  We introduce a double-folded operator that, upon iterative application,
generates a dynamical system with two types of trajectories: a cyclic one and,
another that grows endlessly on parabolas. These trajectories produce two
distinct partitions of the set of lattice points in the plane.
  Our object is to analyze these trajectories and to point out a few special
arithmetic properties of the integers they represent.
  We also introduce and study the parabolic-taxicab distance, which measures
the fast traveling on the steps of the stairs defined by points on the
parabolic trajectories whose coordinates are based on triangular numbers.
