Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices

Previous Article
DCDS Home
This Issue
Next Article
Hausdorff dimension of self-affine limit sets with an invariant direction

November 2008, 21(4): 977-1013. doi: 10.3934/dcds.2008.21.977

Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices

1.	Department of Mathematics, Indiana University, Bloomington, IN 47405, United States, United States

Received June 2007 Revised February 2008 Published May 2008

Abstract
Related Papers

We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set $\mathcal{SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to $\mathcal{SC}_q$.

Keywords: Birational maps., Degree growth.

Mathematics Subject Classification: Primary: 37F10; Secondary: 14J9.

Citation: Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977

[1]	Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683
[2]	Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[3]	Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[4]	Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[5]	Zalman Balanov, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 983-1016. doi: 10.3934/dcds.2006.15.983
[6]	C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267
[7]	Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905
[8]	Jaume Llibre, Claudia Valls. Centers for polynomial vector fields of arbitrary degree. Communications on Pure & Applied Analysis, 2009, 8 (2) : 725-742. doi: 10.3934/cpaa.2009.8.725
[9]	Joseph Bayara, André Conseibo, Moussa Ouattara, Artibano Micali. Train algebras of degree 2 and exponent 3. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1371-1386. doi: 10.3934/dcdss.2011.4.1371
[10]	Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[11]	Cristóbal Camarero, Carmen Martínez, Ramón Beivide. Identifying codes of degree 4 Cayley graphs over Abelian groups. Advances in Mathematics of Communications, 2015, 9 (2) : 129-148. doi: 10.3934/amc.2015.9.129
[12]	Junbo Jia, Zhen Jin, Lili Chang, Xinchu Fu. Structural calculations and propagation modeling of growing networks based on continuous degree. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1215-1232. doi: 10.3934/mbe.2017062
[13]	Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469
[14]	Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765
[15]	Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407
[16]	Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975
[17]	Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015
[18]	Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861
[19]	Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
[20]	Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences