Symbolic dynamics of piecewise-linear maps

In this paper we develop the theory of symbolic dynamics on piecewise-linear maps. We prove several results concerning periodic points and admissible periodic sequences and show how this theory is used on maps which are composed of signum functions by means of two examples in signal processing, namely digital filters with overflow nonlinearity and sigma-delta modulators. For example, we show that in the double-loop sigma-delta modulator with a two-bit quantizer, the set of initial conditions which generate periodic output has zero measure for any constant input, in contrast to the single-loop sigma-delta modulator