Continuous pointwise ergodicity for semigroup actions on locally compact spaces

What the math gives to ML

The paper provides a transferable equivalence between continuity of a state-to-invariant-measure map and mean ergodicity of the Koopman representation, including trajectories that escape to infinity. Its useful neural-network asset is a state-dependent asymptotic projection: features can be decomposed into invariant components and transient coboundary components without requiring a predetermined averaging scheme. This suggests a recurrent or world-model regularizer that learns continuous invariant summaries of trajectories, explicitly distinguishing convergence to an attractor from escape or instability.