• polynomial dynamics
• periodic and preperiodic points
• canonical heights
• fully invariant sets
• Narkiewicz's properties
• number fields

This thesis discusses about algebraic dynamic, that is, the application of algebraic number theory to dynamical systems, expecially to polynomial ones.

In the first part we deal with periodic and preperiodic orbits. After showing the most classical results about finiteness properties, we discuss the problem of the maximum length of a finite orbit and give a complete classification of periodic and preperiodic orbits in case of a quadratic number field.

In the second part we study fully invariant sets for polynomial mappings and state some particular finiteness properties, called "Narkiewicz's properties", focusing particularly on the relationships among them.