Universal form of renormalizable knots in symbolic dynamics of bimodal maps

摘要

A general relation between knot theory and symbolic dynamics is studied for bimodal maps in this paper. When symbolic sequences of maps are expressed as knots, it is easy to see that the knot of a renormalizable sequence is composed of a bunch of periodic flows. In this setting, the generation of renormalizable knots can be simply operated in geometry and explicitly calculated in algebra. In this paper we provide a universal form of renormalizable knots, which can be decomposed into a sequence of elementary templates; especially, it is independent of the traditional *-products of symbolic dynamics. We present some examples and list the elementary templates of period not bigger than 6.