  After investigating by examples the unusual and striking elementary
properties of the Penrose tilings and the Arnold cat map, we associate a finite
symbolic dynamics with finite grammar rules to each of them. Instead of
studying these Markovian systems with the help of set-topology, which would
give only pathological results, a noncommutative approximately finite
C*-algebra is associated to both systems. By calculating the K-groups of these
algebras it is demonstrated that this noncommutative point of view gives a much
more appropriate description of the phase space structure of these systems than
the usual topological approach.
  With these specific examples it is conjectured that the methods of
noncommutative geometry could be successfully applied to a wider class of
dynamical systems.
