Related papers: Degree-growth of monomial maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the…

We consider the ergodic theory of plane rational maps that preserve the natural holomorphic volume form on the algebraic torus. Specifically we construct natural invariant probability measures for a large class of such maps by intersecting…

Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $\mu_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapsto\mu_f$ is known to be continuous on $Rat_d$, and it is shown…

Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree…

For a finite dimensional vector space equipped with a $\mathbb C$-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of…

We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of…

Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic…

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic…

Let X be a complex projective manifold and f a dominating rational map from X onto X. We show that the topological entropy h(f) of f is bounded from above by the logarithm of its maximal dynamical degree.

We confront two integrability criteria for rational mappings. The first is the singularity confinement based on the requirement that every singularity, spontaneously appearing during the iteration of a mapping, disappear after some steps.…

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of…

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure…

In this paper we investigate fixed-point numbers and entropies of endomorphisms on abelian varieties. It was shown quite recently that the number of fixed-points of an iterated endomorphism on a simple complex torus is either periodic or…

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by…

In 1965 Adler, Konheim and McAndrew defined the topological entropy for continuous self-maps of compact spaces. Topological entropy is very well-understood for endomorphisms of compact Abelian groups. A fundamental result in this context is…

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…

We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…

Various limit-free formulas are given for the computation of the algebraic and the topological entropy, respectively in the settings of endomorphisms of locally finite discrete groups and of continuous endomorphisms of totally disconnected…

We give a brief survey on the entropy of holomorphic self maps $f$ of compact K\"ahler manifolds, and rational dominating self maps $f$ of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii…