A comparison of categorical and topological entropies on Weinstein manifolds

TL;DR

The paper studies a Weinstein manifold $W$ equipped with a compactly supported exact symplectic automorphism $\phi$ and compares two entropy notions for the induced dynamics: the topological entropy $h_{top}(\phi)$ and the categorical entropy $h_{cat}(\phi)$ of the auto-equivalence on the wrapped Fukaya category. The authors relate $h_{cat}$ to the growth of Lagrangian Floer homology and connect $h_{top}$ to the growth of Lagrangian Floer cochain complexes, using Crofton-type inequalities and a Lagrangian tomograph to prove the central bound $h_{cat}(\phi) \le h_{top}(\phi)$. They also explore the compact Fukaya category case under additional duality-like assumptions, and introduce barcode entropy as a finer invariant that sits between the two, showing $h_{cat} \le h_{bar} \le h_{top}$ and outlining fundamental questions about equalities among these entropies. The paper provides two illustrative examples demonstrating the potential strictness of the inequality and the value of barcode entropy, and raises open questions about when these bounds become equalities. Overall, the work extends the interplay between dynamical entropy and Floer-theoretic invariants, offering new tools (Crofton-type bounds, Lagrangian tomography, barcode entropy) to study symplectic dynamics through categorical data.

Abstract

Let $W$ be a symplectic manifold, and let $φ:W \to W$ be a symplectic automorphism. Then, $φ$ induces an auto-equivalence $Φ$ defined on the Fukaya category of $W$. In this paper, we prove that the categorical entropy of $Φ$ bounds the topological entropy of $φ$ from below where $W$ is a Weinstein manifold and $φ$ is compactly supported. Moreover, being motivated by the work of Cineli, Ginzburg, and Gurel, we propose a conjecture which generalizes a result in dynamical system.

Figures (1)

• Figure 1: The interior of the black dotted circle is the base of a Lefschetz fibration $\pi$. The star marked points are the singular values and the black dot is $-\infty$. We note that the stop $\Lambda$ is given by $\Lambda = \pi^{-1}(-\infty)$. One can choose $G$ such that $\pi(G)$ is the union of all black curves. Similarly, $\pi\left(\varphi_0(G)\right)$ is the union of all red curves. Let $\pi(W)$ be contained in the interior of blue dotted circle. Then, $\left(\varphi_0(G),\phi^n(G)\right)$ is a good pair for all $n \in \mathbb{Z}$.

Theorems & Definitions (45)

• Remark 1.1
• Theorem 1.2: =Theorem \ref{['thm main']}
• proof : Sketch of Proof
• Theorem 1.3: =Theorem \ref{['thm main theorem for compact']}
• Proposition 1.4: =Propositions \ref{['prop bar vs top']} and \ref{['prop bar vs cat']}
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: Dinaburg, Bowen
• Theorem 2.5: Yomdin