NOWHERE-DIFFERENTIABLE ATTRACTORS

The notion of nowhere-differentiable attractors is illustrated with four prototype equations, that is, maps of invertible type. Four classes of nowhere-differentiable attractors can be distinguished so far: the (nongeneric) continuous-nonchaotic-nonfractal type; the (nongeneric) continuous-fractal type; the (generic) singular-continuous-fractal type; and the (generic) continuous-fractal-in-a-projection type. The history of all four classes is linked with the name of J. A. Yorke in different ways. Even though continuous fractal nowhere-differentiable attractors do not exist generically, the hypothesis that the fractal geometry of nature may be a consequence of the fact that nature is a differentiable dynamical system is strengthened. Attractors with nowhere-differentiable generic projections can mimic the whole richness of fractal pictures. (C) 1995 John Wiley and Sons, Inc.