A simplexwise Patterson-Sullivan inequality and entropy-metric Hessian

Definitions and mathematical statement

The source asks whether \[ i(T,\mu_{T'})\,i(T',\mu_T)\ge1\qquad(T,T'\in X_n) \] and, failing a global answer, where it holds. The phrase "as above" refers to missing context: the record does not define $X_n$, the current map, or its normalization. The aid's interpretation uses Outer space of free simplicial metric $F_n$-trees, with finite marked core-graph quotients and $n\ge2$, and Patterson--Sullivan currents normalized by $i(T,\mu_T)=1$. A geodesic current is a flip-invariant $F_n$-invariant Radon measure on pairs of distinct boundary points. The tree-current pairing extends translation length, $i(T,\eta_w)=\|w\|_T$ for counting currents. Normalization is essential to the numerical constant one; projectively rescaling a tree with its self-normalized current leaves the cross-product unchanged.

The aid reports a simplexwise result. Fix one finite marked core graph $\mathcal G$, and consider two strictly positive edge-length vectors on it, with trees in the same open Outer-space simplex. Let $h$ denote volume entropy, the exponential growth rate for distances in the universal-cover metric. It has scaling $h(c x)=h(x)/c$. Choose entropy-one representatives $x,y$, so $h(x)=h(y)=1$, and define directed Bregman divergence on the positive length cone by \[ D_h(y\Vert x)=h(y)-h(x)-dh_x(y-x). \] The aid reports the exact factorization for the normalized cross-product \[ I(x,y):=i(T_x,\mu_{T_y})i(T_y,\mu_{T_x}) =(1+D_h(y\Vert x))(1+D_h(x\Vert y)). \] It reports $I(x,y)\ge1$, with equality exactly for the same projective metric on this fixed marked graph, equivalently $x=y$ among entropy-one representatives. No claim is made for zero edge lengths or different marked-graph simplices.

It further identifies one half of the logarithmic diagonal Hessian with the established entropy metric on the simplex. To state the normalization explicitly, on the entropy-one hypersurface put $g_{h,x}(v,v)=D^2h_x(v,v)$ for tangent $v$ with $dh_x(v)=0$. The reported identity is \[ \frac12\left.\frac{d^2}{dt^2}\right|_{t=0} \log I(x,y(t))=g_{h,x}(v,v), \qquad y(0)=x,\ y'(0)=v,\ h(y(t))=1. \] Equivalently the second-variable diagonal Hessian is twice this entropy metric. The exact conventions of the cited established metric and Patterson--Sullivan assignment must be checked, since their defining references are not supplied. A local Hessian identity does not itself construct a global Weil--Petersson metric or settle positivity across simplex changes.

The source motivation is a WP-type metric on $X_n$, perhaps on an $\operatorname{Out}(F_n)$-invariant subspace or manifold. Kasra Rafi discusses Teichmüller-space currents and Bonahon's metric, and warns that an analogous quadratic-differential/Lipschitz construction fails positivity. That warning concerns the comparison construction, not a supplied counterexample to the displayed Outer-space inequality. The text then introduces a following question from Martin Lustig with a second part added by "Arnaud," mentioning an uncertain "Huber[?]" and Besson attribution for decompositions of ${\mathbb{R}}$-trees and dual laminations. That adjacent context lacks the actual next question and is not folded into the present inequality. The aid reports the global cross-simplex question unresolved; all partial formulas are attributed unreviewed aid claims.

Short English statement

With explicit self-normalization, the aid reports the Patterson--Sullivan product inequality on each open marked-graph simplex, an exact Bregman factorization, and a Hessian equal to twice the entropy metric. The global inequality and a global WP-type construction remain unresolved in the supplied account.

Sources

• Local UnsolvedMath registry, permanent ID $20001654$, source code catalogue entry; aim-geometric-group-theory-notes.json, notes[162], tag problem.
• AIM workshop "The geometry of the outer automorphism group of a free group," 2010, Problem 5.1: https://aimath.org/WWN/outerauto/outerauto.pdf. Source type PDF; statement status reconstructed_unverified. The supplied extracted text was used; no PDF was accessed or compiled.
• The aid names Hilion's symmetric Patterson--Sullivan product and the established entropy metric without authoritative full citations. Original discussion names Kasra Rafi, Bonahon, Martin Lustig, "Arnaud," uncertain "Huber[?]," and Besson. The absent preceding definitions and following question are not reconstructed. Original literature field is empty.
• Research attempt 1: partial_result, status at run open, rigor label proved, novelty confidence low; registry status partially_solved, classification PARTIAL-PROGRESS. Created and updated 14 August 2026. Machine-generated aid, not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.