Symmetries of star products and metric universalities in 1D quadri-modal maps

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.