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Degree-growth of monomial maps

Published online by Cambridge University Press:  01 October 2007

BORIS HASSELBLATT  and
JAMES PROPP
BORIS HASSELBLATT
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA (email: bhasselb@tufts.edu)
JAMES PROPP
Affiliation:
Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell,MA 01854, USA (email: propp@jamespropp.org)
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Abstract

For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

Type
SURVEY
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1375 - 1397
Copyright
Copyright © Cambridge University Press 2007

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