A dynamical system $f \colon [0,1] \to [0,1]$ is called turbulent if there exist two closed subintervals $I, J \subset [0,1]$ such that $I \cap J$ contains at most one point, and

$$I \cup J \subseteq f(I) \cap f(J).$$

Although it is not immediately clear from the definition, turbulence is another way of approaching chaotic behaviour for dynamical systems on intervals.

In this lecture we prove:

• If $f \colon [0,1] \to [0,1]$ is transitive then $f^2$ is turbulent.
• If $f$ is turbulent then $\mathsf{h}_{\operatorname{top}}(f) \ge \log 2$.

Putting these two results together completes the proof of the main result from the last lecture: if $f \colon [0,1] \to [0,1]$ is transitive then $\mathsf{h}_{\operatorname{top}}(f) \ge \log \sqrt{2}$.