Non-negative integral matrices with given spectral radius and controlled dimension

A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 42; no. 10; pp. 3246 - 3269
Main Author YAZDI, MEHDI
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2022
Subjects
Algebra
Apexes
Arithmetic
Codes
Fields (mathematics)
Graph theory
Number theory
Original Article
Real numbers
37B40
entropy
Perron numbers
37B10
15B48
non-negative matrices
Perron—Frobenius theory
15B36
shifts of finite type
15A18
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Summary:A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2021.93