Related papers: A $1$-cohomologically hyperbolic birational map of…

A dominant rational self-map on a projective variety is called $p$-cohomologically hyperbolic if the $p$-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over $\overline{\mathbb{Q}}$, we study…

We construct a birational map of $\mathbb{P}^d$ ($d\geq6$) whose intermediate dynamical degrees are all trancendental.

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics…

We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.

We study the birational maps of $\mathbb{P}^3_\mathbb{C}$. More precisely we describe the irreducible components of the set of birational maps of bidegree $(3,3)$ (resp. $(3,4)$, resp. $(3,5)$).

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space…

We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type $(p,q)$, are branched covers and is modeled on transcendental…

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk\"ahler birational…

We construct a hyperbolic three-manifold with trivial finite type invariants up to a given degree.

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…

Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup…

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical…

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental H\'enon maps offers the potential of combining ideas from transcendental dynamics in one variable,…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

In the last section of \cite{CompHyp} it is proved that the map $\mathcal{I}$ is a finite-sheeted covering map between $\mathcal{P}$ and $\mathcal{M}$. As $\mathcal{M}$ is simply connected it is deduced that $\mathcal{I}$ is a…

We study the boundaries of non-univalent simply connected Baker domains of transcendental maps (both entire and meromorphic), of hyperbolic and simply parabolic type. We prove non-ergodicity and non-recurrence for the boundary map, and…

We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of…

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are…

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the…