We present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of Wick contractions. The method gives exact results for monomials of low order. For higher–order monomials, it leads to an error of order 1/Nα where N is the dimension of the matrix and where α is independent of the degree of the monomial. We give a lower bound on the integer α and sho...

The theory of substitution sequences and their higher-dimensional analogues is intimately connected with symbolic dynamics. By systematically studying the factors (in the sense of dynamical systems theory) of a substitution dynamical system, one can reach a better understanding of spectral and topological properties. We illustrate this point of view by means of some characteristic examples, inc...

We propose a theory of deterministic chaos for discrete systems, based on their representations in symbolic history spaces Ω = (B ∞ , T ∆). These are spaces of semi-infinite sequences, as the one-sided shift spaces, but endowed with a more general topology T ∆ which we call a semicausal topology. We show that Ω is a metrizable Cantor set which embeds the chaotic attractor Λ. We discuss metrical...

A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks. A symbolic representation of the dynamics in this equation is given as a directed graph on an N-dimensional hypercube. This provides a formal link with discrete neural networks such as the original Hopfield models. Analytic criteria are given to establish steady states and limit cycle osci...