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Lower bound for the Perron–Frobenius degrees of Perron numbers

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

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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References

Boyle, M. and Lind, D.. Small polynomial matrix presentations of non-negative matrices. Linear Algebra Appl. 355 (2002), 49–70.10.1016/S0024-3795(02)00313-0CrossRefGoogle Scholar
Fried, D.. Growth rate of surface homeomorphisms and flow equivalence. Ergod. Th. & Dynam. Sys. 5(4) (1985), 539–563.Google Scholar
Gantmacher, F. R. and Brenner, J. L.. Applications of the Theory of Matrices. Dover Publications, Mineola, NY, 2005.Google Scholar
Hamenstädt, U.. Typical properties of periodic Teichmüller geodesics: stretch factors. Preprint, 2019; http://www.math.unibonn.de/people/ursula/tracefield.pdf.Google Scholar
Hubbard, J. H.. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 2. Matrix Editions, Ithaca, NY, 2016.Google Scholar
Leininger, C. J. and Reid, A. W.. Pseudo-Anosov homeomorphisms not arising from branched covers. Preprint, 2017, arXiv:1711.06881.Google Scholar
Lind, D. A.. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4(2) (1984), 283–300.10.1017/S0143385700002443CrossRefGoogle Scholar
McMullen, C.. Slides for dynamics and algebraic integers: perspectives on Thurston’s last theorem, 2014, http://www.math.harvard.edu/ctm/expositions/home/text/talks/cornell/2014/slides/slides.pdf.Google Scholar
Penner, R. C.. Bounds on least dilatations. Proc. Amer. Math. Soc. 113(2) (1991), 443–450.10.1090/S0002-9939-1991-1068128-8CrossRefGoogle Scholar
Thurston, W.. Entropy in dimension one. Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday. Princeton University Press, Princeton, NJ, 2014.Google Scholar