In this lecture we introduce the notion of an expansive dynamical system. Roughly speaking, a system is expansive if the forward orbits of any two distinct points moves apart a fixed distance.

We prove that (on compact metric spaces) no reversible system is ever expansive, and thus introduce the notion of a weakly expansive system[1].

Finally, we show that for (weakly) expansive systems the topological entropy is always finite.

This result probably won't shock you, since we have yet to see an example of a dynamical system whose entropy is infinite. However rest assured such systems do exist: an example is on Problem Sheet F.

  1. Warning: This terminology is not entirely standard. ↩︎

Comments and questions?