Genus and Gonality of Small Curves, Dynamical Uniform Boundedness, and Bifurcation

Abstract

We prove the Gonality Conjecture in arithmetic dynamics: for any non-isotrivial one-parameter algebraic family of rational maps on $\mathbb{P}^1_{\mathbb{C}}$, the gonality of distinct dynatomic curves tends to infinity. More generally, outside the flexible Latt\`es family, every small sequence of horizontal curves has gonality tending to infinity, and its genus grows superlinearly with its degree over the parameter curve. We also obtain higher-dimensional analogues under natural bifurcation and multiplier-genericity hypotheses. As applications, we prove uniform boundedness results for iterated preimages over number fields and geometric uniform boundedness results for preperiodic points over function fields. The proof combines arithmetic equidistribution, woven currents, and bifurcation theory; the bifurcation mechanism is what forces the growth of genus and gonality.