Uniqueness of Maximal Entropy Measures for Surface Maps

Definitions and mathematical statement

Let $f:X\dashrightarrow X$ be a meromorphic self-map of a projective compact complex surface in the source's small-topological-degree setting. Conventionally, for a dominant map, its topological degree $\lambda_2(f)$ counts preimages of a generic point with multiplicity, its first dynamical degree $\lambda_1(f)$ measures exponential growth of pullbacks of divisor classes, and small topological degree means \[ \lambda_2(f)<\lambda_1(f). \] The exact standing hypotheses, such as the model on which dynamical pullbacks and energy are considered, are not supplied.

Assume the report's finite-dynamical-energy hypotheses and let $\mu$ be the invariant probability measure constructed there. The original statement describes $\mu$ as a measure of maximal entropy. The question is whether it is the unique such measure. Writing $\mathcal P_f(X)$ for the admissible invariant Borel probability measures, the intended uniqueness statement is \[ \forall\nu\in\mathcal P_f(X),\qquad h_\nu(f)=h_\mu(f)=\sup_{\eta\in\mathcal P_f(X)}h_\eta(f) \quad\Longrightarrow\quad \nu=\mu. \] Here $h_\nu(f)$ is measure-theoretic entropy. Because $f$ is meromorphic, invariance must be formulated on measures for which the map is defined almost everywhere and the pushforward is meaningful. The record does not define the class precisely. No finite-energy condition is silently imposed on every competing measure: finite dynamical energy is a hypothesis for the source construction of $\mu$, while the question asks uniqueness among maximal-entropy measures, not just among measures produced by that construction. Nor is the competitor restricted to ergodic or mixing measures.

Finite dynamical energy is a technical source condition whose definition and exact assumptions are absent. The source does not reproduce the measure construction or specify a topological-entropy convention for meromorphic maps, so no replacement energy criterion or numerical entropy identity is invented.

The corrected-verified note reports that the official report leaves uniqueness open under this energy hypothesis. The supplied triage checked 2026-08-21 labels the question open and says later papers establish maximal entropy and related structure without the general uniqueness claim. These are dated attributed statements, not new literature verification.

Short English statement

Under the report's finite-dynamical-energy hypotheses, ask whether its invariant measure for a small-topological-degree surface map is the only measure of maximal entropy. Competing measures are not silently restricted to the construction's finite-energy class.

Sources

• Local UnsolvedMath registry, permanent ID 30001090, source code catalogue entry; supplied complete record and corrected-verified report-check note. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• Trends and Developments in Complex Dynamics (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/48, pp. 2703--2762: https://doi.org/10.4171/owr/2008/48. Proposed-year metadata gives 2008.
• Diller--Dujardin--Guedj, small topological degree III: https://arxiv.org/abs/0806.0146.