9. Expansive Dynamical Systems
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.
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.
Warning: This terminology is not entirely standard. ↩︎
Comments and questions?