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February  2012, 32(2): 565-586. doi: 10.3934/dcds.2012.32.565

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, Portugal
2. 

CIMA-UE and Departmental Area of Mathematics, ISEL - Lisbon Superior Engineering Institute, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal

Received  October 2010 Revised  July 2011 Published  September 2011

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.
Keywords: Lorenz knots, genus, renormalization, braid index..
Mathematics Subject Classification: Primary: 57M25, 37E05; Secondary: 37E20, 57M2.
Citation: Nuno Franco, Luís Silva. Genus and braid index associated to sequences of renormalizable Lorenz maps. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 565-586. doi: 10.3934/dcds.2012.32.565
References:
[1]

J. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz's equations, Topology, 22 (1983), 47-82. doi: 10.1016/0040-9383(83)90045-9.  Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems" (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Mathematics, 819, Springer, Berlin, (1980), 18-34.  Google Scholar

[3]

W. de Melo and M. Martens, Universal models for Lorenz maps, Ergod. Th and Dynam. Sys., 21 (2001), 833-860.  Google Scholar

[4]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, 1993.  Google Scholar

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J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Am. Math. Soc., 303 (1987), 97-108. doi: 10.1090/S0002-9947-1987-0896009-2.  Google Scholar

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R. Ghrist, P. Holmes and M. Sullivan, "Knots and Links in Three-Dimensional Flows," Lecture Notes in Mathematics, 1654, Springer-Verlag, Berlin, 1997.  Google Scholar

[7]

P. Holmes, Knotted periodic orbits in suspensions of Smale's horseshoe: Period multiplyind and cabled knots, Physica D, 21 (1986), 7-41.  Google Scholar

[8]

L. Silva and J. Sousa Ramos, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243.  Google Scholar

[9]

M. St. Pierre, Topological and measurable dynamics of Lorenz maps, Dissertationes Mathematicae (Rozprawy Matematyczne), 382 (1999), 134 pp.  Google Scholar

[10]

S. Waddington, Asymptotic formulae for Lorenz and horseshoe knots, Comm. Math. Phys., 176 (1996), 273-305. doi: 10.1007/BF02099550.  Google Scholar

[11]

R. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. doi: 10.1007/BF02684770.  Google Scholar

[12]

R. Williams, The structure of Lorenz attractors, "Turbulence Seminar" (eds. A. Chorin, J. Marsden and S. Smale) (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Mathematics, 615, Springer, Berlin, (1977), 94-112.  Google Scholar

show all references

References:
[1]

J. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz's equations, Topology, 22 (1983), 47-82. doi: 10.1016/0040-9383(83)90045-9.  Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems" (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Mathematics, 819, Springer, Berlin, (1980), 18-34.  Google Scholar

[3]

W. de Melo and M. Martens, Universal models for Lorenz maps, Ergod. Th and Dynam. Sys., 21 (2001), 833-860.  Google Scholar

[4]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, 1993.  Google Scholar

[5]

J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Am. Math. Soc., 303 (1987), 97-108. doi: 10.1090/S0002-9947-1987-0896009-2.  Google Scholar

[6]

R. Ghrist, P. Holmes and M. Sullivan, "Knots and Links in Three-Dimensional Flows," Lecture Notes in Mathematics, 1654, Springer-Verlag, Berlin, 1997.  Google Scholar

[7]

P. Holmes, Knotted periodic orbits in suspensions of Smale's horseshoe: Period multiplyind and cabled knots, Physica D, 21 (1986), 7-41.  Google Scholar

[8]

L. Silva and J. Sousa Ramos, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243.  Google Scholar

[9]

M. St. Pierre, Topological and measurable dynamics of Lorenz maps, Dissertationes Mathematicae (Rozprawy Matematyczne), 382 (1999), 134 pp.  Google Scholar

[10]

S. Waddington, Asymptotic formulae for Lorenz and horseshoe knots, Comm. Math. Phys., 176 (1996), 273-305. doi: 10.1007/BF02099550.  Google Scholar

[11]

R. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. doi: 10.1007/BF02684770.  Google Scholar

[12]

R. Williams, The structure of Lorenz attractors, "Turbulence Seminar" (eds. A. Chorin, J. Marsden and S. Smale) (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Mathematics, 615, Springer, Berlin, (1977), 94-112.  Google Scholar

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