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In our final lecture on topological dynamics, we discuss a remarkable theorem of Furstenberg that classifies a major type of topological dynamical system – distal systems – in terms of highly structured (from an algebraic point of view) systems, namely towers of isometric extensions. This theorem is also a model for an important analogous result in ergodic theory, the Furstenberg-Zimmer structure theorem, which we will turn to in a few lectures. We will not be able to prove Furstenberg’s structure theorem for distal systems here in full, but we hope to illustrate some of the key points and ideas.

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