Certain dynamical systems, e.g., geophysical fluid flows, can be so complicated that it is difficult to provide concise mathematical models for their behavior. Nevertheless, simulations and experiments provide us with data in which we can recognize simple patterns that resemble systems that are well studied and understood. My research aimed to extract regions of such simpler behaviors and analyze how micro-structures fit together to form larger coherent structures, taking an approach alternative to classical geometric dynamical systems analysis.

I start with the assumption that one can access only *averages of functions* along trajectories. Although this seems like a big constraint, the averages along trajectories are computationally more robust than trajectories themselves, which is important when numerical and measurement errors cannot be neglected, e.g., in chaotic regimes. Moreover, description through trajectory averages turns out to be a natural setting for comparison of behaviors of different trajectories.

To each initial condition, we assign a set of measurements, formed by averaging a continuous function basis along the trajectory emanating from the initial condition. The collection of such measurements for all initial conditions is termed \emph{the ergodic quotient}. The generalized inverse problem that I have studied asks what we can say about geometric and spectral invariants of the dynamical system, knowing only the ergodic quotient.

For measure-preserving systems on compact manifolds, the ergodic quotient is the set of weak representatives of ergodic measures for the system. Their supports are ergodic sets, the smallest invariant sets in the state space. Therefore, through the ergodic quotient, one can access geometric- and operator-theoretic invariants using the analytic machinery of sequence spaces.

To study geometry, the sequence space is endowed with a weighted euclidean metric, corresponding to a negative-index Sobolev norm on the space of invariant measures. In this topology, we can formulate conditions on connectedness and continuity of the ergodic quotient which extend the notion of Reeb-Fomenko graphs, used in Morse theory of continuous integrals of motion for integrable systems.

In applied settings, we work with a truncated set of functions to average. Even then, the intrinsic dimension of the ergodic quotient is lower than the dimension of the ambient space. Applying a manifold-learning technique, the Diffusion Maps, we extract a low-dimensional parametrization of the ergodic quotient by the modes of diffusion along it. Such parametrization can be used to extract coherent structures, for visualization or further analysis. As diffusion modes are scale-ordered, we can visualize coarse features with relatively few modes used.

As a proof-of-concept, I implemented the visualization of coherent structures based on the ergodic quotient in a computer code. Analyzing two systems, the Chirikov standard map (see Figure 1) and the Arnold-Beltrami-Childress flows (see Figure 2), it was confirmed that the results conform to the known features of those flows. Additionally, we analyzed two fluid flows whose features were less known: in both cases, the procedure proved useful, uncovering unknown invariant structures in the state spaces.