# 6. Furstenberg's Theorem

We begin by proving *Furstenberg's Theorem*, which tells us that if $f \colon X \to X$ is a weakly mixing system then any $n$-fold product

$$ f \times \cdots \times f \colon X \times \cdots \times X \to X \times \cdots \times X $$

is automatically transitive.

Afterwards we introduce *linear *dynamical systems on Banach spaces, and construct an example of a linear dynamical system that is weakly mixing but not mixing.

Here are today's notes:

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