Related papers: On the Degree Growth of Birational Mappings in Hig…

This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly,…

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. The classification elements of infinite order with a…

We compute the degree complexity of a family of birational mappings of the plane with high order singularities.

We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of these birational maps. We find the all the maps in…

We prove that the degrees of the iterates ${\rm deg}(f^n)$ of a birational map satisfy $\liminf({\rm deg}(f^n))<+\infty$ if and only if the sequence ${\rm deg}(f^n)$ is bounded, and that the growth of ${\rm deg}(f^n)$ can not be arbitrarily…

This work deals with a special case of family of birational maps f : C2 -> C2 dynamically classified in [9]. In this work we study the zero entropy sub families of f. The sequence of degrees dn associated to the iterates of f is found to…

We construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such…

The dynamical degree of a dominant rational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$ is the quantity $\delta(f):=\lim(\text{deg} f^n)^{1/n}$. We study the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make a…

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer…

The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational…

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.

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are…

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…

In this paper, we develop several tools to study the degree growth and stabilization of monomial maps. Using these tools, we can classify semisimple three dimensional monomial maps by their dynamical behavior.

Bedford asked if there exists a birational self map $f$ of the complex projective plane such that for any automorphism $A$ of the complex projective plane $A\circ f$ is not conjugate to an automorphism. Blanc gave such a $f$ of degree $6$…

Let f be a rational mapping of a space X . The complexity of (f,X) as a dynamical system is measured by the dynamical degrees $\delta_p(f)$, $1\le p\le {\rm dim}(X)$. We give the definition of the dynamical degrees show how they are…

We compute the degree complexity of the family of birational maps considered in \cite{bedford-kim-tuyen-abarenkova-maillard} for all exceptional cases. Some interesting properties of the family are also given.

We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of…

The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can…

We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$. We prove that when conditioning such maps…