Let 𝒪_K be the ring of integers of an imaginary quadratic field K. Recently, Ji and Xie proved that every rational map f :ℂ→ℂ of degree d ≥ 2 whose multipliers all lie in 𝒪_K is a power map, a Chebyshev map or a Lattès map. Their proof relies on a result from non-Archimedean dynamics obtained by Rivera-Letelier. In the present note, we show that one can avoid using this result by considering a differential equation instead. Our proof of Ji and Xie’s result also applies to the case of entire maps. Thus, we also show that every nonaffine entire map f :ℂ→ℂ whose multipliers all lie in 𝒪_K is a power map or a Chebyshev map.

When ω is a primitive n-th root of unity, the quadratic polynomial F(z) = ωz (1 -z) and the entire map F(z) = ωz e^-z both have a parabolic fixed point at 0. Their parabolic multiplicity is equal to 1, that is, F^∘ n(z) = z ( 1 +c z^n +𝒪(z^n+1) ) with c ≠ 0. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in ℤ/(n -1) ℤ, and which is new in the polynomial case and requires working in the p-adic field ℚ_p for a suitable prime p such that the order of 2 in (ℤ/p ℤ)^× is exactly n.

We prove that for all degree d≥ 2 and all bounded type irrational θ, in the space of monic polynomials having a period 1 Siegel disk Δ of rotation number θ, the maximum locus of the conformal radius of Δ with respect to its fixed point contains polynomials having all critical points on the boundary of Δ. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.

Given a number field $\mathbb{K} \subset \mathbb{C}$ that is not contained in $\mathbb{R}$, we prove the existence of a dense set of entire maps $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose preperiodic points and multipliers all lie in $\mathbb{K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb{Q}$.

A meromorphic connection on the tangent bundle of a Riemann surface induces a complex affine structure on the complement of the poles. Local models for Fuchsian singularities are already known. In this paper, we introduce a complete set of local invariants for a meromorphic connection and provide local models for a complex affine structure in a punctured neighborhood of an irregular singularity. Generalizing a construction attributed to Veech, we introduce the Delaunay decomposition of a compact Riemann surface endowed with a meromorphic connection with irregular singularities. In particular, we give upper bounds on the complexity of the decomposition.

In the context of holomorphic families of \mathbb{P}^{k} endomorphisms, we show that various notions of stability are equivalent. This allows us to both extend and simplify the architecture of the proof of certain results of Berteloot–Bianchi–Dupont (2018).

This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of $\mathbb{C}^2$.

We propose a set of questions on the dynamics of Hénon maps from the real, complex, algebraic and arithmetic points of view.

Among the connected components of the interior of the Mandel-brot set are those that are hyperbolic. These components consist of parameters c E C for which the critical point z0 = 0 of fc : z 7 -> z2 + c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z0 is periodic. For a given n > 1, the Gleason polynomial for period n is the monic polynomial Gam, E Z[c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if Gam, factors over Z. In this article, we factor Gam, modulo 2. We prove the following remarkable fact: the number of irreducible factors of Gam, modulo 2 is equal to the number of real roots of Gam,.

Recently, Noytaptim and Petsche proved that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c(z):=z^2+c$ is postcritically finite are $0$, $-1$ and $-2$. In this note, we show that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c$ has a parabolic cycle are $\frac14$, $-\frac34$, $-\frac54$ and $-\frac74$.

abstract:We study the geometry of certain algebraic curves in the moduli space of cubic polynomials, and in the moduli space of quadratic rational maps. Given $k\geq 0$, ($k\neq 1$ in the case of quadratic rational maps), we show that the set of conjugacy classes of maps with a prefixed critical point of preperiod $k$, is an algebraic curve that is irreducible (over $\Bbb{C}$). We then study a closely related question concerning the irreducibility (over $\Bbb{Q}$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely arithmetic; they rely on a result providing sufficient conditions under which irreducibility over $\Bbb{C}$ is equivalent to irreducibility over $\Bbb{Q}$, and on a generalized Eisenstein criterion for irreducibility.

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree two circle maps F λ : R / Z → R / Z defined by F λ ( x ) ≔ 2 x + a + b π sin ( 2 π x ) with λ ≔ ( a , b ) ∈ R / Z × ( 0 , 1 ) . We prove that if F λ ◦ n − i d has a zero of multiplicity three in R / Z , then there is a system of local coordinates ( α , β ) : W → R 2 defined in a neighborhood W of λ , such that α ( λ ) = β ( λ ) = 0 and F μ ◦ n − i d has a multiple zero with μ ∈ W if and only if β 3 ( μ ) = α 2 ( μ ). This shows that the tips of tongues are regular cusps.

