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Linear immersions (or Koopman eigenmappings) of a nonlinear system have wideapplications in prediction and control. In this work, we study the propertiesof linear immersions for nonlinear systems with multiple omega-limit sets.While previous research has indicated the possibility of discontinuousone-to-one linear immersions for such systems, it has been unclear whethercontinuous one-to-one linear immersions are attainable. Under mild conditions,we prove that any continuous immersion to a class of systems including linearsystems collapses all the omega-limit sets, and thus cannot be one-to-one.Furthermore, we show that this property is also shared by approximate linearimmersions learned from data as sample size increases and sampling intervaldecreases. Multiple examples are studied to illustrate our results.

Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings