Unicritical Polynomials Research Articles

Preperiodic points of polynomial dynamical systems over finite fields

For a prime p, positive integers [Formula: see text], and a polynomial f with coefficients in [Formula: see text], let [Formula: see text]. As n varies, the [Formula: see text] partition the set of strictly preperiodic points of the dynamical system induced by the action of f on [Formula: see text]. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of [Formula: see text] lying in a given [Formula: see text]. Moreover, when we generalize our definition of [Formula: see text], we obtain both upper and lower bounds for the resulting averages.

Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity

Abstract In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ , where p is a prime number and where the orbit of $0$ is finite. For example, if $p=2$ and $0$ is periodic under $T^2+c$ with $c\in \mathbb {R}$ , we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in $\mathbb {Q}(y)[T]$ .

Totally real algebraic integers in short intervals, Jacobi polynomials, and unicritical families in arithmetic dynamics

AbstractWe classify all postcritically finite unicritical polynomials defined over the maximal totally real algebraic extension of . Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion that allows one to give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than 4.

Analytic structures and harmonic measure at bifurcation locus

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci Md for unicritical polynomials fc(z)=zd+c. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove C1+α2d+α−ϵ-conformality, α=2−HD(Jc0), of the parameter-phase space similarity maps ϒc0(z):C↦C at typical c0∈∂Md and establish that globally quasiconformal similarity maps ϒc0(z), c0∈∂Md, are C1-conformal along external rays landing at c0 in C∖Jc0 mapping onto the corresponding rays of Md. This conformal equivalence leads to the proof that the z-derivative of the similarity map ϒc0(z) at typical c0∈∂Md is equal to 1/T(c0), where T(c0)=∑n=0∞(D(fc0n)(c0))−1 is the transversality function.The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on ∂Md, [27]. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in ∂Md effectively blocking a good access of ∂Md from the outside.

Misiurewicz polynomials and dynamical units, part I

We study the dynamics of the unicritical polynomial family [Formula: see text]. The [Formula: see text]-values for which [Formula: see text] has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when [Formula: see text] is a prime power and [Formula: see text] is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of [Formula: see text]. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases.

A finiteness property of postcritically finite unicritical polynomials

A finiteness property of postcritically finite unicritical polynomials

New families satisfying the dynamical uniform boundedness principle over function fields

We extend a technique, originally due to the first author and Poonen, for proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic dynamics over function fields of positive characteristic. The original method applied to unicritical polynomials for which the characteristic does not divide the degree. We show that many new 1-parameter families of polynomials satisfy the SUBP, including the family of all quadratic polynomials in even characteristic. We also give a new family of non-polynomial, non-Lattès rational functions that satisfies the SUBP.

Accessibility and porosity of harmonic measure at bifurcation locus

We study hyperbolic geodesics running from ∞ $\infty$ to a generic point, by the harmonic measure with the pole at ∞ $\infty$ , on the boundary of the connectedness locus M d ${\cal M}_d$ for unicritical polynomials f c ( z ) = z d + c $f_c(z)=z^d+c$ . It is known that a generic parameter c ∈ ∂ M d $c\in \partial {\cal M}_d$ is not accessible within a John angle and ∂ M d $\partial {\cal M}_d$ spirals round them infinitely many times in both directions. We prove that almost every point from ∂ M d $\partial {\cal M}_d$ is asymptotically accessible by a flat angle with apperture decreasing slower than ( log ∘ ⋯ ∘ log dist ( c , ∂ M d ) ) − 1 $(\log \circ \dots \circ \log {\rm dist}\,(c,\partial {\cal M}_d))^{-1}$ for any iterate of log $\log$ . This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary β > 0 $\beta >0$ , the bifurcation locus is not β $\beta$ -porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.

Quadratic and cubic Newton maps of rational functions

The dynamics of all quadratic Newton maps of rational functions is completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three of any rational function is conformally conjugate to a unicritical polynomial (i.e., with exactly one finite critical point). However, there are cubic Newton maps which are conformally conjugate to other polynomials. The Julia set of such a Newton map is shown to be a closed curve. It is a Jordan curve whenever the Newton map has two attracting fixed points.