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher-degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we re-prove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point, which eventually maps to a repelling fixed point.

Letf:C -> Cbe a postcritically finite rational map, and letQ(C)be the space of meromorphic quadratic differentials onC with simple poles. We study the set of eigenvalues of the pushforward operatorf*:Q(C)-> Q(C). In particular, we show that whenf:C -> Cis a unicritical polynomial of degreeDwith periodic critical point, the eigenvalues off*:Q(C)-> Q(C)are contained in the annulus14D<|lambda|<1and belong to1DUwhereUis the group of algebraic units.

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $\alpha$, to depend on $\alpha$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.

Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of postcritical ray-equivalence classes for the modified mating, but it is enough to implement the pullback map for the formal mating. The proof combines local estimates and the Selinger extension of the Thurston Algorithm to augmented Teichmüller space. The latter is illustrated with several examples of canonical obstructions and canonical strata, including a relation between matings of conjugate polynomials and their core

In this article, we first study arithmetical properties of postcritically finite unicritical polynomials fa : z 7→ azD +1 with D ≥ 2. In particular, we answer a question of Milnor, showing that there exist non Galois conjugate parameters a1 ∈ C and a2 ∈ C such that fa1 and fa2 have critical orbits periodic with the same period. We also answer a question of Baker and DeMarco, proving that the set of parameters a ∈ C such that 0 and 1 are simultaneously (pre)periodic for qa : w 7→ w2 + a is equal to {0,−1,−2}. Introduction We study polynomials f : C → C of degree D ≥ 2 from a dynamical point of view, i.e., we consider sequences {zn}n≥0 defined by iteration: z0 ∈ C and zn := f(zn−1) = f(z0). This sequence is called the orbit of z0 for f . The point z0 is periodic if there is an integer n ≥ 1 such that f(z0) = z0. If p is the smallest integer with this property, we call it the period of z0. The point z0 is (pre)periodic if there exists a (smallest) integer k ≥ 0 such that f(z0) is periodic of period p. We say that k is the preperiod and that p is the period. Consider the polynomials fa defined by fa(z) = az D + 1, a ∈ C. For a 6= 0, those are polynomials of degree D with a unique critical point at 0. We are interested in the sets AD ⊂MD defined by AD := { a ∈ Cr{0} ; 0 is (pre)periodic for fa } and MD := { a ∈ C ; the orbit of 0 for fa is bounded } . If a ∈ AD, we say that fa is postcritically finite. The set AD is the set of Misiurewicz parameters and the setMD is the Multibrot set (a generalization of the Mandelbrot set in degree D). We shall first prove a Kronecker type result, where the set of roots of unity is replaced by AD, and the unit disk is replaced by MD. Proposition 1. If a is an algebraic integer such that a and all its Galois conjugates are contained in MD, then a ∈ AD ∪ {0}. Conversely, according to Milnor [M2, Theorem 3.2], if a ∈ AD, then • a is an algebraic integer • its Galois conjugates are in AD, • the product of a and its Galois conjugates divides D and • if 0 is periodic for fa with period p ≥ 2, then a is an algebraic unit. This research was supported in part by the ANR grant Lambda ANR-13-BS01-0002.

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a closely related question concerning the irreducibility (over $\mathbb Q$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely algebraic.

Fix $D\geq 2$ and consider the unicritical polynomial $f_a:\mathbb C \to \mathbb C$ defined by $f_a(z) = az^D+1$. We say that $0$ is (pre)periodic under iteration of $f_a$ if $f_a^{\circ (k+n)}(0) = f_a^{\circ k}(0)$ for some integers $k\geq 0$ and $n\geq 1$. If $k$ and $n$ are minimal, then $k$ is the preperiod and $n$ is the period. Recently, Goksel proved that if $D$ is prime, then two parameters $a_1\in \mathbb C$ and $a_2\in \mathbb C$ for which $0$ is preperiodic with period $1$ and with the same preperiod $k\geq 2$ are Galois conjugate; he also proved that when $D=2$, the result extends to the case of period $2$. We give a new proof of this result and extend it to the case of periods $1$ and $2$ for arbitrary prime power degrees, i.e., $D= p^e$ for some prime $p$. We also extend the result to the case of period $3$ in degree $D=2$.