Nonlinear rotations on a lattice

Abstract

We consider a prototypical two-parameter family of invertible maps of Z 2 , representing rotations with decreasing rotation number. These maps describe the dynamics inside the island chains of a piecewise affine discrete twist map of the torus, in the limit of fine discretisation. We prove that there is a set of full density of points which, depending of the parameter values, are either periodic or escape to infinity. The proof is based on the analysis of an interval-exchange map over the integers, with infinitely many intervals. 1 arXiv:1709.08904v1 [math.DS]

Nonlinear Rotations on a 3-Dimensional Lattice

“…In this map, the parameter 𝛼 represents the "stretching" of the elliptic-type orbits in horizontal direction and the parameter 𝛽 can be viewed as the "shrinking" of the elliptic-type orbits in the vertical direction. The behavior of the discrete points ∅ defined in equation ( 4) is discussed in detail in the paper by Alwani and Vivaldi (2018). They have proved that if 𝛼 ̅ = 𝛼 𝑔𝑐𝑑(𝛼,2𝛽)…”

“…This research is motivated by the question of what happens to the trajectory of this 2-dimensional (2-D) space orbits in the 3dimensional (3-D) space? Basically, in paper (Alwani & Vivaldi, 2018), this 2-D lattice is reduced to the 1-dimensional (1-D) lattice by using Poincaré surface of section that is defined by Σ = {(𝑋, 0) ∈ ℤ 2 : 𝑋 ≥ 0}. We will modify this 2-D map defined in equation ( 4) by adding "extra" Z-axis in the map to investigate the orbits in the 3-D space.…”