Dynamics of vertical real rhombic Weierstrass elliptic functions

Single and double toral band Fatou components in meromorphic dynamics

“…(3) We restrict our attention to the map f (x) = ℘(x) − a, where x ∈ (0, λ 1 ) and a > a 0 . (4) As a increases, there is a unique fixed point, This method goes over to real triangular lattices with g 3 > 0 since the Schwarzian derivative of ℘ Λ (and therefore ℘ Λ + b, b ∈ C) is negative [23]. We can modify the above procedure to obtain nonperiodic horizontal toral bands that iterate to attracting cycles.…”

“…The dynamics of ℘ and ℘ also have some connections to iteration of rational maps of the Riemann sphere, as was known to Lattès as early as 1918. The earliest work on iterating these functions with a view to understanding their Julia and Fatou sets seems to have been started in the early 1990s in a series of papers starting with [1], and including extensive results by Kotus and Urbánski [24] and [25], and a plethora of examples and results appearing in [14]- [23]. Iteration of meromorphic functions has been widely studied by many, for example [1]- [4], [6], [9], and [10].…”

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“…(3) We restrict our attention to the map f (x) = ℘(x) − a, where x ∈ (0, λ 1 ) and a > a 0 . (4) As a increases, there is a unique fixed point, This method goes over to real triangular lattices with g 3 > 0 since the Schwarzian derivative of ℘ Λ (and therefore ℘ Λ + b, b ∈ C) is negative [23]. We can modify the above procedure to obtain nonperiodic horizontal toral bands that iterate to attracting cycles.…”

“…The dynamics of ℘ and ℘ also have some connections to iteration of rational maps of the Riemann sphere, as was known to Lattès as early as 1918. The earliest work on iterating these functions with a view to understanding their Julia and Fatou sets seems to have been started in the early 1990s in a series of papers starting with [1], and including extensive results by Kotus and Urbánski [24] and [25], and a plethora of examples and results appearing in [14]- [23]. Iteration of meromorphic functions has been widely studied by many, for example [1]- [4], [6], [9], and [10].…”