Global scaling behaviors and chaotic measure characterized by the convergent rates of period-p-tupling bifurcations.

Abstract

The Derrida-Gervois-Pomeau composition of the sequences Q* is a topological conjugate compression operation. On the basis of the numerical calculation for the quadratic map f(lambda)(x) = 1 - lambda x(2), it is found that Q* has the uniform compression ratio delta(-1)(Q), i.e., the reciprocal of the universal convergent rate. A series of global scaling behaviors relating to the periodic windows, the window bands, and the steps of equal topological entropy class (ETEC) are revealed to be independent of the sequences. In particular, the geometric interpretation for the Yorke-Grebogi-Ott-Tedeschini-Lalli normalized crisis value of ''windows'' mu(c) = 9/4 is given clearly through the scaling for the ETEC steps. The universal positions of superstable points in all the periodic windows are determined by the window scaling ratios gamma(PDB)=1 for the period-doubling bifurcation (PDB) sequences and gamma(N)=1/3 for the non-PDB sequences. The approximate analytic formula of the chaotic measure is obtained by employing the convergent rates delta of periodic sequences. The singularity spectrum and the generalized fractal dimension of the chaotic set are also calculated. These results imply that the metric universality of the periodic sequences may be a very good complete description for the topological space of two symbols.