Global metric regularity of the devil's staircase of topological entropy

Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps

The dual-star composition rule of doubly superstable (DSS) sequences presents a complete renormalizable algebraic structure for studying Feigenbaum's metric universality and self-similar classification of DSS sequences in symbolic dynamics of bimodal maps of the interval. Here an important feature is that the complete combinations of up- and down-star products create all the generalized Feigenbaum's routes of transitions to chaos. These routes can be classified into two types: one consists of countably infinitely many regular routes which preserve Feigenbaum's metric universality; another consists of uncountably infinitely many universal nonscaling routes described by the irregularly mixed dual-star products, which break Feigenbaum's asymptotically convergent metric universality although they are structurally universal. The combinatorial complexity of dual-star products may increase the grammatical complexity of languages of symbolic dynamics. Moreover, it is found that there exists a global regularity between the fractal dimensions d and the scaling factors [alpha(C),alpha(D)] for Feigenbaum-type attractors: d(Z)log(/Z/)/alpha(C)(Z)alpha(D)(Z)/=beta((2)), where beta((2)) is independent of the concrete DSS sequences Z.