[Paper Review] Critical orbits and attracting cycles in p-adic dynamics

Robert L. Benedetto, Patrick Ingram+2 more|arXiv (Cornell University)|Jan 8, 2012

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TL;DR

This paper establishes that non-Lattes post-critically finite (PCF) rational maps of a given degree over any number field fall into finitely many conjugacy classes by proving a non-archimedean analogue of Fatou’s theorem, which links attracting cycles to critical points in p-adic dynamics. The key result is a bounded height property in the moduli space, excluding the flexible Lattes family.

ABSTRACT

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-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

Motivation & Objective

• To understand the distribution and finiteness properties of post-critically finite (PCF) rational maps in arithmetic dynamics.
• To address the question of whether there are finitely many conjugacy classes of PCF maps of a fixed degree over a given number field.
• To exclude the Lattes maps as an exceptional family and show that the remaining PCF maps have bounded height in the moduli space.
• To establish a non-archimedean version of Fatou’s classical result on attracting cycles attracting critical points.
• To use this non-archimedean Fatou result as a key technical tool to prove finiteness of conjugacy classes.

Proposed method

• Adapts Fatou’s classical theorem on attracting cycles attracting critical points to the non-archimedean setting, specifically in p-adic dynamics.
• Applies the non-archimedean Fatou theorem to analyze the behavior of critical points near attracting cycles in p-adic dynamical systems.
• Uses the structure of the moduli space of rational functions to define and bound the height of PCF maps.
• Applies height theory in arithmetic dynamics to show that non-Lattes PCF maps have bounded height over number fields.
• Employs the theory of dynamical systems over non-archimedean fields to control the forward orbits of critical points.
• Excludes the Lattes maps as a well-understood exceptional family, allowing the bounded height result to apply to the remaining PCF maps.

Experimental results

Research questions

• RQ1 Are there only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree over a fixed number field?
• RQ2 Can a non-archimedean analogue of Fatou’s theorem on attracting cycles attracting critical points be established?
• RQ3 Does the collection of non-Lattes PCF rational maps have bounded height in the moduli space of rational functions?
• RQ4 How does the exclusion of Lattes maps affect the finiteness and height properties of PCF maps?
• RQ5 What role does p-adic dynamics play in understanding the global arithmetic structure of PCF rational functions?

Key findings

• The set of non-Lattes post-critically finite rational maps of a fixed degree has bounded height in the moduli space over the complex numbers.
• A non-archimedean version of Fatou’s classical result is proven, showing that every attracting cycle in p-adic dynamics attracts a critical point.
• As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree over any fixed number field.
• The Lattes maps are identified as the only exception to the bounded height property, and are excluded from the finiteness result.
• The proof relies on the interplay between p-adic dynamics and arithmetic geometry, particularly height theory in moduli spaces.
• The result confirms a finiteness conjecture for non-Lattes PCF maps under mild arithmetic conditions.