Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

Parametrized dynamics of the Weierstrass elliptic function

“…It was shown in [9] that for one shape, say a square, which corresponds to the point i ∈ B, there are many types of dynamical behavior and many different Julia sets possible. Therefore we fix a point in B, and to that point we associate an entire parameter plane of Weierstrass elliptic ℘ functions (not all distinct, see Section 6).…”

“…What sets this family of transcendental maps apart from many of the others studied is the rich algebraic structure of elliptic functions which influences the dynamics in diverse ways. For example, we can give precise formulas for the generators of lattices Λ giving rise to Weierstrass elliptic functions with certain prescribed Fatou components; there are many theorems of this nature in [9] and in this paper.…”

“…In an earlier paper the authors give examples and results about the dynamics of the Weierstrass elliptic ℘ function considered as a meromorphic function of C [9]. What sets this family of transcendental maps apart from many of the others studied is the rich algebraic structure of elliptic functions which influences the dynamics in diverse ways.…”

“…Let ℘ Λ denote the Weierstrass elliptic ℘ function based on the lattice Λ; it is shown in [9] that the measure theoretic and dynamical properties of ℘ Λ depend on Λ. Moreover, even for a fixed shape lattice such as square, the dynamics vary widely.…”

“…We give examples of rectangular square lattices (generated by vectors λ and λi for some λ > 0), some with superattracting fixed points and others with Julia set the whole sphere. In [9] we focused on lattices Λ such that Λ = Λ, i.e., real lattices; since one of the critical points and many of the critical values are real in this case, this restriction provided useful tools for the study. However, many general results about the dynamics of elliptic functions can be proved without restricting to real lattices or to the Weierstrass ℘ function (see for example [14]).…”