Sensitive dependence on initial conditions

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 ap-plies 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—χ-systems—as a better abstract framework for a general theory of chaotic dynamics. The vague notion of Chaos has attracted a great deal of attention in recent years and several authors have tried to formalize it in various ways. One popular such attempt uses the definition of “sensitive dependence on initial conditions”. A chaotic system is defined, according to this school, to be a compact metric space