27 October 2025
C. Hinsley
To calculate the topological entropy of a unimodal map, you have in general two choices: you can either calculate the Milnor-Thurston kneading determinant, or you can compute the Markov graph associated to the corresponding Hofbauer tower. Each technique relies on computing the iterates of the turning point in the map, but they have their own advantages. The kneading determinant is algorithmically easier to calculate, but is somewhat conceptually divorced from the dynamics of intervals in the map. On the other hand, the Hofbauer tower has a clear dynamical relationship with the original map, and one can easily reason about approximations of topological entropy by only going up to a certain number of iterates of the critical point (this can be done with the kneading determinant approach as well, but it is much harder to ascertain facts about the truncation error).
In the case of a multimodal map, the kneading determinant approach has been studied extensively; its calculation is relatively straightforward and it is the natural tool one reaches for when needing to study the topological entropy, especially when the critical points are eventually periodic (or have eventually periodic kneading invariants). But the problem of how one lifts a multimodal map to a Markov graph by a similar means to the Hofbauer approach for unimodal maps has not been studied; as close as I have seen the literature come to this is Bruin’s inducing scheme approach. I set out last Tuesday to see if the notion of a Hofbauer tower can be extended to multimodal (continuous) maps, and it turned out that the answer is yes. The conceptual clarity offered by Hofbauer towers as opposed to Milnor-Thurston kneading determinants in the unimodal case also extends to the multimodal case, so I think it’s worth describing how one constructs these towers for multimodal maps.
Before beginning, I should note that my sketches of map graphs in cobweb diagrams are incorrect at points not actually lying on the cobweb. The graphs are only meant to serve intuition but they can be adjusted in an obvious way to satisfy certain properties in the full (untruncated) cobwebs that we require. Another way of saying this is that, if you were to continue drawing the cobwebs for further iterates, you might find that some of the things I claim about the Hofbauer towers beyond the levels drawn are not true; you can nevertheless adjust the maps’ graphs so that these claims are true.
Suppose $f: [0, 1] \to [0, 1]$ maps laps (continuous monotone intervals in $[0, 1]$ maximal under inclusion) onto unions of laps. In particular, if this is true then the turning points (endpoints of laps) aren’t mapped into the interior of any lap and we say $f$ is a Markov map. We can construct a Markov graph having the laps of $f$ as vertices and having an edge from lap $I$ to lap $I'$ if $I' \subseteq f(I)$. Note that $I$ may equal $I'$. Here's an example of a Markov map:
In this map, there are three laps $I_1$, $I_2$, $I_3$ in order from left to right. One only needs to note that there are two turning points to determine this. How do we know it's Markov? Observe that the boundary and the turning points each map either onto the boundary or a turning point. That's sufficient, but we could slightly relax this condition if it were true on some restriction of the map to a subinterval of the domain still containing all the turning points.
Now consider to where each lap is sent by $f$:
We can show this with the Markov graph:
The adjacency matrix of the Markov graph is
$$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}. $$
The topological entropy of $f$ turns out to be the logarithm of the spectral radius of the adjacency matrix $A$ of the associated Markov graph:
$$ h_\mathrm{top}(f) = \ln\rho(A). $$