Degree-growth of monomial maps

By Boris Hasselblatt, James Gary Propp. Ergodic Theory and Dynamical Systems, 2006

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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.

Subfields: Geometry and Topology, Mathematical Physics, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/fdde13d2f84ee2f3ce83bb9d1e998596f4158ea3

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Citations in the corpus

Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, background): “Dynamical degrees were initially studied by Russakovskii and Shiffman [53] and Arnol′d [5] in the 1990s, and they have since attracted considerable attention; see for example [8, 10, 11, 18, 22, 23, 24, 26, 33, 44, 43, 45, 47, 61].”
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (2012, background): “Monomial maps are not, in general, algebraically stable, but their dynamical degrees are known to be algebraic integers [23].”
Examples of dynamical degree equals arithmetic degree (2012, background): “Monomial maps are not, in general, algebraically stable, but their dynamical degrees are known to be algebraic integers [12].”
Current trends and open problems in arithmetic dynamics (2018, background): “4 (Hasselblatt–Propp, 2007, [106]; Bedford–Kim, 2008, [14]).”