English

Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points

Dynamical Systems 2026-07-21 v1

Abstract

Let kk be an algebraically closed, complete non-Archimedean field of residue characteristic 00. Let ff be a polynomial of degree at least 22 over kk which does not have potential good reduction. We prove that if gg is any other polynomial with the same Julia set, then ff and gg must be dynamically related. As a consequence, we show that for any two complex polynomials f,gf,g of degree at least 22, 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.

Cite

@article{arxiv.2607.19252,
  title  = {Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points},
  author = {Chen Gong and Jit Wu Yap},
  journal= {arXiv preprint arXiv:2607.19252},
  year   = {2026}
}

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R2 v1 2026-07-22T20:50:49.637Z