Statistical properties of bidimensional patterns generated from delayed and extended maps

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Abstract
The space-time chaotic patterns associated with a class of dynamical systems ranging from delayed to extended maps are investigated. All the systems are constructed in such a way that the corresponding two-dimensional (2D) representation is characterized by the same updating rule in the bulk. The main difference among them is the direction of the ‘‘time’’ axis in the plane. Despite the different causality relations among the various models, the resulting patterns are shown to be statistically equivalent. In particular, the Kolmogorov-Sinai entropy density assumes always the same value. Therefore, it can be considered as an absolute indicator, measuring the amount of disorder of a 2D pattern. The Kaplan-Yorke dimension density is instead rule dependent: this indicator alone cannot be used to quantify the degrees of freedom of a given pattern; one must further specify the direction of propagation in the plane.