Low-dimensional non-linear dynamical systems and generalized entropy

Low-dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to i nitial conditions at the edge of chaos which can be naturally formulated wi thin the generalized Tsallis statistics prescription which is characterized by the entropic index q. General scaling arguments provide a direct relati on between the entropic index q and the scaling exponents associated with t he extremal sets of the multifractal critical attractor. The above result c omes in favor of recent conjectures that. Tsallis statistics is the natural frame for studying systems with a fractal-like structure in the phase-spac e. Power-law sensitivity in high-dimensional dissipative and Hamiltonian sy stems are also discussed within the present picture.