Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields

Abstract

In this paper we study the dynamics of rational maps induced by endomorphisms of ordinary elliptic curves defined over finite fields.

Dynamics of the a-map over residually finite Dedekind domains and applications

“…These polynomials have many interesting properties and appear in diverse areas such as network coding theory [19] and finite projective geometries; see for example Chapter 3.4 of [3] and the notes at the end of this chapter for more properties and applications of these polynomials. The hanging trees attached to some periodic points of Chebyshev polynomials and maps induced by endomorphism of elliptic curves considered in [17], both over finite fields, can also be explained from our main result.…”

“…Let's consider an example taken from [17] (Example 4.1) Example 4.4. Let E : Y 2 = X 3 − X be the elliptic curve defined over F 73 and consider the endomorphism α(x, y) = (α 1 (x), yα 2 (x)) with α 1 (x) = −3(x 10 −3x 8 +5x 6 −5x 4 +3x 2 −1)…”

“…Dynamic of rational maps over finite fields such as Rédei functions [10] and maps of the form x → k(x + x −1 ) [15,16] have also been considered. The dynamic of certain maps associated with endomorphism of ordinary elliptic curves over finite fields was dealt in [17]. A survey on iteration of functions over finite fields is given in [4].…”

On the functional graph of the power map over finite groups

“…Elementary trees appear in a wide range of context when we describe the dynamics of different classes of maps, see for example [5,10,13,14,15,16,20]. They will play a key role also in our description of the functional graph associated with the power map over flower groups.…”

Density of Periodic Points for Lattès maps over Finite Fields

“…Ugolini [11] studied the phase portraits of Lattès maps associated to ordinary elliptic curves over finite fields. The methods of [11] are substantively equivalent to those we use to prove Theorem 1.2, but there appears to be no direct overlap in results.…”