A tensor approach to higher order expectations of quantized chaotic trajectories. I. General theory and specialization to piecewise affine Markov systems

The problem of computing any-order expectations of trajectories generated by discrete-time one-dimensional chaotic systems is addressed by means of a suitable generalization of the Perron-Frobenius operator and its quantization. Tools from tensor algebra are introduced and analytical expressions for the special case of piecewise-affine Markov maps are obtained. Results are further specialized for a family of maps with quite general features. As an example application, some cross- and self-interference terms are computed, which are involved in the evaluation of the performance of chaos-based DS-CDMA systems in an asynchronous multipath environment.