Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.

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