The aim of this survey is to illustrate various connections that exist between word combinatorics and arithmetic discrete geometry through the discussion of some discretizations of elementary Euclidean objects (lines, planes, surfaces). We focus on the role played by dynamical systems (toral rotations mainly) that can be associated in a natural way with these discrete structures. We show how cl...

We introduce an equivalence relation between functions from to . By describing a symbolic dynamical system in terms of forbidden words, we prove that the -equivalence class of the function that counts the minimal forbidden words of a system is a topological invariant of the system. We show that the new invariant is independent from previous ones, but it is not characteristic. In the case of sof...

The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a genera...

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possess...