Order isomorphism under higher power shift and generic form of renormalization group equations in multimodal maps

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.