Maximal entropy for measure-preserving nonuniformly expanding circle maps

Determine whether there exists a degree-two measure-preserving nonuniformly expanding map whose metric entropy equals the maximal value $ln 2$.

Background

The paper constructs boundary maps that are nonuniformly expanding, preserve Lebesgue measure, and have entropy arbitrarily close to ln2ln 2, the topological-entropy upper bound for degree-two circle maps.

It remains unresolved whether this upper bound is attained by any measure-preserving nonuniformly expanding degree-two map. The question concerns exact attainment, not merely approximation by maps with entropy approaching the maximum.

References

It is unknown to the author whether there exists a measure-preserving non-uniformly expanding map of degree 2 such that its entropy reaches the maximum value $\ln 2$.

— Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps  (2609.00601 - Jiang, 1 Sep 2026) in Section 3, subsection “SRB entropy near or in the boundary of uniformly expanding maps,” Example 2