Genus and braid index associated to sequences of renormalizable Lorenz maps

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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

Abstract
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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 and 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.
[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.
[3]	W. de Melo and M. Martens, Universal models for Lorenz maps, Ergod. Th and Dynam. Sys., 21 (2001), 833-860.
[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.
[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.
[6]	R. Ghrist, P. Holmes and M. Sullivan, "Knots and Links in Three-Dimensional Flows," Lecture Notes in Mathematics, 1654, Springer-Verlag, Berlin, 1997.
[7]	P. Holmes, Knotted periodic orbits in suspensions of Smale's horseshoe: Period multiplyind and cabled knots, Physica D, 21 (1986), 7-41.
[8]	L. Silva and J. Sousa Ramos, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243.
[9]	M. St. Pierre, Topological and measurable dynamics of Lorenz maps, Dissertationes Mathematicae (Rozprawy Matematyczne), 382 (1999), 134 pp.
[10]	S. Waddington, Asymptotic formulae for Lorenz and horseshoe knots, Comm. Math. Phys., 176 (1996), 273-305. doi: 10.1007/BF02099550.
[11]	R. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. doi: 10.1007/BF02684770.
[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.

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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.
[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.
[3]	W. de Melo and M. Martens, Universal models for Lorenz maps, Ergod. Th and Dynam. Sys., 21 (2001), 833-860.
[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.
[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.
[6]	R. Ghrist, P. Holmes and M. Sullivan, "Knots and Links in Three-Dimensional Flows," Lecture Notes in Mathematics, 1654, Springer-Verlag, Berlin, 1997.
[7]	P. Holmes, Knotted periodic orbits in suspensions of Smale's horseshoe: Period multiplyind and cabled knots, Physica D, 21 (1986), 7-41.
[8]	L. Silva and J. Sousa Ramos, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243.
[9]	M. St. Pierre, Topological and measurable dynamics of Lorenz maps, Dissertationes Mathematicae (Rozprawy Matematyczne), 382 (1999), 134 pp.
[10]	S. Waddington, Asymptotic formulae for Lorenz and horseshoe knots, Comm. Math. Phys., 176 (1996), 273-305. doi: 10.1007/BF02099550.
[11]	R. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S., 50 (1979), 73-99. doi: 10.1007/BF02684770.
[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.

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