In this thesis, the structure and the distribution of periodic (and preperiodic) orbits in certain discrete dynamical systems on the torus are studied.

The first part is concerned with the action of toral endomorphisms on the rational lattices of the torus.

The connection between fixed point counts on the whole torus and on particular lattices is discussed, and the period lengths on a given lattice are studied in a group-theoretic setting. For non-invertible endomorphisms, the structure of the trees induced on certain lattices, as well as on the union of all rational lattices is investigated.

Furthermore, the symmetry properties of toral endomorphisms

are studied in terms of their (local) symmetry and reversing symmetry groups.

The second part is devoted to the study of

the Casati-Prosen map for rational parameter pairs,

a two-parameter family of random reversible maps that preserve certain rational lattices of the torus.

Based on exact computations, conjectures concerning the period distribution for certain parameter values are stated and numerical evidence in support of these conjectures is given.