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

2005/12/21 by Eric Bedford, Bedford, Eric, Kyounghee Kim +1 · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.math/0512507

openalex publication_date 2005/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider two (densely defined) involutions on the space of q× q matrices; I(x_ij) is the matrix inverse of (x_ij), and J(x_ij) is the matrix whose ijth entry is the reciprocal x_ij^-1. Let K=I∘ J. The set \cal SC_q of symmetric, cyclic matrices is invariant under K. In this paper, we determine the degrees of the iterates Kⁿ=K∘...∘ K restricted to \cal SC_q.

Cited by

• On the geometry of a 4-dimensional extension of a q-Painlevé I equation with symmetry type A₁⁽¹⁾
• Linear Recurrences in the Degree Sequences of Monomial Mappings

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