Repulsive periodic-point equidistribution conjecture for non-Archimedean rational maps

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Fix a rational map RR over KK of degree d≥2d\geq 2. A fixed point xx is called repulsive if either x∈C⁡R∩HKx\in \operatorname{\mathcal{C}}_R\cap \mathbb{H}_K, or x∈PK1x\in \mathbb{P}^1_K and the multiplier of RR at xx has norm greater than 11. For n∈N∗n\in\mathbb{N}^*, let RFix⁡(Rn)\operatorname{RFix}(R^n) be the set of repulsive fixed points of RnR^n, and let ρR\rho_R denote the canonical measure of RR. Repulsive periodic-point equidistribution conjecture. One has

1dn∑x∈RFix⁡(Rn)deg⁡Rn(x) δx⟶ρRas n→+∞.\frac{1}{d^n}\sum_{x\in\operatorname{RFix}(R^n)}\deg_{R^n}(x)\,\delta_x\longrightarrow\rho_R\quad\text{as }n\to+\infty.

For arbitrary KK with positive Lyapunov exponent, this extends the known equidistribution results for repulsive periodic points; the conjecture is proved in the paper when RR is non-wild and χ(R)>0\chi(R)>0, while the general non-moderate case remains delicate.

References

Primary source

Charles Favre and Juan Rivera-Letelier, “Rigidité, expansion et entropie en dynamique non-archimédienne (Rigidity, expansion and entropy in non-Archimedean dynamics)”, arXiv:2504.20280 (2026).

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