2026/07/21 by Chen Gong, Jit Wu Yap
#math.DS #math.NT
Let k be an algebraically closed, complete non-Archimedean field of residue characteristic 0. Let f be a polynomial of degree at least 2 over k which does not have potential good reduction. We prove that if g is any other polynomial with the same Julia set, then f and g must be dynamically related. As a consequence, we show that for any two complex polynomials f,g of degree at least 2, either their sets of preperiodic points coincide, or the number of their common preperiodic points is uniformly bounded above by a constant depending only on the degrees, thereby answering a conjecture of DeMarco--Krieger--Ye for polynomials. We also establish relative results, allowing us to prove special cases of the DeMarco--Mavraki conjecture.