[en] Star products in symbolic dynamics of 1D quadri-modal maps are presented, the complexity of substitution rules is discussed besides their inherent cyclic and dual properties. Feigenbaum's metric universalities in bifurcations of period-n-tupling sequences are calculated by the new numerical method of the word-lifting technique for quadri-modal maps. It is known that symmetries of dynamic behavior are pretty different between even-modal maps and odd-modal maps, the former has central symmetric property in phase space. This paper provide a complete example to obtain star products of even-modal maps

[en] A framework of the sequence renormalization in symbolic space, which degenerates for order patterns of block shifts, is presented in this paper. The order isomorphism of both the higher power shift on a block sequence and the shift on a sequence is the topological and metric foundation of renormalization group (RG) operator. Based on the order relation with many compositions, the generic form of RG equations in multimodal maps is derived

[en] By means of star products and high precision numerical calculation, an abnormal phenomenon is found in period-p-tupling bifurcation processes in one-dimensional trimodal maps. A route of transition to chaos, presented by a right-associative non-normal star product, breaks the Feigenbaum's metric universality, namely, the conventional Feigenbaum's successive rates exhibit a strong divergence. To overcome the divergence, an approximate scheme of accelerating convergence is proposed; and the Feigenbaum scenario is included as a special case in the new bifurcation scenario. It will provide access to understanding non-normal star products and their corresponding renormalization