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

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 $\mathcal{SC}_q$.
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]

Dinh T. Tran, John A. G. Roberts. Linear degree growth in lattice equations. Journal of Computational Dynamics, 2019, 6 (2) : 449-467. doi: 10.3934/jcd.2019023

[2]

Inês Cruz, Helena Mena-Matos, Esmeralda Sousa-Dias. The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020010

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Christian Bläsche, Shawn Means, Carlo R. Laing. Degree assortativity in networks of spiking neurons. Journal of Computational Dynamics, 2020, 7 (2) : 401-423. doi: 10.3934/jcd.2020016

[10]

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

[11]

Jérôme Buzzi. The degree of Bowen factors and injective codings of diffeomorphisms. Journal of Modern Dynamics, 2020, 16: 1-36. doi: 10.3934/jmd.2020001

[12]

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

[13]

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

[14]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315

[15]

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

[16]

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

[17]

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

[18]

Wei Gao, Juan Luis García Guirao, Mahmoud Abdel-Aty, Wenfei Xi. An independent set degree condition for fractional critical deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 877-886. doi: 10.3934/dcdss.2019058

[19]

Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071

[20]

Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]