Maximal-entropy measure conjecture for non-Archimedean rational maps

From papers

Fix a rational map RR over KK of degree d2d\geq 2, and for nNn\in\mathbb{N}^* let RFix(Rn)\operatorname{RFix}(R^n) be the set of repulsive fixed points of RnR^n. Let μR\mu_R be a measure of maximal metric entropy for RR. Maximal-entropy measure conjecture. The rational map RR admits a unique measure of maximal metric entropy μR\mu_R, and

1#RFix(Rn)xRFix(Rn)δxμRas n+.\frac{1}{\#\operatorname{RFix}(R^n)}\sum_{x\in\operatorname{RFix}(R^n)}\delta_x\longrightarrow\mu_R\quad\text{as }n\to+\infty.

When RR is non-wild and χ(R)=0\chi(R)=0, the source states that this conjecture is solved affirmatively; in the non-moderate case, the conjecture remains delicate.

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Sources & referencesView supporting material

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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