Rational Self-map Research Articles

Arithmetic degrees for dynamical systems over function fields of characteristic zero

We study the arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We interpret the notion of arithmetic degree and study related problems over function fields geometrically. We give another proof of the theorem that the arithmetic degree at any point is smaller than or equal to the dynamical degree. We also suggest a sufficient condition for the arithmetic degree to coincide with the dynamical degree, and prove that any self-map has many points whose arithmetic degrees are equal to the dynamical degree. We also study dominant rational self-maps on projective spaces in detail.

On endomorphisms of hypersurfaces

For any prime $p \geq 5$, we show that generic hypersurface $X_{p} \subset \mathbf{P}^{p}$ defined over $\mathbf{Q}$ admits a non-trivial rational dominant self-map of degree $> 1$, defined over ${\mathbf{\bar {Q}}}$. A simple arithmetic application of this fact is also given.

On a dynamical version of a theorem of Rosenlicht

Consider the action of an algebraic group $G$ on an irreducible algebraic variety $X$ all defined over a field $k$. M. Rosenlicht showed that orbits in general position in $X$ can be separated by rational invariants. We prove a dynamical analogue of this theorem, where $G$ is replaced by a semigroup of dominant rational self-maps of $X$. Our semigroup $G$ is not required to have the structure of an algebraic variety and can be of arbitrary cardinality.

Rational self-maps of moduli spaces

We discuss some examples of geometrically meaningful rational self-maps of moduli space of curves of low genus and homogeneous forms.

Rational surface maps with invariant meromorphic two-forms

Let \(f:S\dashrightarrow S\) be a rational self-map of a smooth complex projective surface \(S\) and \(\eta \) be a meromorphic two-form on \(S\) satisfying \(f^*\eta = \delta \eta \) for some \(\delta \in \mathbf {C}^*\). We show that under a mild topological assumption on \(f\), there is a birational change of domain \(\psi : X\dashrightarrow S\) such that \(\psi ^*\eta \) has no zeros. In this context, we investigate the notion of algebraic stability for \(f\), proving that \(f\) can be made algebraically stable if and only if it acts nicely on the poles of \(\eta \). We illustrate this last result in the case \(\eta = \frac{dx\wedge dy}{xy}\), where we translate our stability result into a condition on whether a circle homeomorphism associated to \(f\) has rational rotation number.

On the Stein factorization of resolutions of two-dimensional Keller maps

We study the resolutions of rational self-maps of the projective plane that come from hypothetical counterexamples to the two-dimensional Jacobian Conjecture and establish several strong restrictions on its structure, in particular, its Stein factorization. We prove that all curves at infinity there go through a single point, and that the di-critical log-ramification divisor is ample.

On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties

Abstract Let f : X ⤏ X ${f:X\dashrightarrow X}$ be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K, let δ f be the dynamical degree of f, and let h X : X ( K ¯ ) → [ 1 , ∞ ) ${h_X:X({\bar{K}})\rightarrow [1,\infty )}$ be a Weil height relative to an ample divisor. We prove that for every ϵ > 0 ${\epsilon >0}$ there is a height bound h X ∘ f n ≪ ( δ f + ϵ ) n h X , $ h_X\circ f^n \ll (\delta _f+\epsilon )^n h_X, $ valid for all points whose f-orbit is well-defined, where the implied constant depends only on X, hX , f, and ϵ. An immediate corollary is a fundamental inequality α ¯ f ( P ) ≤ δ f ${\overline{\alpha }_f(P)\le \delta _f}$ for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence f * D ≡ β D ${f^*D\equiv \beta D}$ for some β > δ f ${\beta >\sqrt{\delta _f}}$ , we prove that the canonical height h ^ f , D = lim β - n h D ∘ f n $ {{\hat{h}}_{f,D}=\lim \beta ^{-n}h_D\circ f^n} $ converges and satisfies h ^ f , D ∘ f = β h ^ f , D ${{\hat{h}}_{f,D}\circ f=\beta {\hat{h}}_{f,D}}$ and h ^ f , D = h D + O ( h X ) ${{\hat{h}}_{f,D}=h_D+O(\sqrt{h_X})}$ . We also prove that the arithmetic degree α f ( P ) ${\alpha _f(P)}$ , if it exists, gives the main term in the height counting function for the f-orbit of P. We conjecture that α ¯ f ( P ) = δ f ${\overline{\alpha }_f(P)=\delta _f}$ whenever the f-orbit of P is Zariski dense and describe some cases for which we can prove our conjecture.

Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

AbstractLet φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit 𝒪φ(P) of P. We then prove our conjectures for monomial maps.

Remarks on endomorphisms and rational points

AbstractLet X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

Existence of non-preperiodic algebraic points for a rational self-map of infinite order

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 would follow immediately from the theory of canonical heights, but it does not work very well for rational self-maps. We provide an elementary proof following an argument by Bell, Ghioca and Tucker (arxiv:0808.3266).

On fixed points of rational self-maps of complex projective plane

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the set of fixed point free rational self-maps of $\pp^2$ is closed (modulo degenerate maps) in some natural topology on the space of rational self-maps of $\pp^2$.

Invariant Elliptic Curves as Attractors in the Projective Plane

Let f be a rational self-map of ℙ2 which leaves invariant an elliptic curve $\mathcal{C}$ with strictly negative transverse Lyapunov exponent. We show that $\mathcal{C}$ is an attractor, i.e. it possesses a dense orbit and its basin has strictly positive measure.

A computation of invariants of a rational self-map

Je demontre la stabilite algebrique et calcule les degres dynamiques de l'auto-application rationnelle (construite par C. Voisin) de la variete des droites sur une cubique dans ℙ 5 .

Ascoli-Arzelà type theorem for rational iterations on the complex projective plane

For a dominant algebraically stable rational self-map of the complex projective plane of degree at least 2, we will consider three different definitions of the Fatou set and show the equivalence of them (Ascoli-Arzela type theorem). As a corollary, it follows that all Fatou components are Stein. This is an improvement of an early result by Fornaess and Sibony.

Monopoles, non-linear ? models, and two-fold loop spaces

In this paper we study the topology of\(\hat M_k\), the moduli spaces ofSU(2) monopoles associated with the Yang-Mills-Higgs and Bogomol'nyi equations, and ℜ(m)k, non-linear σ models from quantum field theory. Beautiful work of Donaldson [18, 19], Hitchin [24, 25] and Taubes [37, 39, 40] shows that gauge equivalence classes of monopoles correspond to based rational self-maps of the Riemann sphere. Similarly, the non-linear σ models we consider here are based harmonic maps from the Riemann sphere to complex projectivem space. In seminal work, Segal [35] studied ℛ(m)k, the space of based rational maps from the Riemann sphere to complex projectivem space of a fixed degreek. Any element of ℛ(m)k is clearly an element of Ωk2CP(m), the space of all based continuous maps from the Riemann sphere to complex projectivem space of a fixed degreek, and this assignment gives rise to the natural inclusion of ℛ(m)k in Ωk2CP(m). Segal showed that these natural inclusions are homotopy equivalences through dimensionk(2m − 1). As the topology of the two-fold loop space Ω2CP(m) is well understood, Segal's result gives a very efficient way to explicitly determine the low dimensional topology of ℛ(m)k. Thus iterated loop spaces have much to say about the topology of monopoles and non-linear σ models.