This presentation aims at illustrating what could a a set-theorist's point of view contribute to the exploration of mathematical models of biological systems.

These models usually take the form of highly complex dynamical systems. One standard measure of the complexity of such a system is topological entropy $h$.

Two of the textbook definitions of this notion look like this:

\begin{equation*}

h = \lim_{\eps \rightarrow 0^+}\limsup_{T \rightarrow \infty} \frac{\ln sep(\eps, d_T)}{T} \quad \mbox{and} \quad h = \lim_{\eps \rightarrow 0^+}\limsup_{T \rightarrow \infty} \frac{\ln span(\eps, d_T)}{T}.

\end{equation*}

They were proposed around 1970 independently by Bowen and Dinaburg.

An obvious question is whether we can replace lim sup in this definition by lim. One would expect that there should have been a known counterexample by now. But there wasn't, until the presenter and his former Ph.D. student Dr. Ying Xin constructed one in 2017.

The construction was inspired by some techniques that the presenter remembered from his time in the Toronto Set Theory Seminar.