Entropy-volume context and an exact scaling/cover ledger

Definitions and mathematical statement

This extract is background, not a complete independent open question. It reports an entropy-degree theorem and ends before the next section's question.

For a closed connected Riemannian $n$-manifold, define volume entropy by \[ h_{\mathrm{vol}}(g)=\lim_{r\to\infty}\frac1r \log\operatorname{Vol}_{\widetilde g}(B_{\widetilde g}(\widetilde x,r)),\qquad E(M,g)=h_{\mathrm{vol}}(g)^n\operatorname{Vol}(M,g). \] The universal-cover growth rate is independent of the base point. The source recalls Gromov's entropy-degree conjecture: for a continuous map $f:(M,g)\to(N,h)$ of nonzero degree between closed connected oriented $n$-manifolds, with $(N,h)$ of constant negative curvature, \[ E(M,g)\geq|\deg f|E(N,h). \] Its constant-curvature equality wording says that $g$ also has constant negative curvature and $f$ is homotopic to a covering with $|\deg f|$ sheets, allowing its orientation sign.

The passage then reports Besson--Courtois--Gallot's Theorem 5.3.2 extending the target to negatively curved locally symmetric metrics. The standard rigidity formulation in real dimension $n\geq3$ says equality holds precisely when $f$ is homotopic to a homothetic Riemannian covering: there are a covering $F\simeq f$ and $c>0$ such that $g=c^2F^*h$. The source extract does not state the dimension restriction or the full extended equality clause. These must be checked against the cited theorem; in particular, the stronger homothetic-cover conclusion must not be extended to arbitrary surface markings. When the target is a general rank-one locally symmetric space, equality means this homothetic-cover condition, not necessarily constant real sectional curvature.

The supplied aid gives an auxiliary exact ledger. Whenever the entropies involved are positive, define \[ R(f)=\frac{E(M,g)}{|\deg f|E(N,h)},\qquad\delta(f)=\log R(f). \] Independent homotheties of domain and target leave $R$ unchanged. For composable nonzero-degree maps $f:M\to N$ and $q:N\to P$, with the same intermediate metric in both ratios, \[ R(q\circ f)=R(f)R(q),\qquad\delta(q\circ f)=\delta(f)+\delta(q). \] For a compatible covering square $q\circ f'=f\circ p$, where $p:M'\to M$ and $q:N'\to N$ have sheet numbers $a,b$, use the pullback metrics. Then $E(M')=aE(M)$, $E(N')=bE(N)$, and $b\deg f'=a\deg f$; consequently $R(f')=R(f)$. Connected components and orientations must be chosen consistently. Nonnegativity of $\delta$, and its vanishing exactly for homothetic coverings, require the applicable entropy-rigidity theorem's hypotheses; they are not claims for all maps or target metrics.

The historical context distinguishes topological entropy $h_{\mathrm{top}}$, Liouville metric entropy $h_{\mathrm{Liou}}$ using normalized invariant Liouville measure, and volume entropy. It reports locally symmetric metrics as critical for the first two at fixed volume, minimizers of topological entropy, but not maximizers of Liouville entropy in higher dimensions. It says an earlier Conjecture 5.3.1 holds for deformations of constant-curvature metrics, without including that conjecture's statement. The final paragraph reports Hölder regularity of the Anosov splitting, at least $C^1$ regularity in dimension two or under quarter pinching, and typical failure of $C^1$ regularity in higher dimensions. It stops at "It is natural to ask:"; no missing regularity question is invented.

Short English statement

This record contains entropy-rigidity context and an incomplete transition to another question. The supplied aid records exact scaling, composition and covering identities for a normalized entropy-degree ratio. It does not identify an additional unresolved problem, despite the registry's open label.

Sources

• Keith Burns and Vladimir S. Matveev, Open Problems and Questions About Geodesics, section 5.3 and the opening of section 5.4: https://aimath.org/pastworkshops/geodesicsproblems.pdf.
• Supplied labels: [Flaminio1995], [Ka-Kn-We1991], [Gromov1983], Besson--Courtois--Gallot [Be-Co-Ga1995] (including section 9), [Eberlein2001], and [Hasselblatt1994]. The passage also refers to Katok's paper without a complete entry. Full bibliographic details are not supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001891, code catalogue entry; canonical aim-geometry-notes.json, notes[228], source item 5.3; updated 14 August 2026.
• Research attempt 1: context only, problem status at run not a problem, rigor label proved, novelty confidence low. The current registry nevertheless says open and classifies it as open triage. The aid is machine generated, not peer reviewed, and requires independent verification before citation.