arXiv subjects

Kenneth Jacobs

Publications and source records attributed to Kenneth Jacobs.

6 recordsLinked to original sources

arXiv2018

Hyperbolic Equivariants of Rational Maps

Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. In this article, we introduce two new equivariants associated to a rational map $f\in K(z)$. These objects naturally live on a real hyperbolic space, and carry information about the action of $f$ on $\mathbb{P}^1(K)$. When $K=\mathbb{C}$ we relate the asymptotic behavior of these equivariants to the conformal barycenter of the measure of maximal entropy. We also give a complete description of these objects for rational maps of degree $d=1$. The constructions in this article are based on work of Rumely in the context of rational maps over non-Archimedean fields; similarities between the two theories are highlighted throughout the article.

math.DS↗

arXiv2016

Iteration and the Minimal Resultant

Let $K$ be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let $\varphi\in K(z)$ have degree $d\geq 2$. We characterize maps for which the minimal resultant of an iterate $\varphi^n$ is given by a simple formula in terms of $d$, $n$, and the minimal resultant of $\varphi$. We show that such maps are precisely those with reduction outside of an indeterminacy locus $I(d)$ and which also have semi-stable reduction for every iterate $\varphi^n$. We give two equivalent ways of describing such maps, one measure theoretic and the other in terms of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps. As an application, we are able to give an explicit formula for the minimal value of the diagonal Arakelov-Green's function of a map satisfying the conditions of the main theorem. We illustrate our results with some explicit calculations in the case of the Latt\`es maps.

math.DS↗

arXiv2015

Lower Bounds for non-Archimedean Lyapunov Exponents

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 exponential rate of growth along a typical orbit of $\phi$. When $\phi$ is defined over $\mathbb{C}$, the Lyapunov exponent is bounded below by $\frac{1}{2}\log d$. In this article, we give a lower bound for $L(\phi)$ for maps $\phi$ defined over non-Archimedean fields $K$. The bound depends only on the degree $d$ and the Lipschitz constant of $\phi$. For maps $\phi$ whose Julia sets satisfy a certain boundedness condition, we are able to remove the dependence on the Lipschitz constant.

math.DS↗

arXiv2015

Quantitative Logarithmic Equidistribution of the Crucial Measures

Let $K$ be a algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value. Let $\phi\in K(z)$ with $\textrm{deg}(\phi)\geq 2$. In this paper we establish uniform logarithmic equidistribution of the crucial measures $\nu_{\phi^n}$ attached to the iterates of $\phi$. These measures were introduced by Rumely in his study of the Minimal Resultant Locus of $\phi$. Our equidistribution result comes from a bound on the diameter of points in $\textrm{supp}(\nu_{\phi^n})$ that depends only on $n$ and $\phi$. We also show that the sets $\textrm{MinResLoc}(\phi^n)$ are bounded independent of $n$, and we give an explicit bound for the radius of a ball about $\zeta_{\textrm{Gauss}}$ containing $\textrm{Bary}(\mu_\phi)$.

math.DS↗

arXiv2015

Configuration of the Crucial Set for a Quadratic Rational Map

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of $\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $\varphi$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.

math.NT↗

arXiv2014

An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line

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 of $\phi^n$ at points $x$ in $\textbf{P}^1_{\textrm{K}}$, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov-Green's function $g_{\mu_{\phi}}(x,x)$ attached to the canonical measure of $\phi$. Following this, we are able to prove an equidistribution result for Rumely's crucial measures $\nu_{\phi^n}$, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of $\phi$.

math.DS↗