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Sensitive dependence on initial conditions. (English) Zbl 0790.58025
Summary: It is shown that the property of sensitive dependence on initial conditions in the sense of {Guckenheimer} follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity, minimality and positive topological entropy. In light of these investigations and several examples which we exhibit, we propose a natural family of dynamical systems -- $\chi$-systems -- as a better abstract framework for a general theory of chaotic dynamics.

37A99Ergodic theory
37B99Topological dynamics
54H20Topological dynamics
20D05Finite simple groups and their classification
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