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Lower bound for the Perron–Frobenius degrees of Perron numbers

Part of: Special matrices Topological dynamics Algebraic number theory: global fields

Published online by Cambridge University Press:  14 January 2020

MEHDI YAZDI Open the ORCID record for MEHDI YAZDI [Opens in a new window]
MEHDI YAZDI*
Affiliation:
University of Oxford, Mathematics, Andrew Wiles Building, Woodstock Road, OxfordOX2 6GG, UK email yazdi@maths.ox.ac.uk
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Abstract

Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.

Keywords

symbolic dynamicsPerron numbersPerron–Frobenius degreenon-negative matricesentropy

MSC classification

Primary: 37B10: Symbolic dynamics 37B40: Topological entropy 15B48: Positive matrices and their generalizations; cones of matrices 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1264 - 1280
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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