Elliptic Curves as Attractors in ℙ²Part 1: Dynamics

Abstract

A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe "blowout" bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, the elliptic curve (a topological torus) can never have a trapping neighborhood, yet it can have an attracting basin of large measure (perhaps even of full measure). We also describe examples where there appear to be Herman rings (that is topological cylinders mapped to themselves with irrational rotation number) with open attracting basin. In some cases we provide proofs, but in other cases the discussion is empirical, based on numerical computation.

Bifurcations in the elementary Desboves family

“…The family above (referred to as the elementary Desboves family) was studied by Bonifant-Dabija [BD02] and Bonifant-Dabija-Milnor [BDM07], who revealed the dynamical richness of these maps. One of their main properties is that the small Julia set J 2 (λ) of f λ (the support of the equilibrium measure) and its large Julia set J 1 (λ) (the complement of the Fatou set, which coincides with the support of the The first author was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002 and by the FIRB2012 grant "Differential Geometry and Geometric Function Theory", RBFR12W1AQ 002.…”

“…For other values of λ, it can give a parabolic petal or an attracting disc in J 2 (λ). It is even possible to have a set of positive Lebesgue measure covered by such attracting discs, see [BDM07,Theorem 6.3] and [Taf10]. In particular, the small Julia set has positive measure for these parameters.…”

“…See [BD02] and [BDM07] for a more detailed description. We start considering the classical Desboves map given by…”

“…We will now consider a well-chosen perturbation of f D , still leaving the curve C invariant. A natural way to do this is to consider the following Desboves family (studied in [BDM07]), parametrized by (a, b, c) ∈ C 3 . We set Φ(x, y, z) := x 3 + y 3 + z 3 for convenience.…”

Semigroup Actions on Sets and the Burnside Ring

“…Attractors of finite state automata. It is well known that the notion of attractors and attracting sets play an important role in physics, geometry and in particular in the theory of dynamical systems, see for example [1], [10], [2]. We define analogues notion for finite state automata.…”