The critical height is a moduli height

TL;DR

This paper proves Silverman's conjecture that the critical height of a rational function is comparable to any ample Weil height on the moduli space, except on the Lattes locus, linking dynamical complexity to arithmetic geometry.

Contribution

It establishes that the critical height is at least proportional to any ample Weil height on the moduli space, confirming a conjecture by Silverman.

Findings

Critical height is comparable to Weil heights on moduli space.

Proves Silverman's conjecture regarding the relationship between critical height and Weil height.

Except on the Lattes locus, the critical height and Weil height are essentially proportional.

Abstract

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 well-defined on moduli space, but bears no obvious relation to the arithmetic geometry of that space as a variety. At a conference in 2010, Silverman conjectured that this disconnect is superficial, and that in fact the critical height should be at least commensurate to any ample Weil height on the moduli space, except on the Lattes locus (where that has no hope of being true). Here, we prove that conjecture.