Rational Functions Of Degree Research Articles

Tame rational functions: Decompositions of iterates and orbit intersections

Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A(x)-A(y)=0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A^{\circ d}, d\geq 1, into compositions of rational functions can be obtained from decompositions of a single iterate A^{\circ N} for N large enough.

J-stability in non-archimedean dynamics

Let Cv be a complete, algebraically closed non-archimedean field, and let f∈Cv(z) be a rational function of degree d≥2. If f satisfies a bounded contraction condition on its Julia set, we prove that small perturbations of f have dynamics conjugate to those of f on their Julia sets.

Schwarzian Versus a Family of Moving Parabolic Points

In this article, the trajectories of meromorphic quadratic differentials whose coefficients are Schwarzian derivatives of rational maps are studied. We conjecture that the graphs consisting of the so-called negative-real critical Schwarzian trajectories are finite for all rational maps of degree at least two. For any given rational map f, we construct a family of parabolic rational maps { g w } w having a parabolic fixed point at the non-critical point w of f. If w is a zero of the Schwarzian Sf , we prove that the orientations of the negative-real critical trajectories of S f ( z ) d z 2 landing at w coincide with the attracting directions of gw at w. The critical curve in every immediate parabolic basin of w is an arc starting at w along the attracting direction and ending at a critical point of f, which is a pole of Sf . We conduct some numerical experiments to compare the critical curves in the immediate parabolic basins with the negative-real critical Schwarzian trajectories, which support our conjecture.

Rational functions of degree four that permute the projective line over a finite field

Recently, rational functions of degree three that permute the projective line over a finite field were determined by Ferraguti and Micheli. In the present paper, using a different method, we determine all rational functions of degree four that permute

On a Characterization of Polynomials Among Rational Functions in Non-Archimedean Dynamics

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.

The Image Size of Iterated Rational Maps over Finite Fields

Abstract Let $\varphi : {{\mathbb{P}}}^1( {{\mathbb{F}}}_q)\to{{\mathbb{P}}}^1( {{\mathbb{F}}}_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets $\varphi ^n( {{\mathbb{P}}}^1( {{\mathbb{F}}}_q))$ as a function of $n$. This is done using properties of Galois groups of iterated maps, whose connection to the size of image sets is established via the Chebotarev Density Theorem. We apply our results in the following setting. For a rational map defined over a number field, consider the reduction of the map modulo each prime of the number field. We use our results to give explicit bounds on the proportion of periodic points in the residue fields.

Commuting rational functions revisited

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

A special class of infinite measure-preserving quadratic rational maps

ABSTRACTWe show the existence of a set of positive measure in parameter space, each element corresponding to a rational map of degree , for which there exists an invariant infinite σ-finite measure equivalent to a conformal measure. These maps are conservative and exact with respect to the invariant measure, and are perturbations of generalized Boole maps and inner functions. For quadratic Boole maps of with real coefficients we show that the Krengel entropy provides a complete invariant for c-isomorphism classes of quadratic maps of this type.

Finite index theorems for iterated Galois groups of cubic polynomials

Let K be a number field or a function field. Let $$f\in K(x)$$ be a rational function of degree $$d\ge 2$$ , and let $$\beta \in {\mathbb {P}}^1(\overline{K})$$ . For all $$n\in \mathbb {N}\cup \{\infty \}$$ , the Galois groups $$G_n(\beta )={{\mathrm{Gal}}}(K(f^{-n}(\beta ))/K(\beta ))$$ embed into $${{\mathrm{Aut}}}(T_n)$$ , the automorphism group of the d-ary rooted tree of level n. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $$[{{\mathrm{Aut}}}(T_\infty ):G_\infty (\beta )]<\infty $$ . When f is a cubic polynomial and K is a function field of transcendence degree 1 over an algebraic extension of $${\mathbb {Q}}$$ , we resolve this problem by proving a list of necessary and sufficient conditions for finite index. This is the first result that gives necessary and sufficient conditions for finite index, and can be seen as a dynamical analog of the Serre Open Image Theorem. When K is a number field, our proof is conditional on both the abc conjecture for K and Vojta’s conjecture for blowups of $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$ . We also use our approach to solve some natural variants of the finite index problem for modified trees.

The maximal entropy measure of Fatou boundaries

We look at the maximal entropy measure (MME) of the boundaries of connected components of the Fatou set of a rational map of degree $≥ 2$. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.

Bounds for preperiodic points for maps with good reduction

Let K be a number field and let ϕ in K(z) be a rational function of degree d≥2. Let S be the set of places of bad reduction for ϕ (including the archimedean places). Let Per(ϕ,K), PrePer(ϕ,K), and Tail(ϕ,K) be the set of K-rational periodic, preperiodic, and purely preperiodic points of ϕ, respectively. The present paper presents two main results. The first result is a bound for |PrePer(ϕ,K)| in terms of the number of places of bad reduction |S| and the degree d of the rational function ϕ. This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that |Per(ϕ,K)|≥4 (resp. |Tail(ϕ,K)|≥3), we prove bounds for |Tail(ϕ,K)| (resp. |Per(ϕ,K)|) that depend only on the number of places of bad reduction |S| (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when |Per(ϕ,K)|<4 (resp. |Tail(ϕ,K)|<3).

Integral points of bounded degree on the projective line and in dynamical orbits

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\varphi(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

Counting number fields in fibers (with an Appendix by Jean Gillibert)

Let X be a projective curve over $${\mathbb Q}$$ and $${t\in {\mathbb Q}(X)}$$ a non-constant rational function of degree $${n\ge 2}$$ . For every $${\tau \in {\mathbb Z}}$$ pick $${P_\tau \in X(\bar{\mathbb Q})}$$ such that $${t(P_\tau )=\tau }$$ . Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$ is of degree at least $$e^{cN/\log N}$$ over $${\mathbb Q}$$ , where $${c>0}$$ depends only on X and t. In this paper we extend this result, replacing $${\mathbb Q}$$ by an arbitrary number field.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

AbstractMilnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

On semiconjugate rational functions

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of such functions, then either $B$ can be obtained from $A$ by a certain iterative process, or $A$ and $B$ can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.

$C^*$-algebras generated by multiplication operators and composition operators with rational symbol

Let $R$ be a rational function of degree at least two, let $J_R$ be the Julia set of $R$ and let $\mu^L$ be the Lyubich measure of $R$. We study the C$^*$-algebra $\mathcal{MC}_R$ generated by all multiplication operators by continuous functions in $C(J_R)$ and the composition operator $C_R$ induced by $R$ on $L^2(J_R, \mu^L)$. We show that the C$^*$-algebra $\mathcal{MC}_R$ is isomorphic to the C$^*$-algebra $\mathcal{O}_R (J_R)$ associated with the complex dynamical system $\{R^{\circ n} \}_{n=1} ^\infty$.

On the Belyi Functions of Planar Circular Maps

A map (S,G) is a closed Riemann surface S with an embedded graph G such that S \ G amounts to the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. The purpose of this article is to demonstrate a method of finding a Belyi function for planar circular maps and a way to plot a planar circular map by its Belyi function. Also we present a list of planar circular maps having no more than five edges, their Belyi functions, and their plots. We remark that the Belyi function for a planar circular map with E edges obtained with the help of our method is a rational function of degree E.

Portraits of preperiodic points for rational maps

AbstractLet K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ there exists a place $\mathfrak{p}$ of K such that the point α has preperiod m and minimum period n under the action of ϕ. This answers a conjecture made by Ingram–Silverman [13] and Faber–Granville [8]. We prove a similar result, under suitable modification, also when ϕ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple (c0, . . ., cd−2) ∈ kd−1 and for almost all pairs (mi, ni) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ for i = 0, . . ., d − 2, there exists a polynomial f ∈ k[z] of degree d in normal form such that for each i = 0, . . ., d − 2, the point ci has preperiod mi and minimum period ni under the action of f.

A Note on the Maximum Number of Zeros of $$r(z) - \overline{z}$$ r ( z ) - z ¯

An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006) states that the complex harmonic function $$r(z) - \overline{z}$$ , where $$r$$ is a rational function of degree $$n \ge 2$$ , has at most $$5 (n - 1)$$ zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $$r(z) - \overline{z}$$ no more than $$5 (n - 1) - 1$$ zeros can occur. Moreover, we show that $$r(z) - \overline{z}$$ is regular, if it has the maximal number of zeros.

Attaining Potentially Good Reduction in Arithmetic Dynamics

Let K be a non-archimedean field, and let φ ∈ K(z) be a rational function of degree d ≥ 2. If φ has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that φ is conjugate over L to a map of good reduction. In particular, if d = 2 or d is less than the residue characteristic of K, the bound is d + 1. If K is discretely valued, we give examples to show that our bound is sharp. Fix the following notation throughout this paper.