Attracting cycles in $p$-adic dynamics and height bounds for postcritically finite maps

@article{Benedetto2012AttractingCI,
  title={Attracting cycles in \$p\$-adic dynamics and height bounds for postcritically finite maps},
  author={Robert L. Benedetto and Patrick Ingram and Rafe Jones and Alon Y. Levy},
  journal={Duke Mathematical Journal},
  year={2012},
  volume={163},
  pages={2325-2356}
}
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non… 
View PDF on arXiv
37 Citations
Background Citations
5
Methods Citations
3
Results Citations
3

37 Citations

Iterated monodromy groups of rational functions and periodic points over finite fields

Let q be a prime power and φ a rational function with coefficients in a finite field Fq. For n ≥ 1, each element of P1(Fqn) is either periodic or strictly preperiodic under iteration of φ. Denote by…

Finiteness and liftability of postcritically finite quadratic morphisms in arbitrary characteristic

We show that for any integer n and any field k of characteristic different from 2 there are at most finitely many isomorphism classes of quadratic morphisms from the projective line over k to itself…

The critical height is a moduli height

The critical height of a rational function (with algebraic coefficients) is a natural measure of dynamical complexity, essentially an adelic analogue of the Lyapunov exponent. Coordinate-free, it is…

Two New Normals Forms for Polynomial Endomorphisms of the Projective Line with Applications to Postcritically Finite Maps

We explore two normal forms for polynomial endomorphisms of the projective line. The first is a normal form for degree 3 polynomials in terms of the multipliers of the fixed points. This normal form…

Approximation of non-archimedean Lyapunov exponents and applications over global fields

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…

Cyclotomic and abelian points in backward orbits of rational functions

We prove several results on backward orbits of rational functions over number fields. First, we show that if K is a number field, φ ∈ K(x) and α ∈ K then the extension of K generated by the abelian…

The McMullen Map in Positive Characteristic

There is no generally convergent purely iterative root-finding algorithm over a non-archimedean field whose residue characteristic is larger than either thedegree of the algorithm or the degree of the polynomial whose roots the algorithm finds, generalizing McMullen's result over the complex numbers.

Discreteness of postcritically finite maps in p-adic moduli space

Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let $f_c(z)$ be a one-parameter family of rational…

A new equivariant in nonarchimedean dynamics

Let K be a complete, algebraically closed, nonarchimedean valued field, and let f(z) be a rational function in K(z) of degree d at least 2. We show there is a natural way to assign non-negative…

Constraining Images of Quadratic Arboreal Representations

In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite…

References

SHOWING 1-10 OF 48 REFERENCES

Topology and geometry of the Berkovich ramification locus for rational functions, II

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions $$\varphi \in k(z)$$. Here we show the ramification locus is contained in a…

A finiteness result for post-critically finite polynomials

We show that the set of complex points in the moduli space of polynomials of degree d corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows…

Multiplier Spectra and the Moduli Space of Degree 3 Morphisms on P1

The moduli space of degree $d$ morphisms on $\mathbb{P}^1$ has received much study. McMullen showed that, except for certain families of Latt\`es maps, there is a finite-to-one correspondence (over…

Families of Rational Maps and Iterative Root-Finding Algorithms

In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is…

Several approaches to non-archimedean geometry

and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C. Serre's Harvard…

Thurston’s pullback map on the augmented Teichmüller space and applications

Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σf of a finite-dimensional Teichmüller space. We prove that this…

An algebraic characterization of expanding Thurston maps

Let $f: S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its…

The Image of an Arboreal Galois Representation

Much is known regarding images of p-adic Galois representations coming from subquotients of étale cohomology groups of varieties over number fields. In particular, the Mumford-Tate conjecture gives…

On Thurston's pullback map

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the…

A census of quadratic post-critically finite rational maps defined over Q

We find all quadratic post-critically finite (PCF) rational maps defined over the rationals. We describe an algorithm to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve…