Böttcher coordinates at fixed indeterminacy points

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.

Linear recurrences in the degree sequences of monomial mappings

Let A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

Degree-growth of monomial maps

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.