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A dynamical system is one of the most physically interpretable mathematical notions. It consists of a "phase-space" X and a flow or semi-group of transformations on X. One of the main theoretical developments was to shift from a phase space perspective and study the effect of the dynamics on the space of measurements or observables on the $X$ instead. This enables the vast machinery of Operator theory to become applicable to dynamics. I will present how this leads to a sequence of transformations of measure preserving dynamics (m.p.d.) into unitary semi-groups on Hilbert spaces, and then finally into spectral measures on the unit circle. The overall transformation is functorial, meaning that it perfectly encodes the hierarchical structure within any m.p.d. into algebraic information. I will discuss the advantages and shortcomings of this transformation. The focus will be on obtaining approximations of the original system from approximations of the spectral measure itself. |