Related papers: Valiron and Abel equations for holomorphic self-ma…

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this…

This paper provides a fixed point theorem and iterative construction of a common fixed point for a general class of nonlinear mappings in the setup of uniformly convex hyperbolic spaces. We translate a multi-step iteration, essentially due…

We survey several results in value distribution theory for parabolic Riemann surfaces. Let Y be a parabolic Riemann surface, i.e. subharmonic functions defined on Y are constant. We discuss Nevanlinna's theory for holomorphic maps f from Y…

We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence,…

We get three basic results in algebraic dynamics: (1). We give the first algorithm to compute the dynamical degrees to arbitrary precision. (2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower…

We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space…

We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real…

We present an interesting connection between Einstein-Friedmann equations for the models of universe filled with scalar field and the special form of Abel equation of the first kind. This connection works in both ways: first, we show how,…

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\phi$ into a…

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing"…

We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we…

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing…

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex…

We obtain a complete classification of complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$, for which the dimension of the group of holomorphic automorphisms is equal to $n^2$.

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs B_n of bounded linear operators on Hilbert spaces. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on…

We consider 5-dimensional gauged supergravity coupled to Abelian vector multiplets, and we look for supersymmetric solutions for which the 4-dimensional K\"ahler base space admits a holomorphic isometry. Taking advantage of this isometry,…

We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric…

This paper is devoted to study the semilinear elliptic system of H\'enon-type \begin{eqnarray*} -\Delta_{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v) \\ -\Delta_{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v) \end{eqnarray*} in the hyperbolic space…

Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to…

In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we…