相关论文: Birational maps with transcendental dynamical degr…

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called…

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of…

We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of…

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers xi_1,...,xi_t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers…

We prove that the group of birational automorphisms of a geometrically irreducible algebraic surface over a finite field is Jordan. We show that the analogous statement fails in higher dimensions. Finally, we prove that groups of birational…

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n…

By Liouville's theorem, in dimensions 3 or more conformal transformations form a finite-dimensional group, an apparent drastic departure from the 2-dimensional case. We propose a derived enhancement of the conformal Lie algebra which is an…

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Anders\'en-Lempert Theory to prove that symplectic completions always exist, and we prove the…

A particular case of a causal set is considered that is a directed dyadic acyclic graph. This is a model of a discrete pregeometry on a microscopic scale. The dynamics is a stochastic sequential growth of the graph. New vertexes of the…

A birational transformation is one which leaves invariant an ordinary differential equation, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation…

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can…

We investigate the plane dynamical system given by the secant map applied to a polynomial $p$ having at least one multiple root of multiplicity $d>1$. We prove that the local dynamics around the fixed points associated to the roots of $p$…

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of…

We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. The construction involves an iteration procedure on an infinite-dimensional…

This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l}…

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^d$ to $\mathbb{R}^d$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou…

Let X be a smooth projective complex variety, of dimension 3, whose Hodge numbers h^{3,0}(X), h^{1,0}(X) both vanish. Let f: X--> X be a birational map that induces an isomorphism on (dense) open subvarieties U,V of X. Then we show that the…

This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…