Monomial Maps and Algebraic Entropy

In this note, we study some of the simplest algebraic self-maps of projective spaces. These maps, which we call monomial maps, are in one-to-one correspondence with nonsingular integer matrices, and are closely related to toral endomorphisms. In Theorem 1 we give a lower bound for the topological entropy of monomial maps and in Theorem 2 we give a formula for algebraic entropy (as defined by Bellon and Viallet). Theorems 1 and 2 imply that the algebraic entropy of a monomial map is always less than or equal to its topological entropy, and that the inequality is strict if the defining matrix has two or more eigenvalues outside the unit circle. Also, Theorem 2 implies that the algebraic entropy of a monomial map is the logarithm of an algebraic integer. This provides new corroboration of Bellon and Viallet’s conjecture that the algebraic entropy of every rational map is the logarithm of an algebraic integer. However, a simple example shows that a more detailed conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map from projective space to itself need not satisfy a linear recurrence relation with constant coefficients.