Discrete Geometry and Symbolic Dynamics

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 classical techniques in symbolic dynamics applied to some codings of such discretizations allow one to obtain results concerning the enumeration of configurations and their statistical properties. Note that we have no claim to exhaustivity: the examples that we detail here have been chosen for their simplicity. Let us first illustrate this interaction with Figure 1.1 below where a piece of an arithmetic discrete plane in R3 is depicted, as well as its orthogonal projection onto the antidiagonal plane ∆: x1+x2+x3 = 0 in R 3, which can be considered as a piece of a tiling of the plane by three kinds of lozenges, and lastly, its coding as a two-dimensional word over a three-letter alphabet.