Non integrable dynamics : time quantitative results Very little is known about non integrable dynamical systems a subject on the borderline of mathematics and physics. This is an attempt to give a first coherent description of a new and extremely exciting aspect of this subject This is a book on some flat dynamical systems which can be read without any background of ergodic theory The only technical requirement is a basic understanding of some basic linear algebra and elementary number theory The authors take great care in introducing the ideas at a leisurely pace and often explain the main ideas by studying some examples The ideas are further illustrated by over 200 figures The book is accessible to a beginning graduate student as well as any interested experienced researcher