Unambiguity of sofic entropy across sofic approximations

Ascertain whether, for any measure-preserving action of a sofic group, the sofic entropy along every sofic approximation is either equal to a single, approximation-independent value or equals −∞. Establish the independence of finite sofic entropy values from the choice of sofic approximation, or provide a counterexample.

Background

Sofic entropy, introduced by Bowen, depends on a chosen sofic approximation of the acting group. Whether different approximations yield the same entropy value is a major question in ergodic theory. The paper proves an analogous unambiguity for the introduced almost periodic entropy, motivating the corresponding open problem for sofic entropy.

References

In ergodic theory, an important open problem asks whether the sofic entropy of a measure-preserving system along any sofic approximation must always equal either one particular value or $-\infty$.

— Notions of entropy for unitary representations (2412.13751 - Austin, 18 Dec 2024) in Subsection 'Some consequences of Theorem C' (Section 10.4)