The infinite number of generalized dimensions of fractals and strange attractors

Abstract We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities Dq. We prove that lim q→0 Dq = fractal dimension (D), limq→1Dq = information dimension (σ) and Dq=2 = correlation exponent (v). Dq with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > Dq for any q′ > q. For homogeneous fractals Dq = Dq. A particularly interesting dimension is Dq=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D∞.