Related papers: The Arakelov-Zhang pairing and Julia sets

Given two rational maps $\varphi$ and $\psi$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\varphi,\psi>$ which is closely related to the canonical height functions $h_\varphi$ and $h_\psi$…

We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational…

Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\textbf{P}^1$ denote the Berkovich projective line over $K$. The Lyapunov exponent for a rational map $\phi\in K(z)$ of degree $d\geq 2$ measures the…

Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\phi\in K(z)$ with $\textrm{deg}(\phi) \geq 2$. In this paper we consider the family of functions $\textrm{ordRes}_{\phi^n}(x)$, which measure the resultant…

Let ${\mathbb X}$ be a compact, connected, Riemannian manifold (without boundary), $\rho$ be the geodesic distance on ${\mathbb X}$, $\mu$ be a probability measure on ${\mathbb X}$, and $\{\phi_k\}$ be an orthonormal system of continuous…

We introduce a metric of mutual energy for adelic measures associated to the Arakelov-Zhang pairing. Using this metric and potential theoretic techniques involving discrete approximations to energy integrals, we prove an effective bound on…

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $\phi\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $\phi$ is the natural restriction of $\mathbb{C}_p$-Julia set,…

Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H\"older continuous potential $\phi: J(f)\rightarrow \R$ and the pressure $P(f,\phi). In the case where $\sup_{J(f)}\phi<P(f,phi)$, the uniqueness and…

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial…

Stability arguments suggest that the Kaluza-Klein (KK) internal scalar field, $\Phi$, should be coupled to some external fields. An external bulk real scalar field, $\psi$, minimally coupled to gravity is proved to be satisfactory. At low…

Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X…

Let $\mathbbm{P}^{1,an}$ be the Berkovich projective line over a complete, algebraically closed, non-Archimedean field. Let $\phi$ be a degree $\geq 2$ rational map with potential good reduction, acting on $\mathbbm{P}^{1,an}$. In this…

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…

For a hyperbolic rational map $f$ with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set $J_f$ from a family of energies of associated graph maps. Concretely, the dynamics of $f$…

We establish a coupling between the $\mathcal{P}(\phi)_2$ measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result…

The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces. We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves,…

Let $X$ be an algebraic variety equipped with a dominant rational self-map $\phi:X\to X$. A new quantity measuring the interaction of $(X,\phi)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,\phi)$…

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_t: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$, parameterized by $t$ in a…

Private set intersection (PSI) allows two mutually untrusting parties to compute an intersection of their sets, without revealing information about items that are not in the intersection. This work introduces a PSI variant called…

We extend a newly developed formal system for the description of astrophysical maps. In this formalism, we consider the difference between maps to be the distance between elements of a pseudometric space (the space of all such maps). This…