
Previous Article
A geometric criterion for the nonuniform hyperbolicity of the KontsevichZorich cocycle
 JMD Home
 This Issue
 Next Article
New light on solving the sextic by iteration: An algorithm using reliable dynamics
1.  Mathematics Department, The California State University at Long Beach, Long Beach, CA 908401001 
One such group is the Valentiner action $\mathcal{V}$isomorphic to the alternating group $\mathcal{A}_6$on the complex projective plane. A previous algorithm that solved sixthdegree equations harnessed the dynamics of a $\mathcal{V}$equivariant. However, important global dynamical properties of this map were unproven. Revisiting the question in light of the reflection group conjecture led to the discovery of a degree31 map that is critical on the 45 lines of reflection for $\mathcal{V}$. The map's critical finiteness provides a means of proving its possession of the previous elusive global properties. Finally, a sexticsolving procedure that employs this map's reliable dynamics is developed.
References:
[1] 
S. Crass and P. Doyle, Solving the sextic by iteration: A complex dynamical approach,, Internat. Math. Res. Notices, (1997), 83. doi: 10.1155/S1073792897000068. 
[2] 
S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics,, Experiment. Math., 8 (1999), 209. 
[3] 
S. Crass, A family of critically finite maps with symmetry,, Publ. Mat., 49 (2005), 127. 
[4] 
S. Crass, Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168,, J. Mod. Dyn., 1 (2007), 175. doi: 10.3934/jmd.2007.1.175. 
[5] 
, S. Crass,, Available from: \url{www.csulb.edu/~scrass/math.html}, (). 
[6] 
J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension I,, Complex analytic methods in dynamical systems (Rio de Janeiro, 222 (1994), 201. 
[7] 
C. McMullen, Julia, (computer program)., Available from: \url{www.math.harvard.edu/~ctm/programs.html}, (). 
[8] 
H. Nusse and J. Yorke, "Dynamics: Numerical Explorations,", Second edition. Accompanying computer program Dynamics 2 coauthored by Brian R. Hunt and Eric J. Kostelich, (1998). 
[9] 
G. Shephard and T. Todd, Finite unitary reflection groups,, Canad. J. Math., 6 (1954), 274. doi: 10.4153/CJM19540283. 
[10] 
K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces,, Publ. Mat., 51 (2007), 333. 
show all references
References:
[1] 
S. Crass and P. Doyle, Solving the sextic by iteration: A complex dynamical approach,, Internat. Math. Res. Notices, (1997), 83. doi: 10.1155/S1073792897000068. 
[2] 
S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics,, Experiment. Math., 8 (1999), 209. 
[3] 
S. Crass, A family of critically finite maps with symmetry,, Publ. Mat., 49 (2005), 127. 
[4] 
S. Crass, Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168,, J. Mod. Dyn., 1 (2007), 175. doi: 10.3934/jmd.2007.1.175. 
[5] 
, S. Crass,, Available from: \url{www.csulb.edu/~scrass/math.html}, (). 
[6] 
J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension I,, Complex analytic methods in dynamical systems (Rio de Janeiro, 222 (1994), 201. 
[7] 
C. McMullen, Julia, (computer program)., Available from: \url{www.math.harvard.edu/~ctm/programs.html}, (). 
[8] 
H. Nusse and J. Yorke, "Dynamics: Numerical Explorations,", Second edition. Accompanying computer program Dynamics 2 coauthored by Brian R. Hunt and Eric J. Kostelich, (1998). 
[9] 
G. Shephard and T. Todd, Finite unitary reflection groups,, Canad. J. Math., 6 (1954), 274. doi: 10.4153/CJM19540283. 
[10] 
K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces,, Publ. Mat., 51 (2007), 333. 
[1] 
Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 3439. 
[2] 
Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385400. doi: 10.3934/amc.2012.6.385 
[3] 
Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139152. doi: 10.3934/amc.2014.8.139 
[4] 
Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems  B, 2016, 21 (10) : 33013314. doi: 10.3934/dcdsb.2016098 
[5] 
Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, ThiênHiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495505. doi: 10.3934/proc.2007.2007.495 
[6] 
John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 57. 
[7] 
Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637646. doi: 10.3934/jmd.2009.3.637 
[8] 
James Benn. Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group. Journal of Geometric Mechanics, 2016, 8 (1) : 112. doi: 10.3934/jgm.2016.8.1 
[9] 
Bastian Laubner, Dierk Schleicher, Vlad Vicol. A combinatorial classification of postsingularly finite complex exponential maps. Discrete & Continuous Dynamical Systems  A, 2008, 22 (3) : 663682. doi: 10.3934/dcds.2008.22.663 
[10] 
Jussi Behrndt, A. F. M. ter Elst. The DirichlettoNeumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems  S, 2017, 10 (4) : 661671. doi: 10.3934/dcdss.2017033 
[11] 
Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete & Continuous Dynamical Systems  A, 2010, 27 (4) : 15711586. doi: 10.3934/dcds.2010.27.1571 
[12] 
Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete & Continuous Dynamical Systems  S, 2016, 9 (4) : 12011234. doi: 10.3934/dcdss.2016049 
[13] 
Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems  A, 1998, 4 (3) : 405430. doi: 10.3934/dcds.1998.4.405 
[14] 
Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 7278. doi: 10.3934/proc.1998.1998.72 
[15] 
Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems  B, 2007, 8 (1) : 161173. doi: 10.3934/dcdsb.2007.8.161 
[16] 
Antonio Ambrosetti, Massimiliano Berti. Homoclinics and complex dynamics in slowly oscillating systems. Discrete & Continuous Dynamical Systems  A, 1998, 4 (3) : 393403. doi: 10.3934/dcds.1998.4.393 
[17] 
Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems  B, 2015, 20 (5) : 13931404. doi: 10.3934/dcdsb.2015.20.1393 
[18] 
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically selfsimilar dynamics. Conference Publications, 2005, 2005 (Special) : 131141. doi: 10.3934/proc.2005.2005.131 
[19] 
Doron Levy, Tiago Requeijo. Modeling group dynamics of phototaxis: From particle systems to PDEs. Discrete & Continuous Dynamical Systems  B, 2008, 9 (1) : 103128. doi: 10.3934/dcdsb.2008.9.103 
[20] 
Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557590. doi: 10.3934/cpaa.2017028 
2016 Impact Factor: 0.706
Tools
Metrics
Other articles
by authors
[Back to Top]