Characterization of almost-everywhere time-average values

Determine, for a function f in L^1(μ) and an ergodic invariant measure μ of a generalized dynamical system, which possible value of the trajectory-wise time average can be realized almost everywhere.

Background

Proposition 5.1 establishes that, for an ergodic invariant measure, each possible value of the time average of a function is either very common or very rare in the sense that the set of initial points from which some trajectory realizes that value has measure zero or one. The concluding remarks ask whether one can determine which of these values is realized almost everywhere for a given integrable function and ergodic measure.

References

For an ergodic system, a possible value of the time average is either very common or very rare by Proposition \ref{prop_rare}. A natural question here is: Given $f \in L1(\mu)$, where $\mu$ is ergodic, is it possible to determine which value can be realized as the time average almost everywhere?

— Ergodicity of dynamical systems without uniqueness of orbits  (2609.11087 - Suda, 10 Sep 2026) in Section 5, Concluding Remarks