In this lecture we define what it means for a dynamical system $f \colon X \to X$ to be (topologically) mixing.

This is a natural strengthening of the notion of transitivity: instead of asking that for any two open non-empty sets $U$ and $V$ there exists some $k$ such that $f^k(U) \cap V \ne \emptyset$, mixing requires that this intersection is nonempty for all sufficiently large $k$.

We then introduce an intermediate notion of a weakly mixing system, which is a dynamical system $f$ such that $f \times f$ is transitive on $X \times X$. This condition sits in between transitivity and mixing:

$$\text{mixing} \qquad \Rightarrow \qquad \text{weakly mixing} \qquad \Rightarrow \qquad \text{transitive}.$$

Next lecture we will see that neither of these implication signs may be reversed.

We can unify the notions of transitivity, mixing, and weakly mixing in terms of return times. Given two non-empty open subsets $U$ and $V$, the set of return times for $f$ is defined as:

$$\operatorname{ret}_f(U,V) := \left\{ k \ge 0 \mid f^k(U) \cap V \ne \emptyset \right\}.$$

Then we have:

The dynamical system is: If the set of return times is always:
Transitive Non-empty (and in fact, infinite, by Problem A5)
Weakly mixing Contains arbitrarily long intervals
Mixing Cofinite (i.e. its complement is finite)

The proof that weakly mixing systems can be characterised as thus will come next lecture after we have proved Furstenberg's Theorem.

Here are today's notes.