Family Of Rational Maps Research Articles

On a Degenerating Limit Theorem of DeMarco–Faber

One of our aims is to complement the proof of DeMarco–Faber’s degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line, degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco–Faber’s more conceptual transfer principle between those dynamics.

Discreteness of postcritically finite maps in 𝑝-adic moduli space

Let p ≥ 2 p \geq 2 be a prime number and let C p \mathbb {C}_p be the completion of an algebraic closure of the p p -adic rational field Q p \mathbb {Q}_p . Let f c ( z ) f_c(z) be a one-parameter family of rational functions of degree d ≥ 2 d\geq 2 , where the coefficients are meromorphic functions defined at all parameters c c in some open disk D ⊆ C p D\subseteq \mathbb {C}_p . Assuming an appropriate stability condition, we prove that the parameters c c for which f c f_c is postcritically finite (PCF) are isolated from one another in the p p -adic disk D D except in certain trivial cases. In particular, all PCF parameters of the family f c ( z ) = z d + c f_c(z)=z^d+c are p p -adically isolated.

A generalization of Floater–Hormann interpolants

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on γ, an additional positive integer parameter. For γ=1, the original Floater–Hormann interpolants are obtained. When γ>1 we prove that the new rational functions share a lot of the nice properties of the original Floater–Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum (h∗) and maximum (h) distance between two consecutive nodes. It turns out that, in contrast to the original Floater–Hormann interpolants, for all γ>1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h∼h∗). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants (γ>1) uniformly converge to the interpolated function f, for any continuous function f and all γ>1. The same is not ensured by the original FH interpolants (γ=1). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater–Hormann interpolants.

Non-statistical rational maps

We show that in the family of degree $$d\geqslant 2$$ rational maps of the Riemann sphere, the closure of strictly postcritically finite maps contains a (relatively) Baire generic subset of maps displaying maximal non-statistical behavior: for a map f in this generic subset, the set of accumulation points of the sequence of empirical measures of almost every point in the phase space, is equal to the largest possible one, that is the set of all f-invariant measures. We also introduce an abstract setting to study non-statistical dynamics and sufficient conditions for their existence in a general family of maps. These sufficient conditions are related to the notions of “statistical instability” and “statistical bifurcation” for which we give a general formalization in the first section. The proof of our result in the family of rational maps is based on a transversality argument which allows us to control the behavior of the orbits of critical points for maps close to strictly postcritically finite rational maps. Using this property and doing careful perturbations, we show that the closure of strictly postcritically finite maps satisfies the sufficient conditions introduced in the abstract setting.

Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x, y, and z, using creative telescoping, yielding modular forms expressed as pullbacked 2F1 hypergeometric functions, can be obtained much more efficiently by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p=xyz. In other cases where creative telescoping yields pullbacked 2F1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2F1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve.

Bispectrality and biorthogonality of the rational functions of q-Hahn type

We introduce families of rational functions that are biorthogonal with respect to the q-hypergeometric distribution. A triplet of q-difference operators X, Y, Z is shown to play a role analogous to the pair of bispectral operators of orthogonal polynomials. The recurrence relation and difference equation take the form of generalized eigenvalue problems involving the three operators. The algebra generated by X, Y, Z is akin to the algebras of Askey–Wilson type in the case of orthogonal polynomials. The actions of these operators in three different bases are presented. Connections with Wilson's ϕ910 biorthogonal rational functions are also discussed.

Periodic points of polynomials over finite fields

Fix an odd prime p p . If r r is a positive integer and f f is a polynomial with coefficients in F p r \mathbb {F}_{p^r} , let P p , r ( f ) P_{p,r}(f) be the proportion of P 1 ( F p r ) \mathbb {P}^1\left (\mathbb {F}_{p^r}\right ) that is periodic with respect to f f . We show that as r r increases, the expected value of P p , r ( f ) P_{p,r}(f) , as f f ranges over quadratic polynomials, is less than 22 / ( log ⁡ log ⁡ p r ) 22/\left (\log {\log {p^r}}\right ) . This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.

Consensus of Julia Sets of Potts Models on Diamond-Like Hierarchical Lattice

The limit sets of zeros of the partition function for λ-state Potts models on diamond-like hierarchical lattice are the Julia sets of functions in a family of rational functions. In this paper, the consensus problem of Julia sets generated by λ-state Potts models on diamond-like hierarchical lattice is studied. Two types of the consensus problem of Julia sets are considered, one is with a leader and the other is with no leaders. Based on these two types, two different control protocols are proposed respectively to make systems achieve consensus of Julia sets. The simulations confirm the efficacy of control protocols.

Geometric limit of Julia set of a family of rational functions with odd degree

For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by with . Also, when we show that the geometric limit of the Julia set and filled Julia set of the family exists and is the unit circle.

Dynamics of a family of rational maps concerning renormalization transformation

Considering a family of rational maps Tnλ concerning renormalization transformation, we give a perfect description about the dynamical properties of Tnλ and the topological properties of the Fatou components F(Tnλ). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J(Tnλ)) about real parameter λ.

