Counting non-attracting subtrees and the equidistribution of repelling points

Abstract: Consider the action of a rational function on the Berkovich projective line over an algebraically closed field complete with respect to a nontrivial non-archimedean absolute value. We say a connected component of the fixed locus is a "non-attracting subtree" if it contains no attracting points. We define a "degree of a non-attracting subtree," in terms of the local degrees of the rational function at its points. We show that the degrees of non-attracting subtrees sum up to the degree of the rational function. This counting result is obtained by studying the potential of a discrete measure charging each non-attracting subtree with mass equal to its degree. We prove that as we iterate the rational function these measures equidistribute to the equilibrium measure. Under certain conditions, notably in the case of polynomials, this reduces to the equidistribution of repelling points. We also show that repelling type II points become negligible under iteration when the Lyapunov exponent is positive. These results give a partial and a complete answer to questions of Favre and Rivera-Letelier.

• The paper proves the exact counting identity ∑Γ degR Γ = deg R by constructing a potential whose Laplacian encodes non-attracting subtrees, then shows the normalized measures τRⁿα converge to the canonical equilibrium measure νR.
• Repelling points equidistribute for polynomials, for connected Julia sets under an indifferent-subtree negligibility condition, and for rational maps with positive Lyapunov exponent, while the general conjecture remains open.
• The results replace irregular or non-isolated Berkovich repelling-point counts with finite-degree subtree counts and isolate indifferent and shared-repelling subtrees as the main obstacles to a general proof.

Context and motivation

Let $K$ be an algebraically closed field, complete with respect to a nontrivial non-archimedean absolute value, and let $R \in K(z)$ be a rational function. Favre and Rivera-Letelier developed the ergodic theory of the dynamical system induced by $R$ on the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}$, proving that iterated preimages of almost any point equidistribute toward the canonical equilibrium measure $\nu_R$, and that periodic $K$-points equidistribute under conditions later relaxed by Okuyama. Their Question 1, restated as Conjecture 1 in their later work, asks whether repelling periodic points also equidistribute: writing $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$, the conjecture predicts $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$ weakly on $\mathbb{P}^1_{\mathrm{Berk}}$ (2608.14200).

The paper observes a structural obstruction to transplanting the complex-dynamical proof strategy. In the complex case, one proves equidistribution of all periodic points and then discards non-repelling cycles using Fatou's theorem, which guarantees finitely many such cycles. In the non-archimedean setting this argument fails: repelling fixed points of $R^n$ may lie in $R \in K(z)$0, where they are generally not isolated and can form uncountable, non-discrete fixed loci. Moreover, the number of repelling fixed points is not governed by a uniform counting formula: it equals $R \in K(z)$1 for polynomials and for maps of good reduction, but can reach $R \in K(z)$2 in Rumely's examples. The paper's strategy is therefore to replace individual repelling points with an object admitting a regular count — the connected components of the non-attracting fixed locus — and then control the discrepancy between the two.

Non-attracting subtrees and the counting theorem

The fixed locus $R \in K(z)$3 decomposes into connected components, called non-attracting subtrees; an element $R \in K(z)$4 is repelling if it contains a repelling fixed point and indifferent otherwise. The paper assigns to each component the degree

$R \in K(z)$5

which is a well-defined positive integer because only finitely many points of $R \in K(z)$6 satisfy $R \in K(z)$7. For $R \in K(z)$8, the central object is the measure

$R \in K(z)$9

where $R$0 denotes the retraction of $R$1 onto $R$2.

The main counting theorem states that for any nonconstant $R$3, the measure $R$4 is finite of total mass $R$5, and consequently

$R$6

This is proved by potential-theoretic methods. The paper introduces a function $R$7, defined "diagonally" as the number of preimages of $R$8 (counted with local degree) whose only path to $R$9 passes through $\mathbb{P}^1_{\mathrm{Berk}}$0, and shows that $\mathbb{P}^1_{\mathrm{Berk}}$1 is locally constant and vanishes outside a quasi-finite subgraph. Integrating $\mathbb{P}^1_{\mathrm{Berk}}$2 along segments from $\mathbb{P}^1_{\mathrm{Berk}}$3 yields a potential $\mathbb{P}^1_{\mathrm{Berk}}$4 in the Baker–Rumely space $\mathbb{P}^1_{\mathrm{Berk}}$5, whose Laplacian satisfies

$\mathbb{P}^1_{\mathrm{Berk}}$6

The proof of the key locality property of $\mathbb{P}^1_{\mathrm{Berk}}$7 relies on Okuyama's notion of the depth $\mathbb{P}^1_{\mathrm{Berk}}$8 of a direction and on the identity $\mathbb{P}^1_{\mathrm{Berk}}$9.

Two structural corollaries follow immediately. If $\nu_R$0 has potential good reduction, then $\nu_R$1 consists of a single subtree, and $\nu_R$2 is explicitly $\nu_R$3 plus, when $\nu_R$4, a mass $\nu_R$5 at the unique point $\nu_R$6 of the Julia set. The converse fails: in Rumely's degree-$\nu_R$7 examples with $\nu_R$8 type II repelling fixed points of local degree $\nu_R$9, the counting theorem forces $K$0, yet for $K$1 the Julia set has at least $K$2 points, so $K$3 cannot have potential good reduction. This shows that the counting formula detects potential good reduction only one way.

Equidistribution of the measures $K$4

The second main theorem establishes that for $K$5 and any $K$6, the normalized measures $K$7 converge weakly to $K$8.

