Intrinsic semistable reduction loci for the iterations of non-archimedean quadratic rational functions and moduli-theoretic applications
We study the intrinsic semistable reduction loci for the iterations of quadratic rational functions over algebraically closed complete non-archimedean fields. Using a reduction-theoretic slope formula for the hyperbolic resultant function and the non-archimedean argument principle, we establish a precise eventual stationarity theorem when the residue characteristic is different from $2$. In the finite-order case, we define an intrinsic prehole parameter, and for quadratic rational functions with the same reduction order and boundary period, classify the intrinsic reduction data of all iterations at distinguished Fatou boundary points up to simultaneous projective equivalence. For antispecialization algebraic arcs degenerating to an order $p>1$ boundary point of the compactified quadratic moduli space over an algebraically closed field of characteristic $0$, we define an intrinsic antispecialization parameter using the fixed-point multiplier polynomial. We show that this parameter equals the prehole parameter of every Puiseux lift of an antispecialization arc degenerating to the boundary point and that the $p$th iteration specialization determines all iteration specializations. Over $\mathbb{C}$, the two intrinsic parameters coincide with DeMarco's parameter $τ^2$ after an appropriate interpretation. As a global consequence, we construct explicit blow-up models for the iterate compactification over arbitrary algebraically closed fields of characteristic $0$ and describe its boundary intrinsically. Over $\mathbb{C}$, this also recovers DeMarco's compactification.