  This paper has two themes that are intertwined: The first is the dynamics of
certain piecewise affine maps on the Euclidean space that arise from a class of
analog-to-digital conversion methods called Sigma-Delta quantization. The
second is the analysis of reconstruction error associated to each such method.
Sigma-Delta quantization generates approximate representations of functions by
sequences that lie in a restricted set of discrete values. These are special
sequences in that their local averages track the function values closely, thus
enabling simple convolutional reconstruction. In this paper, we are concerned
with the approximation of constant functions only, a basic case that presents
surprisingly complex behavior. An m'th order Sigma-Delta scheme with input x
can be translated into a dynamical system that produces a discrete-valued
sequence (in particular, a 0-1 sequence) q as its output. When the schemes are
stable, we show that the underlying piecewise affine maps possess invariant
sets that tile the Euclidean space up to a finite multiplicity. When this
multiplicity is one (the single-tile case), the dynamics within the tile is
isomorphic to that of a generalized skew translation on the m dimensional
torus. The value of x can be approximated using any consecutive M elements in q
with increasing accuracy in M. We show that the asymptotical behavior of
reconstruction error depends on the regularity of the invariant sets, the order
m, and some arithmetic properties of x. We determine the behavior in a number
of cases of practical interest and provide good upper bounds in some other
cases when exact analysis is not yet available.
