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

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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
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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

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