What does this research mean for the field?

For almost all parameters t, every K-rational preperiodic point of the rational map f_t is the specialization of some K(T)-rational preperiodic point of f. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

February 16, 2026

Distribution of preperiodic points in one-parameter families of rational maps

Key Points

The aim is to investigate the distribution of preperiodic points in one-parameter families of rational maps defined over a number field.
Analyzed preperiodic points outside a set of natural density zero.
Assumed a weak form of the Uniform Boundedness Conjecture.
Calculated the average number of rational preperiodic points.
Provided new estimates for the quadratic family f_t(z) = z^2 + t.
Every K-rational preperiodic point specializes from a K(T)-rational preperiodic point for most t.
Identified new examples where average estimates hold unconditionally.
Obtained specific estimates for preperiodic points in the quadratic family over the field of rational numbers.

Abstract

Let f t fₜ be a one-parameter family of rational maps defined over a number field K K. We show that for all t t outside of a set of natural density zero, every K K -rational preperiodic point of f t fₜ is the specialization of some K (T) K (T) -rational preperiodic point of f f. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K K -rational preperiodic points of f f, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family f t (z) = z 2 + t fₜ (z) = z² + t over the field of rational numbers.

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