# 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…

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## References

SHOWING 1-10 OF 48 REFERENCES

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

- Mathematics
- 2013

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…

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

- Mathematics
- 2011

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

- Mathematics
- 1987

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

- Mathematics
- 2008

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

- Mathematics
- 2010

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

- Mathematics
- 2012

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

- Mathematics
- 2009

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

- Mathematics
- 2009

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

- Mathematics
- 2012

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…