Given a polynomial or a rational map f we associate to it a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier ρ\not=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function ρin the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space.
Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.
In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.
We give a lower bound for the multipliers of repelling periodic points of entire functions. The bound is deduced from a bound for the multipliers of fixed points of composite entire functions.
In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.
We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4^n/p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected
In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
We consider the space of degree $n\ge 2$ rational maps of the Riemann sphere with $k$ distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For $k=2n-2$ and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree $n$ rational maps can be locally parameterized by the multipliers of any $2n-2$ …
We consider a space of complex polynomials of degree $n\ge 3$ with $n-1$ distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent over the field of complex numbers.
It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle.
We present the following result: consider the space of complex polynomials of degree n>2 with n-1 distinct marked periodic orbits of given periods. Then this space is irreducible and the multipliers of the marked periodic orbits considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree n polynomial maps can be locally parameterized by the multipliers of n-1 arbitrary distinct periodic orbits. A detailed proof of this result (together with a proof of a more genera…
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 $K$.
We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We obtain the rationality of the moduli spaces. The rationality of the moduli space of degree $(d,e)$ correspondences is obtained from a representation-theoretic projection to the one for the usual dynamical systems of degree $d+e-1$. We also show that the multiplier maps for the fixed points and the multiplier index theorem (Woods Hole formula) are also reduced through the projection and obtain the reduced…
We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case N=2 has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field obtained by adjoining to C the mth roots of L, for all L.
We establish an approximation of the activity current $T_c$ in the parameter space of a holomorphic family $f$ of rational functions having a marked critical point $c$ by parameters for which $c$ is periodic under $f$, i.e., is a superattracting periodic point. This partly generalizes a Dujardin--Favre theorem for rational functions having preperiodic points, and refines a Bassanelli--Berteloot theorem on a similar approximation of the bifurcation current $T_f$ of the holomorphic family $f$. The proof is based on a dynamical counterpart of this approximation.
We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + ε}$, for some $ε> 0$, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with "small" multipliers. Somehow surprinsingly the proof is based in measure theorical considerations.
We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations.
Let $Λ$ be a quasi-projective variety and assume that, either $Λ$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $Λ$ parametrizes an algebraic family $(f_λ)_{λ\inΛ}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof…
A polygon is called rational if the angle between each pair of sides is a rational multiple of $π.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has ``many'' periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $§^1.$ We will also prove some refinements of Theorem 1: the ``well distribution'' of periodic orbits in…
By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain parameter ranges.
We provide a complete classification of possible graphs of rational preperiodic points of quadratic rational functions defined over the rationals with a rational periodic critical point of period 3, under two assumptions: that these functions have no periodic points of period at least 5 and the conjectured enumeration of points on a certain genus 6 affine plane curve. We show that there are exactly six such possible graphs, and that rational functions satisfying the conditions above have at most eleven rational preperiodic points.
Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in $S$, we give a complete description of triples $(f_1,f_2,p)$ such that $p$ is a rational periodic point for both $f_i\in S$, $i=1,2$. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two n…
We study self-morphisms of smooth real projective algebraic curves that have only real periodic points. In the case of the projective line we provide a convenient characterization of such morphisms. We derive a semialgebraic description of the component of real fibered rational functions all of whose periodic points are real.
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-…
We consider $(1,2)$-rational functions given on the field of $p$-adic numbers $\mathbb Q_p$. In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $(1,2)$-rational function is conjugate to a two-parametric $(1,2)$-rational function. Depending on these two parameters we determine the type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study the ergodicity properties of the function on each invariant set. We de…
The lunar periods of our moon -- the companion of the Earth -- which date back to the Babylonians until around 600 BCE, are 29.53 days for the synodic, 27.55 days for the anomalistic and 27.21 days for the draconitic month. In this paper we define and compute these periods in terms of Floquet multipliers and Conley--Zehnder indices for planar periodic orbits in the spatial Hill lunar problem, which is a limit case of the spatial circular restricted three body problem. For very low energies, we are able to prove analytically the existence of the families of planar direct (family $g$) and retrog…
We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we show that the density of such polynomials with a rational periodic point of any period among all polynomials $f(x)=x^d+c$, $c\in\Q$, is zero. Furthermore, we establish the connection between polynomials $f$ with periodic points and two arithmetic sequences. This yields necessary conditions that must be satisfied by $c$ and $d$ in order for the polynomial $f$ t…
This article addresses the existence of $\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $¶^N$ where each component is a degree 2 homogeneous form in $N+1$ variables which has a $\Q$-periodic point of primitive period $\frac{(N+1)(N+2)}{2} + \lfloor \frac{N-1}{2}\rfloor$. This result is then used to show that for $N$ large enough there exists morphisms of $¶^N$ with $\Q$-rational periodic points with primitive period larger that $c(k)N^k$ for any $k$ and some constant $c(k)$.
We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are $q<\infty$ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by $q$. In particular, there are at most $q$ rationally invisible repelling periodic orbits. The…
In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcat…
This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.
The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type for $n=6$. An explicit description of the $n=6$ surface lets us find several infinite families of quadratic endomorphisms $f: \mathbb{P}^1 \to \mathbb{P}^1$ defined over $\mathbb{Q}$ with a rational periodic point of order $6$. In one of these families, $f$ also has a rational fixed point, for a total of at least $7$ periodic and $7$ preperiodic points. Thi…
We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet mul…
We describe an algorithm for distinguishing hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point. We then illustrate computer images of the hyperbolic components of the parameter spaces V1 - V4, which were produced using our algorithm. We also resolve the singularities of the projective closure of V5 by blowups, giving an alternative proof that as an algebraic curve, the geometric genus of V5 is 1. This explains why we are unable to produce an image for V5.
McMullen proved the moduli space of complex rational maps can be parametrized by the spectrum of all periodic-point multipliers up to a finite amount of data, with the well-understood exception of Lattès maps. We generalize his method to large positive characteristic. McMullen's method is analytic; a modified version of the method using rigid analysis works over a function field over a finite field of characteristic larger than the degree of the map. Over a finite field with such characteristic it implies that, generically, rational maps can indeed be parametrized by their multiplier spectra u…
We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(λ)$ for $λ$ in $\mathbb{C}$, defined by the condition that each $f\in Per_1(λ)$ has a fixed point of multiplier $λ$. We prove that the curve $Per_1(λ)$ contains infinitely many postcritically-finite maps if and only if $λ= 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(λ)$.
We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic …
Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for inf…
We prove existence of infinitely many classical periodic solutions with periodic boundary conditions for a class of monotone semilinear wave equations. Our argument relies on some new estimates for the linear problem with periodic boundary conditions by combining Littewood-Paley techniques, the Hausdorff-Young theorem and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of distributional solutions by differentiating the equations and employing Gagliardo-Nirenberg estimates.
A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this paper presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual "coupling" inputs that abstractly play the role of Lagrange multipliers. The main contribution of this paper is a process---mirroring solving for Lagrange multipliers in robotic systems---wherein we isolate subsystems free of coupling constraints that provably encode the full-ord…
On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties
Osaka Metropolitan University（大阪公立大学）；NTT Institute for Fundamental Mathematics, NTT Communication Science Laboratories, NTT, Inc.（日本电信电话株式会社基础数学研究所、通信科学实验室）
