Fundamental concepts of classical chaos. Part II: Fractals and chaotic dynamics

With the concept of fractals, introduced by B. B. Mandelbrot in the 1970s, geometry assumes again, after Poincaré, a leading role in the theory of dynamical systems and chaos. Dynamical instability and unpredictability in time become inseparable from geometrical complexity and irregularity in space, through self-similarity under scaling. Chaos theory thus acquires its natural setting and description in terms of fractal geometry and symbolic dynamics. Objects of non-integer dimension and distributions with spectra of generalized dimensions become familiar concepts of aesthetic, even philosophical value, while giving researchers at the same time new tools to probe deeper into complex natural phenomena. In this paper, I review in a pedagogical way the main ideas of fractal geometry, multifractal distributions and symbolic dynamics. Of central importance is the connection between temporal and spatial complexity, while important applications of the formalism are also mentioned, particularly in the area of chaotic time series analysis.