The proof is a clean potential-theoretic argument. By Brolin's theorem, the discrete measures $K$9 (for a non-exceptional $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$0) converge to $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$1. When $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$2 has no potential good reduction, $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$3 charges no point, so the Portmanteau theorem gives pointwise convergence of the associated distribution functions $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$4; dominated convergence then yields convergence of the potentials $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$5. Since $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$6 is an Arakelov–Green's function relative to $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$7 for every discrete probability measure $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$8, and the map $$\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\,\delta_\zeta$$9 is continuous by a result of Favre–Jonsson, the Laplacians converge as well. The same argument applied to $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$0, together with the identity $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$1, gives the result. The potential-good-reduction case is handled directly from the explicit form of $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$2.

Equidistribution of repelling points in special cases

The paper decomposes the error between $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$3 and $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$4 into its positive and negative parts,

$(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$5

where $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$6 places mass $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$7 on each indifferent subtree and $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$8 accounts for repelling subtrees containing more than one repelling point. Three theorems analyze cases in which one or both error terms vanish.

Polynomials. If $(\deg R^n)^{-1}\sigma_{R^n} \to \nu_R$9 with $\mathbb{P}^1_{\mathrm{Berk}}$0 and $\mathbb{P}^1_{\mathrm{Berk}}$1 lies in the Fatou component of $\mathbb{P}^1_{\mathrm{Berk}}$2, then $\mathbb{P}^1_{\mathrm{Berk}}$3 for all $\mathbb{P}^1_{\mathrm{Berk}}$4, and hence

$\mathbb{P}^1_{\mathrm{Berk}}$5

The proof uses the fact that for a polynomial, the direction toward $\mathbb{P}^1_{\mathrm{Berk}}$6 is the unique preimage of the direction toward $\mathbb{P}^1_{\mathrm{Berk}}$7 under the map on tangent spaces. This forces each non-attracting subtree $\mathbb{P}^1_{\mathrm{Berk}}$8 to have a unique repelling point, namely $\mathbb{P}^1_{\mathrm{Berk}}$9, and the retraction $R^n$0 agrees with it for $R^n$1 in the basin of $R^n$2. A byproduct is the exact count $R^n$3 for polynomials, stated for lack of a suitable reference.

Connected Julia set. If $R^n$4 is connected, then $R^n$5, and for $R^n$6 in this set the shared-degree error vanishes: $R^n$7 for all $R^n$8. The key lemma shows that if $R^n$9 restricts to the identity on a nonempty open segment between two points, then that segment lies in the Fatou set; since $R \in K(z)$00 is connected, a repelling point $R \in K(z)$01 and $R \in K(z)$02 are joined by a segment in $R \in K(z)$03, forcing $R \in K(z)$04. Under the additional hypothesis that the number of indifferent subtrees of $R \in K(z)$05 is $R \in K(z)$06 — equivalently, $R \in K(z)$07 — repelling points equidistribute. The paper is explicit that the asymptotic smallness of $R \in K(z)$08 is an assumption, not a conclusion, and that an independent asymptotic study of either error term would require new ideas.

Positive Lyapunov exponent. The paper resolves affirmatively Conjecture 3 of Favre and Rivera-Letelier: if $R \in K(z)$09, then

$R \in K(z)$10

The proof combines the counting theorem with Favre and Rivera-Letelier's own result that repelling $R \in K(z)$11-points equidistribute when $R \in K(z)$12. The inequality $R \in K(z)$13 for repelling type II points bounds the mass at non-$R \in K(z)$14-points by twice the total degree of subtrees containing no type I repelling point, which the counting theorem identifies as $R \in K(z)$15. As Favre and Rivera-Letelier noted, this conjecture together with their Theorem B implies both the equidistribution of repelling points (Conjecture 1) and the equidistribution of all repelling periodic points (their Conjecture 2) in the positive-Lyapunov-exponent regime; the paper therefore delivers a complete answer to these questions under the hypothesis $R \in K(z)$16, and a partial answer in general.

Limitations and open questions

The paper is candid that the general Conjecture 1 of Favre and Rivera-Letelier remains open. The reduction $R \in K(z)$17 is exact, but the paper establishes vanishing of the error terms only in the three special cases above; in particular, for connected Julia sets the equidistribution of repelling points is conditional on the unproven smallness of the indifferent-subtree mass $R \in K(z)$18, and the author states that an asymptotic study of either error term in general "would seem to require new ideas." The choice of base point $R \in K(z)$19 also matters: the clean identities for polynomials require $R \in K(z)$20 in the Fatou component of $R \in K(z)$21, and the connected-Julia-set argument requires $R \in K(z)$22 in the non-$R \in K(z)$23-part of $R \in K(z)$24. Whether indifferent subtrees are asymptotically negligible for arbitrary rational functions of degree at least $R \in K(z)$25 is the concrete question the paper leaves unresolved.

Conclusion

The paper introduces the degree of a non-attracting subtree and proves the exact counting identity $R \in K(z)$26, via a Laplacian computation for an explicitly constructed potential $R \in K(z)$27. This yields unconditional equidistribution of the measures $R \in K(z)$28 toward $R \in K(z)$29, and — for polynomials, for maps with connected Julia set under a negligibility hypothesis, and unconditionally for positive Lyapunov exponent — the equidistribution of repelling periodic points, resolving one conjecture of Favre and Rivera-Letelier completely in the expanding case and providing the framework within which the remaining cases can be formulated.