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

Eric Bedford; Kyounghee Kim

doi:10.3934/dcds.2008.21.977

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2008, Volume 21, Issue 4: 977-1013. Doi: 10.3934/dcds.2008.21.977

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

Eric Bedford^1, and
Kyounghee Kim^1,

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

Received: May 31, 2007

Revised: January 31, 2008

Published: September 2008

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Abstract

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:

Degree growth,
Birational maps.

Mathematics Subject Classification: Primary: 37F10; Secondary: 14J99.

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