Connectivity of the Julia sets of singularly perturbed rational maps

We consider a family of rational functions which is given by $$\begin{aligned} f_{\lambda }(z)=\frac{z^n(z^{2n}-\lambda ^{n+1})}{z^{2n}-\lambda ^{3n-1}}, \end{aligned}$$ where $$n\ge 2$$ and $$\lambda \in {\mathbb {C}}^*-\{\lambda :\lambda ^{2n-2}=1\}$$ . When $$\lambda \ne 0$$ is small, $$f_{\lambda }$$ can be seen as a perturbation of the unicritical polynomial $$z\mapsto z^n$$ . It was known that in this case the Julia set $$J(f_\lambda )$$ of $$f_\lambda $$ is a Cantor set of circles on which the dynamics of $$f_\lambda $$ is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that $$J(f_\lambda )$$ is disconnected.

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

In this paper, we investigate the dynamics of the following family of rational maps \begin{document}$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $\end{document} with one parameter $ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $, where $ n\geq 2 $. This family of rational maps is viewed as a singular perturbation of the bi-critical map $ P_{-n}(z) = z^{-n} $ if $ \lambda \neq 0 $ is small. It is proved that the Julia set $ J(f_\lambda) $ is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of $ f_\lambda $ are attracted by the super-attracting cycle $ 0\leftrightarrow\infty $. Furthermore, we prove that there exists suitable $ \lambda $ such that $ J(f_\lambda) $ is a Cantor set of circles but the dynamics of $ f_{\lambda} $ on $ J(f_{\lambda}) $ is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of $ J(f_{\lambda}) $ is also proved if the free critical orbits are not attracted by the cycle $ 0\leftrightarrow\infty $. Finally we give an estimate of the Hausdorff dimension of the Julia set of $ f_\lambda $ in some special cases.

Classification of traveling waves for a quadratic Szegő equation

We give a complete classification of the traveling waves of the following quadratic Szegő equation : \begin{document}$i{\partial _t}u = 2J\Pi \left( {{{\left| u \right|}^2}} \right) + \bar J{u^2},\;\;\;\;u\left( {0, \cdot } \right) = {u_0},$ \end{document} and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.

Generalized Rings Around the McMullen Domain

We consider the family of rational maps given by $$F_\lambda (z)=z^n+\lambda /z^d$$ where $$n, d \in \mathbb {N}$$ with $$1/n+1/d<1,$$ the variable $$z\in {\widehat{{\mathbb {C}}}}$$ and the parameter $$\lambda \in {\mathbb {C}}$$ . It is known that when $$n=d \ge 3$$ there are infinitely many rings $${\mathcal {S}}^k$$ with $$k\in \mathbb {N}$$ , around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter $$\lambda $$ -plane where the Julia sets of $$F_\lambda $$ are Cantor sets of simple closed curves. The rings $${\mathcal {S}}^k$$ converge to the boundary of the McMullen domain as $$k \rightarrow \infty $$ and contain parameter values that lie at the center of Sierpinski holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of $$F_\lambda $$ are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when $$1/n+1/d<1$$ where n is not necessarily equal to d. The number of Sierpinski holes and superstable parameters on $${\mathcal {S}}^1$$ is $$\tau _1^{n,d} = n-1,$$ and on $${\mathcal {S}}^k$$ for $$k> 1$$ is given by $$\tau _k^{n,d} = dn^{k-2}(n-1)-n^{k-1} + 1$$ .

Uniformization of Simply-Connected Ramified Coverings of the Sphere by Rational Functions

We deduce a systemofODEs describing behavior of critical points and poles of a smooth one-parametric family of rational functions uniformizing a given family of ramified coverings of the Riemann sphere.

Some properties of surjective rational maps

We consider the so-called surjective rational maps. We study how the surjectivity property behaves in families of rational maps. Some (counter) examples are provided and a general result is proved.

Dynamical moduli spaces and elliptic curves

In these notes, we present a connection between the complex dynamics of a family of rational functions f t :ℙ 1 →ℙ 1 , parameterized by t in a Riemann surface X, and the arithmetic dynamics of f t on rational points ℙ 1 (k) where k=ℂ(X) or ℚ ¯(X). An explicit relation between stability and canonical height is explained, with a proof that contains a piece of the Mordell–Weil theorem for elliptic curves over function fields. Our main goal is to pose some questions and conjectures about these families, guided by the principle of “unlikely intersections” from arithmetic geometry, as in [53]. We also include a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space 𝕄 d of rational maps of any given degree d>1. These notes are based on four lectures at KAWA 2015, in Pisa, Italy, designed for an audience specializing in complex analysis, expanding upon the main results of [6, 17, 14].

Uniformization of simply connected ramified coverings of the sphere by rational functions

We deduce a system of ODEs describing the behavior of critical points and poles of a smooth one-parametric family of rational functions uniformizing a given family of ramified coverings of the Riemann sphere.

An effective algorithm to compute Mandelbrot sets in parameter planes

In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.

Generalized baby Mandelbrot sets adorned with halos in families of rational maps

We consider the family of rational maps given by where with the variable and the parameter . It is known [1] that when there are small copies of the Mandelbrot set symmetrically located around the origin in the parameter plane. These baby Mandelbrot sets have ‘antennas’ attached to the boundaries of Sierpiński holes. Sierpiński holes are open simply connected subsets of the parameter space for which the Julia sets of are Sierpiński curves. In this paper we generalize the symmetry properties of and the existence of the baby Mandelbrot sets to the case when where n is not necessarily equal to d.