Parametrized dynamics of the Weierstrass elliptic function

Abstract

Abstract. We study parametrized dynamics of the Weierstrass elliptic ℘ function by looking at the underlying lattices; that is, we study parametrized families ℘ Λ and let Λ vary. Each lattice shape is represented by a point τ in a fundamental period in modular space; for a fixed lattice shape Λ = [1, τ] we study the parametrized space kΛ. We show that within each shape space there is a wide variety of dynamical behavior, and we conduct a deeper study into certain lattice shapes such as triangular and square. We also use the invariant pair (g 2 , g 3 ) to parametrize some lattices.

Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

“…So, mkλ is indeed a superattracting periodic point of ℘ Λ and the proof is complete. This theorem/example stemmed form Lemma 7.2 in [HK2]. Theorem 9.3 (with n = 0) and Theorem 9.4 in this paper are other sources of examples in Theorem 15.4.9.…”

“…The proofs of the next two results are analogous to the proof of Theorem 15.7.2 (see Proposition 4.1 in [HKK], and Theorem 8.9 in [HK2]).…”

“…We then, starting with Section 15.5, provide also some different, interesting by their own, and historically first examples of various kinds of Weierstrass ℘ elliptic functions and their modifications. These come from the series of papers [HK1], [HK2], [HK3], [HKK], [HL] by Jane Hawkins and her collaborators. Throughout the whole current chapter we use the notation and terminology introduced in Chapter 11, particularly in Sections 11.3-11.6.…”

Misiurewicz parameters for Weierstrass elliptic functions based on triangle and square lattices

“…It is a wide class of meromorphic functions, periodic with respect to Λ and of order two. We refer to [5,6] for a nice description of dynamical and measure theoretic properties of ℘ Λ depending on the lattice Λ as well as investigation of some specific parametrized families of Weierstrass elliptic functions. For an introduction to the theory of iterating complex functions see e.g.…”

“…2 Dynamics of functions from families W t and W s Let us collect some information about the families W t and W s which will be helpful in our proofs, for more we refer to [6]. First recall that any elliptic function has no asymptotic values so the postsingular set P(f λ ) is the closure of the critical trajectories.…”