Fixed points of Koch maps

We study endomorphisms constructed by Sarah Koch in [Teichmüller theory and critically finite endomorphisms, Advances in Mathematics 248 (2013), 573–617] and we focus on the eigenvalues of the differential of such maps at its fixed points. To each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of C P k {\mathbb {CP}}^k . Koch showed that the eigenvalues of the differentials of such maps along periodic cycles outside the post-critical sets have modulus strictly greater than 1 1 . In this article, we show that the eigenvalues of the differentials at fixed points are either 0 0 or have modulus strictly greater than 1 1 . This confirms a conjecture proposed by the author in his thesis. We also provide a concrete description of such values in terms of the multiplier of a unicritical polynomial.

Minimally critical regular endomorphisms of

AbstractWe study the dynamics of the map $f:\mathbb {A}^N\to \mathbb {A}^N$ defined by $$ \begin{align*} f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b}, \end{align*} $$ for $A\in \operatorname {SL}_N$ , $\mathbf {b}\in \mathbb {A}^N$ , and $d\geq 2$ , a class which specializes to the unicritical polynomials when $N=1$ . In the case $k=\mathbb {C}$ we obtain lower bounds on the sum of Lyapunov exponents of f, and a statement which generalizes the compactness of the Mandelbrot set. Over $\overline {\mathbb {Q}}$ we obtain estimates on the critical height of f, and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.

Unicritical polynomial maps with rational multipliers

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.

A Galois–Dynamics Correspondence for Unicritical Polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell–Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton–Patel, and Vivaldi–Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois–dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm’s work in explicit Hilbert irreducibility.

Dynamical uniform boundedness and the abc-conjecture

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the abc-conjecture. For unicritical polynomials of degree at least five, we require only the standard abc-conjecture.

Near Parabolic Renormalization for Unicritical Holomorphic Maps

Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.

Finite index theorems for iterated Galois groups of unicritical polynomials

Let K K be the function field of a smooth irreducible curve defined over Q ¯ \overline {Q} . Let f ∈ K [ x ] f\in K[x] be of the form f ( x ) = x q + c f(x)=x^q+c , where q = p r , r ≥ 1 , q = p^{r}, r \ge 1, is a power of the prime number p p , and let β ∈ K ¯ \beta \in \overline {K} . For all n ∈ N ∪ { ∞ } n\in \mathbb {N}\cup \{\infty \} , the Galois groups G n ( β ) = G a l ( K ( f − n ( β ) ) / K ( β ) ) G_n(\beta )=\mathrm {Gal}(K(f^{-n}(\beta ))/K(\beta )) embed into [ C q ] n [C_q]^n , the n n -fold wreath product of the cyclic group C q C_q . We show that if f f is not isotrivial, then [ [ C q ] ∞ : G ∞ ( β ) ] > ∞ [[C_q]^\infty :G_\infty (\beta )]>\infty unless β \beta is postcritical or periodic. We are also able to prove that if f 1 ( x ) = x q + c 1 f_1(x)=x^q+c_1 and f 2 ( x ) = x q + c 2 f_2(x)=x^q+c_2 are two such distinct polynomials, then the fields ⋃ n = 1 ∞ K ( f 1 − n ( β ) ) \bigcup _{n=1}^\infty K(f_1^{-n}(\beta )) and ⋃ n = 1 ∞ K ( f 2 − n ( β ) ) \bigcup _{n=1}^\infty K(f_2^{-n}(\beta )) are disjoint over a finite extension of K K .

Expanding Metrics for Unicritical Semihyperbolic Polynomials

We prove that unicritical polynomials \(f(z)=z^d+c\) which are semihyperbolic, i.e., for which the critical point 0 is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric \(\rho (z) |dz|\), where \(\rho (z) = 1+{{\,\mathrm{dist}\,}}(z,P(f))^{-1+1/d}\) has singularities on the postcritical set P(f). We also show that this metric is Hölder equivalent to the usual Euclidean metric.

A question for iterated Galois groups in arithmetic dynamics

AbstractWe formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

Eigenvalues of the Thurston operator

Let f : C ̂ → C ̂ be a postcritically finite rational map, and let Q ( C ̂ ) be the space of meromorphic quadratic differentials on C ̂ with simple poles. We study the set of eigenvalues of the pushforward operator f ∗ : Q ( C ̂ ) → Q ( C ̂ ) . In particular, we show that when f : C → C is a unicritical polynomial of degree D with periodic critical point, the eigenvalues of f ∗ : Q ( C ̂ ) → Q ( C ̂ ) are contained in the annulus 1 4 D < | λ | < 1 and belong to 1 D U where U is the group of algebraic units.

The 𝐴𝐵𝐶-Conjecture implies uniform bounds on dynamical Zsigmondy sets

We prove that the a b c abc -Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the a b c abc -Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.