The dynamics of an iterated linear or affine map of the plane is well-understood and one considers polynomial or rational maps to uncover more exotic features. Another direction is to build maps from one or more linear maps applied according to where the orbit point is in the plane, or applied according to some symbolic sequence in the possible component maps. Such maps can reveal a mixture of regular quasiperiodic motion and chaotic motion. I will focus on maps with rational parameters that induce dynamics on the rational planar lattices and on their reductions mod p that give a dynamics over finite fields. I will describe various aspects of the dynamics. These include arithmetical complexity of rational orbits and, over finite fields, the associated graph theoretic issues related to the function graph of the dynamics and to the Cayley graph of the transformation group of unimodular matrices over FpFp.
[This is joint work with Franco Vivaldi, London]