-
Previous Article
Rational periodic sequences for the Lyness recurrence
- DCDS Home
- This Issue
-
Next Article
Compressible hydrodynamic flow of liquid crystals in 1-D
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 |
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. |
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. |
[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. |
[1] |
Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2022, 27 (2) : 769-797. 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]