Vanishing hyperbolic repulsive-point mass conjecture for non-Archimedean rational maps

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Fix a rational map RR over KK of degree d≥2d\geq 2, let χ(R)\chi(R) be its Lyapunov exponent, let HK\mathbb{H}_K denote the hyperbolic space, and let RFix⁡(Rn)\operatorname{RFix}(R^n) be the set of repulsive fixed points of RnR^n. Vanishing hyperbolic repulsive-point mass conjecture. If χ(R)>0\chi(R)>0, then

lim⁡n→+∞1dn∑x∈RFix⁡(Rn)∩HKdeg⁡Rn(x)=0.\lim_{n\to+\infty}\frac{1}{d^n}\sum_{x\in\operatorname{RFix}(R^n)\cap\mathbb{H}_K}\deg_{R^n}(x)=0.

Together with the preceding equidistribution conjectures, this concerns the asymptotic contribution of repulsive periodic points in the hyperbolic space. The source does not state a general resolution; it records affirmative results in the moderate zero-exponent setting and describes the non-moderate case as 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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