# 5. Mixing and Weakly Mixing Dynamical Systems

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.

Comments and questions?