For a suitable symplectic automorphism $\phi$ of a symplectic manifold $(M,\omega)$, the Floer-theoretic entropy $\mathrm{h}_{\mathrm{Floer}}(\phi)$ measures the logarithmic growth rate of the fixed point Floer homology of iterates of $\phi$. In many cases, the Floer homology naturally carries the structure of a persistence module. Based on this property, Çineli-Ginzburg-Gürel introduced the barcode entropy $\mathrm{h}_{\mathrm{bar}}(\varphi)$ of a Hamiltonian diffeomorphism $\varphi$ on a symplectic manifold and showed that it satisfies
\[ \mathrm{h}_{\mathrm{Floer}}(\varphi) \leq \mathrm{h}_{\mathrm{bar}}(\varphi) \leq \mathrm{h}_{\mathrm{top}}(\varphi).\]
This result was also extended to more general families of symplectic automorphisms recently.
On the other hand, Dimitrov-Haiden-Katzarkov-Kontsevich introduced the notion of categorical entropy $\mathrm{h}_{\mathrm{cat}}(\Psi)$ of an endofunctor $\Psi$ on a triangulated category as a categorical analogue of the topological entropy. It was recently shown that, for ``good" symplectic automorphisms $\phi$ on a symplectic manifold $(M,\omega)$,
\[ \mathrm{h}_{\mathrm{cat}}(\Phi) \leq \mathrm{h}_{\mathrm{bar}}(\phi) \leq \mathrm{h}_{\mathrm{cat}}(\phi),\]
where $\Phi$ is the endofunctor on the Fukaya category of $(M,\omega)$ induced by $\phi$.
In this talk, I will explain these notions of entropy of symplectic automorphisms and the comparison results among them. Furthermore, I will introduce a family of symplectic automorphisms, which are obtained as compositions of Dehn twists along Lagrangian spheres in a symplectic manifold using a Penner-type construction. Then I will explain how to compute their categorical entropies and show that they provide infinitely many examples of symplectic automorphisms with positive categorical entropy. This is based on joint work with Sangjin Lee.
