Berkovich dynamics of twisted rational maps
Hongming Nie, Shengyuan Zhao
该研究定义了非阿基米德域上的扭曲有理映射概念，并探究了这类映射在Berkovich射影直线上的动力学性质。
非阿基米德域$K$上的扭曲有理映射是$K$上的有理函数与$K$的连续自同构的复合。我们探究了部分扭曲有理映射在Berkovich射影直线上的动力学性质。
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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We establish a locally uniform a priori bound on the dynamics of a rational function $f$ of degree $>1$ on the Berkovich projective line over an algebraically closed field of any characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and deduce an equidistribution result for moving targets towards the equilibrium (or canonical) measure $μ_f$, under the no potentially good reductions condition. This partly answers a question posed by Favre and Rivera-Letelier.
In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $\mathbb{R}$-tree are analogues to i…
We show that a rational function $f$ of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of $f$ is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of $f$ is also established.
Connected components of Berkovich fixed locus: Potential good reduction
本文研究了具有潜在良好约简性质的有理映射的固定位的拓扑结构，给出了其连通性和有限性的判据，并提供了固定位连通分量数的上界。
We provide a framework for studying the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over a complete non-archimedean field. We prove a non-archimedean analogue of Mañé, Sad, and Sullivan's $λ$-Lemma and use this to show an equivalence of two stability conditions for families of rational functions parametrized by an open subset of the Berkovich affine line.
The intrinsic reductions and the intrinsic depths in non-archimedean dynamics
本文为非阿基米德动力学中Rumely模理论刻画及退化极限定理提供更简单证明，引入内在约化与方向性内在深度概念，并建立两个关键定理。
We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere must be a countable sum of atoms. For a 1-parameter family f_t of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on the Riemann sphere as the family degenerates. The family f_t may be viewed as a single rational function on the Berkovich projective line over the completion of the field of formal Puiseux series in t, and the limiting measure on the Riemann sphere is the "residual measure" associated to the equilibrium …
We study a question on characterizing polynomials among rational functions of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.
Let K be a complete, algebraically closed nonarchimedean valued field, and let f(z) in K(z) be a rational function of degree d at least 2. We give an algorithm to determine whether f(z) has potential good reduction over K, based on a geometric reformulation of the problem using the Berkovich Projective Line. We show the minimal resultant is is either achieved at a single point in the Berkovich line, or on a segment, and that minimal resultant locus is contained in the tree in spanned by the fixed points and the poles of f(z). When f(z) is defined over the rationals, the algorithm runs in proba…
We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the e-reduction of a Berkovich analytic space, a notion of independent interest.
Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is preperiodic for f. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists epsilon > 0 such that the set of points P in P^1(K) with f-canonical height at most …
Dynamical degrees of twisted rational maps
本文在相对代数动力学框架下拓展了动力学次数理论，证明半共轭的相对动力学次数对应一般纤维上诱导的扭曲有理映射的动力学次数，还证得仿射簇扭曲自同态第一动力学次数的代数性结果。
If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the…
We construct a family of rational map sequences providing an arbitrary large number of dynamically independent rescaling limits of non monomial type. From this, we deduce the existence of a family of rational maps providing a non trivial dynamics on the Berkovich projective line over the field of formal Puiseux series.
Counting non-attracting subtrees and the equidistribution of repelling points
该研究针对非阿基米德完备域上的Berkovich射影直线，计数不含吸引点的非吸引子树，证明其次数和等于有理函数次数，并得到其测度等分布于平衡测度的结果，解答了相关问题。
It is an open problem whether repelling periodic points are dense in the classical Julia set of a non-archimedean rational function of degree more than one. We give a partial positive answer to this question based on a study of a logarithmic equidistribution on the Berkovich projective line over non-archimedean fields.
Continuity of the Julia sets in non-archimedean and hybrid settings
该研究针对完备非阿基米德域上Berkovich空间及带混合范数的复空间参数化的单变量有理函数族，证明了朱利亚集在豪斯多夫拓扑下连续性的类似Mané--Sad--Sullivan定理结果。
We introduce the $f$-crucial function $\operatorname{Crucial}_f$ associated to a rational function $f\in K(z)$ of degree $>1$ over an algebraically closed field $K$ of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely's (resultant) function $\operatorname{ordRes}_f$ in terms of the hyperbolic metric $ρ$ on the Berkovich upper half space $\mathsf{H}^1$ in the Berkovich projective line $\mathsf{P}=\mathsf{P}(K)$. We also obtain geometric formulas for Rumely's weight function $w_f…
A classification of the periodic components of the Fatou set of $p$-adic rational maps. Each such periodic component is either an immediate attracting basin or an open affinoid, where the dynamics is quasi-periodic (the $p$-adic analogues of Siegel discs and Herman rings.) The main tool is a fixed point theorem for connected subsets of Berkovich's projective line (or p-adic hyperbolic space).
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
On the Number of Small Points for Rational Maps
本文证明了存在仅依赖于有理映射次数的一致常数，界定了数域上有理映射的小点数量，给出了 Baker 定理的一致版本，并将 Benedetto 和 Looper 的结果从多项式推广到有理映射。
We study pairs $(f, Γ)$ consisting of a non-Archimedean rational function $f$ and a finite set of vertices $Γ$ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set $Γ$. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we co…
Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $\mathbb{P}^{1,an}$ over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we study more about the activity locus such as its relation to boundedness locus, i.e., the Mandelbrot set, and to the normality of the sequence of the forward orbit.
Rigidité, expansion et entropie en dynamique non-archimédienne (Rigidity, expansion and entropy in non-Archimedean dynamics)
该研究在非阿基米德动力学中证明了有理映射的刚性性质，表明其拓扑熵为整数对数，并建立了李亚普诺夫指数与野分支位置的关系。
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 $ϕ\in K(z)$ of degree $d\geq 2$ measures the exponential rate of growth along a typical orbit of $ϕ$. When $ϕ$ 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 $L(ϕ)$ for maps $ϕ$ defined over non-Archimedean fields $K$. The bound depends only on the degree $d$ and the Lipschitz constant of $ϕ$. For maps $ϕ$ whose Julia sets …
Non-Archimedean techniques and dynamical degenerations
本文利用非阿基米德技术分析复射影线上有理映射序列的退化现象，提出替代 Luo 方法的超限方法，并通过 Berkovich 理论构建混合空间以证明平衡测度的收敛性及李亚普诺夫指数的渐进行为。
We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > δ- ε$, where $…
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