Summary: The best way to investigate the long-time behaviour of dynamical systems is to introduce an appropriate Poincaré mapping P and study its iterates. Two cases of physical interest arise: Conservative and dissipative systems. While the latter has been considered by a great many authors, much less is known for the first one (according to Liouville's theorem, here the mapping leaves a certain measure in phase space invariant). In this paper, we concentrate our attention on compact phase spaces (or, rather, surfaces of section). This assumption is mathematically useful and physically reasonable. We consider the simplest possible (2-dimensional) systems where the phase space is the compact unit disk \(\bar D\) in \({\mathbb{R}}^ 2\). A family of simple area-preserving mappings from \(\bar D\) onto itselves will be given and discussed in detail. It is shown that general characteristics of the dynamics are quite similar to those of e.g. the Hénon-Heiles system, while other features, as the structure of invariant curves, are different.