Entropy and Geometric Growth of Induced Endomorphisms

Definitions and mathematical statement

The source asks for an analogue of Manning's theorem comparing topological entropy of $f:M\to M$ with geometric exponential growth of the induced fundamental-group endomorphism. It does not state a proposed equality or the exact version of Manning's theorem to generalize.

In the standard compact connected manifold setting, for a compatible metric define $d_n(x,y)=\max_{0\leq j<n}d(f^jx,f^jy)$ and let $s_n(\varepsilon)$ be the maximal size of an $\varepsilon$-separated set for $d_n$. Then \[ h_{\mathrm{top}}(f)=\lim_{\varepsilon\downarrow0} \limsup_{n\to\infty}\frac1n\log s_n(\varepsilon). \] Choose a basepoint and a path identifying its image to obtain $\phi=f_*:\pi_1(M)\to\pi_1(M)$; different choices can change this representative by inner automorphism.

One usual candidate group-growth quantity, for a finite generating set $S$ and word length $|\cdot|_S$, is \[ \lambda_S(\phi)=\limsup_{n\to\infty}\frac1n \log\!\left(1+\max_{s\in S}|\phi^n(s)|_S\right). \] A geometric version may instead use shortest representatives of free homotopy/conjugacy classes or lengths of loops in the universal-cover geometry. The excerpt does not specify its intended "geometric exponential growth rate", so these definitions are not declared interchangeable. Basepoint/inner-automorphism, generating-set and geometric normalization issues must be resolved for the chosen invariant.

The requested result is a sharp relation, under specified hypotheses, between $h_{\mathrm{top}}(f)$ and that source-defined growth. A lower bound, an equality in a restricted geometric class, and an equality for minimum entropy in a homotopy class are different assertions; the source selects none. It must not be rewritten as an unsupported equality for all continuous maps. Compactness, manifold class and regularity assumptions on $M,f$ are also not explicit in the standalone sentence; for noncompact spaces entropy requires further conventions.

The literature triage checked 21 August 2026 says a standard inequality does not provide the requested sharp analogy/equality, but does not reproduce that inequality or a complete target theorem. It treats the broad question as open. This dated supplied finding is not an independent literature assessment.

Short English statement

Find a precise sharp comparison between entropy of a map and geometric growth of its induced fundamental-group endomorphism. The growth definition, ambient hypotheses and desired strength are missing; a known inequality is not automatically the requested equality or analogue.

Sources

• Local UnsolvedMath registry, permanent ID 30000543, exact source code catalogue entry; Entropy and Geometric Growth of Induced Endomorphisms. Proposed year 2006, status open. The corrected_verified statement restores the map domain and induced fundamental-group target; claimed source inspection is metadata.
• Supplied report citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/33, pp. 1991--2058, https://doi.org/10.4171/owr/2006/33. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21, citing only "OWR Geometric Group Theory report", https://doi.org/10.4171/owr/2006/33. It says the standard inequality does not settle the requested general sharp relation. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no precise Manning theorem citation is supplied.