one framing the trajectories. The 3D polynomial case is briefly considered.

Talk given by Christian Mira