We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.

For piecewise monotone interval maps, we show that the Kolmogorov-Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.

Recent use of order patterns in time-series analysis shows the need for a corresponding theory. We determine probabilities of order patterns in Gaussian and autoregressive moving-average (ARMA) processes. Two order functions are introduced which characterize a time series in a way similar to autocorrelation. For stationary ergodic processes, all finite-dimensional distributions are obtained from the one-dimensional distribution plus the order structure of a typical time series. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.

A Mandelbrot set M for pairs of complex linear maps was introduced by Barnsley and Harrington in 1985. Bousch proved that M is locally connected, and Odlyzko and Poonen studied the related set of all complex roots of polynomials with coefficients 0 and 1. We give an algorithm to construct this set and study its geometric structure. In contrast to the Mandelbrot set for quadratic maps, it is shown that M is not simply connected.

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