Genus and braid index associated to sequences of renormalizable Lorenz maps

Nuno Franco; Luís Silva

doi:10.3934/dcds.2012.32.565

Article Contents

2012, Volume 32, Issue 2: 565-586. Doi: 10.3934/dcds.2012.32.565

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Genus and braid index associated to sequences of renormalizable Lorenz maps

Nuno Franco^1, and
Luís Silva^2,

1.
CIMA-UE and Department of Mathematics, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora
2.
CIMA-UE and Departmental Area of Mathematics, ISEL - Lisbon Superior Engineering Institute, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa

Received: September 30, 2010

Revised: June 30, 2011

Published: January 2012

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Abstract

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 the ones corresponding to the factors of the $*$-product. 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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Mathematics Subject Classification: Primary: 57M25, 37E05; Secondary: 37E20, 57M27.

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