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Hongming Nie

Publications and source records attributed to Hongming Nie.

At least 19 recordsLinked to original sources
arXiv2025

Zeta function and entropy for non-archimedean subhyperbolic dynamics

Let $K$ be a complete non-archimedean field of characteristic $0$ equipped with a discrete valuation. We establish the rationality of the Artin-Mazur zeta function on the Julia set for any subhyperbolic rational map defined over $K$ with a compact Julia set. Furthermore, we conclude that the topological entropy on the Julia set of such a map is given by the logarithm of a weak Perron number. Conversely, we construct a (sub)hyperbolic rational map defined over $K$ with compact Julia set whose topological entropy on the Julia set equals the logarithm of a given weak Perron number. This extends Thurston's work on the entropy for postcritically finite interval self-maps %of the unit interval to the non-archimedean setting.

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arXiv2024

Compactifications and measures for rational maps

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the discontinuity of the iterate map. We show that the measure of maximal entropy extends continuously to this resolution space. For moduli space, we consider a space which resolves the discontinuity of the iterate map acting on its geometric invariant theory compactification. We show that the measure of maximal entropy, barycentered and modulo rotations, also extends continuously to this resolution space. Thus, answering in the positive a question raised by DeMarco. A main ingredient is a description of limiting dynamics for some sequences.

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arXiv2024

Expanding property and statistical laws for $p$-adic subhyperbolic rational maps

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $ϕ\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $ϕ$ is eventually periodic. We show that subhyperbolic maps in $K(z)$ exhibit expanding property with respect to some (singular) metric. As an application, under a mild assumption, we establish several statistical laws for such maps in $K(z)$ with compact $\mathbb{C}_p$-Julia sets.

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arXiv2023

On a metric view of the polynomial shift locus

We relate generic points in the shift locus $\mathcal{S}_D$ of degree $D\ge 2$ polynomials to metric graphs. Using thermodynamic metrics on the space of metric graphs, we obtain a distance function $ρ_D$ on $\mathcal{S}_D$. We study the (in)completeness of the metric space $(\mathcal{S}_D, ρ_D)$. We prove that when $D \ge 3$, the space $(\mathcal{S}_D, ρ_D)$ is incomplete and its metric completion contains a subset homeomorphic to the space $\mathbb{P}\mathcal{ST}_D^*$ introduced by DeMarco and Pilgrim. This provides a new way to understand the space $\mathbb{P}\mathcal{ST}_D^*$.

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arXiv2023

Berkovich dynamics of twisted rational maps

A twisted rational map over a non-archimedean field $K$ is the composition of a rational function over $K$ and a continuous automorphism of $K$. We explore the dynamics of some twisted rational maps on the Berkovich projective line.

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arXiv2023

Böttcher coordinates at wild superattracting fixed points

Let $p$ be a prime number, let $g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with $r\in\mathbb{Z}_{\geq0}$, and let $ϕ(x)=x+O(x^{2})$ be the Böttcher coordinate satisfying $ϕ(g(x))=ϕ(x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of $ϕ^{-1}(x)$ in $\mathbb{C}_{p}$ is $p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.

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arXiv2022

The basin of infinity of tame polynomials

Let $\mathbb{C}_v$ be a characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in $\mathbb{C}_v[z]$ of degree $d \ge 2$ to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials, tame polynomials with all their critical points in the basin of infinity form the tame shift locus. We show that a tame map $f\in\mathbb{C}_v[z]$ is in the closure of the tame shift locus if and only if the Fatou set of $f$ coincides with the basin of infinity.

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arXiv2022

Nonarchimedean Lyapunov exponents of polynomials

Let $K$ be an algebraically closed and complete nonarchimedean field with characteristic $0$ and let $f\in K[z]$ be a polynomial of degree $d\ge 2$. We study the Lyapunov exponent $L(f,μ)$ of $f$ with respect to an $f$-invariant and ergodic Radon probability measure $μ$ on the Berkovich Julia set of $f$ and the lower Lyapunov exponent $L_f^{-}(f(c))$ of $f$ at a critical value $f(c)$. Under an integrability assumption, we show $L(f,μ)$ has a lower bound only depending on $d$ and $K$. In particular, if $f$ is tame and has no wandering nonclassical Julia points, then $L(f,μ)$ is nonnegative; moreover, if in addition $f$ possesses a unique Julia critical point $c_0$, we show $L_f^{-}(f(c_0))$ is also nonnegative.

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arXiv2021

Siegel disks of the tangent family

We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a forward invariant Siegel disk is unbounded if and only if it contains at least one asymptotic value on the boundary. Our argument is elementary and function-theoretic. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.

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arXiv2020

Quantitative equidistribution of angles of multipliers

We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb C(z)$. For a fixed $K \gg 1$, we show that almost all intervals of length $2π/K$ in $(-π,π]$ contain a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in $K$.

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arXiv2020

Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics

We compute the resultant measures for iterations $P^j$, $j\ge 1$, of a polynomial $P$ of degree $>1$ on the $n$-th level Trucco's trees $\Gamma_n$, $n\ge 0$, in the Berkovich projective line over a non-archimedean field and also determine their barycenters. As applications, we study the asymptotic of those barycenters as $n\to\infty$, and establish a uniform stationarity of Rumely's minimal resultant loci of $P^j$ or equivalently that of the potential semistable reduction loci of $P^j$ as $j\to\infty$. We also establish several equidistribution results for the resultant measures themselves as $n\to\infty$.

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arXiv2020

A Riemannian metric on polynomial hyperbolic components

We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree $d \ge 2$ polynomials. Our metric is constructed by considering the measure-theoretic entropy of a polynomial with respect to some equilibrium state. As applications, we show that the Hausdorff dimension function has no local maximum on such hyperbolic components. We also give a sufficient condition for a point not being a critical point of the Hausdorff dimension function.

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arXiv2020

Indeterminacy loci of iterate maps in moduli space

The moduli space $\mathrm{rat}_d$ of rational maps in one complex variable of degree $d \ge 2$ has a natural compactification by a projective variety $\overline{\mathrm{rat}}_d$ provided by geometric invariant theory. Given $n \ge 2$, the iteration map $Φ_n : \mathrm{rat}_d \to\mathrm{rat}_{d^n}$, defined by $Φ_n: [f] \mapsto [f^n]$, extends to a rational map $Φ_n : \overline{\mathrm{rat}}_d\dashrightarrow \overline{\mathrm{rat}}_{d^n}$. We characterize the elements of $\overline{\mathrm{rat}}_d$ which lie in the indeterminacy locus of $Φ_n$.

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arXiv2019

Perturbations of graphs for Newton maps

We study the convergence of graphs consisting of finitely many internal rays for degenerating Newton maps. We state a sufficient condition to guarantee the convergence. As an application, we investigate the boundedness of hyperbolic components in the moduli space of quartic Newton maps. We prove that such a hyperbolic component is bounded if and only if every element has degree $2$ on the immediate basin of each root.

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arXiv2019

Bounded hyperbolic components of bicritical rational maps

We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.

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arXiv2018

Boundedness of Hyperbolic Components of Newton Maps

We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we characterize the possible points on the boundary at infinity for some other types of hyperbolic components. For general maps, we prove hyperbolic components whose elements have fixed superattracting basins mapping by degree at least three are unbounded.

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arXiv2018

Iteration at the Boundary of Newton Maps

Let $\{N_t\}$ be a holomorphic family of degree $d\ge 3$ Newton maps. By studying the related Berkovich dynamics, we obtain an estimate of the weak limit of the maximal measures of $N_t$. Moreover, we give a complete description of the rescaling limits for $\{N_t\}$.

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