A comparison of categorical and topological entropies on Weinstein manifolds

Let $W$ be a symplectic manifold, and let $\phi :W\to W$ be a symplectic automorphism. Then, $\phi$ induces an auto-equivalence $\mathrm{\Phi }$ defined on the Fukaya category of $W$. In this paper, we prove that the categorical entropy of $\mathrm{\Phi }$ bounds the topological entropy of $\phi$ from below where $W$ is a Weinstein manifold and $\phi$ 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.