# American Institute of Mathematical Sciences

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

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

 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.
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 [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

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
 [1] Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979 [2] Dubi Kelmer. Quadratic irrationals and linking numbers of modular knots. Journal of Modern Dynamics, 2012, 6 (4) : 539-561. doi: 10.3934/jmd.2012.6.539 [3] Sébastien Gautier, Lubomir Gavrilov, Iliya D. Iliev. Perturbations of quadratic centers of genus one. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 511-535. doi: 10.3934/dcds.2009.25.511 [4] Shin Kiriki, Ming-Chia Li, Teruhiko Soma. Geometric Lorenz flows with historic behavior. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7021-7028. doi: 10.3934/dcds.2016105 [5] Fuchen Zhang, Xiaofeng Liao, Guangyun Zhang, Chunlai Mu, Min Xiao, Ping Zhou. Dynamical behaviors of a generalized Lorenz family. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3707-3720. doi: 10.3934/dcdsb.2017184 [6] John Kerin, Hans Engler. On the Lorenz '96 model and some generalizations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021064 [7] Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13 [8] Josep M. Miret, Jordi Pujolàs, Nicolas Thériault. Trisection for supersingular genus $2$ curves in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 375-387. doi: 10.3934/amc.2014.8.375 [9] Steve Limburg, David Grant, Mahesh K. Varanasi. Higher genus universally decodable matrices (UDMG). Advances in Mathematics of Communications, 2014, 8 (3) : 257-270. doi: 10.3934/amc.2014.8.257 [10] David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009 [11] João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641 [12] M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403 [13] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [14] Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210 [15] Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013 [16] Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437 [17] Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 [18] Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773 [19] Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018 [20] Ning Sun, Shaoyun Shi, Wenlei Li. Singular renormalization group approach to SIS problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073

2020 Impact Factor: 1.392