The expansion entropy

In simply: the expansion entropy is a new way to calculate the entropy of a given system.

Expansion entropy uses the linearization of the dynamical system and a notion of a volume on its state space

From a mathematical point of view, we can describe the evolution of a given system $M$ using a map (a function, an application) that acts in the same system $M$: $f: M \rightarrow M$. Every maps $f$ are depending on time, that it could be discrete or continuous.
Using these maps we can construct the so called derivative matrix $Df$, that is constituted by the partial derivatives of $f$ respect the coordinates of the $n$-space $M$.
At this point with $Df$, you can calculate the function $G(Df)$, that is

a local volume growth ratio for the (typically nonlinear) $f$.

or in other words a way to measure the growth of $M$ in time.
Now $G(Df)$ will be integrated on the whole $n$-space and renormalized on the volume, and the new quantity $E(f, S)$, will be used to define the expansion entropy: \[H_0 (f, S) = \lim_{t' \rightarrow \infty} \frac{\ln E_{t', t} (f, S)}{t'-t}\] where $t'$ is the final time, $t$ is the initial time.
In this way the expansion entropy measure the disorder of the system, like the topological entropy, but using the expansion entropy we can define the chaos when $H_0 > 0$.

---

Hunt, B., & Ott, E. (2015). Defining chaos Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (9) DOI: 10.1063/1.4922973 (arXiv)