Fractal dimensions and ƒ(α) spectrum of the Hénon attractor

We measure the generalized fractal dimensions D_q(q≥0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the D_q yields the f{hook}(α) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.