Genus and braid index associated to sequences of renormalizable Lorenz maps

Details Genus and braid index associated to sequences of renormalizable Lorenz maps Genus and braid index associated to sequences of renormalizable Lorenz maps

2008-03-27

arxiv.org/abs/0803.3895v2

Mathematics

Geometric Topology

Scientific paper

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant $(X,Y)*(S,W)^{*n}$, concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.

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