By "dynamical system," I refer to a paired structure (X, f), wherein X represents a mathematical playground and f denotes a self-mapping on the playground. For instance, in the case where X is a vector space, f should be a linear map. The study of dynamical systems (X, f) has been studied extensively when X is a surface, for example, see Thurston's work, Teichmuller theory, geometrical group theory, etc. A natural progression of this inquiry is to extend the investigation to higher dimensional settings, i.e., X is a manifold of greater dimension.

However, the assumption that phenomena observed in lower dimensional spaces will naturally manifest in higher dimensions is often overly optimistic. Thus, the notion of generalizing results from surface theory to dynamical systems (X, f) with dim X > 2 may seem like an ambitious endeavor. Thus, to address this challenge, I study dynamical systems on higher dimensional "symplectic" manifolds. The introduction of this additional structure holds promise for higher dimensional generalization. 

Moreover, the added symplectic structure drives the research into new territory. Since a symplectic manifold has a triangulated category, called Fukaya category, as its powerful invariant, it would be natural to reach out to dynamical systems (X,f) in triangulated categories, in other words, the case that X is a triangulated category and f is an auto-functor. 

To study the question, I need to use many tools, for example, Lefschetz fibrations, barcode entropy, etc. I am also interested in studying those tools and related questions. 

Abstract: In this paper, we formulate the wrapped Fukaya category of plumbing spaces. Plumbing spaces are examples of Weinstein manifolds, which appear naturally in various scenarios; Milnor fibers with simple singularities are plumbing spaces, plumbing spaces have simple Lagrangian skeletons, etc. Thus, it is natural to ask about the Fukaya category of plumbing spaces. 

In the paper, we formulated the Fukaya category of plumbing spaces through a "local-to-global" approach. The key ingredient is computing the wrapped Fukaya category of "plumbing sectors" that serve as a local model for describing Lagrangian skeletons with self-transversally-intersecting points. 

As a corollary of the main theorem, we showed that the Ginzburg dg category of any graded quiver (without potential) is equivalent to the Fukaya category of a specific plumbing space. In other words, we found a geometric model for the Ginzburg dg categories.

Abstract: This paper constitutes the second installment of a series that began with Paper 3. Here, Karabas and I provide explicit descriptions of a cylinder functor and a homotopy colimit functor. The primary objective in constructing these functors is to furnish a computational tool within the realm of dg categories. Consequently, by commencing with "simple" dg categories—particularly semifree dg categories—we can systematically compute their cylinder objects and homotopy colimits in a functorial manner. An application to symplectic topology is also described in the paper, together with a toy example.

Abstract: The notion of pseudo-Anosov autoequivalence is defined by Fan-Filip-Haiden-Katzarkv-Liu, but there are few known examples of it. In the article, we introduce a construction of pseudo-Anosov autoequivalences on N-Calabi-Yau categories associated with trees, motivated by symplectic topology. More precisely, in my thesis, I constructed symplectic automorphisms called "Penner type," which have simple asymptotic behavior. We expected that the Penner type symplectic automorphisms induce pseudo-Anosov autoequivalences because of their simple asymptotic behavior, and the expectation turned out to be true in this article. Moreover, we studied the actions which Penner type autoequivalences induce on the space of stability conditions. The induced actions are hyperbolic under a condition. 

Abstract: We present a study on the dynamical properties of Reeb flows, focusing on the definition of a new invariant within the following context: a contact manifold equipped with a non-degenerate contact one-form and a Liouville filling. By employing the theory of Persistence modules to the symplectic homology of the filling and the corresponding action functional, we establish an entropy-type invariant that we call barcode entropy. We demonstrate that the barcode entropy remains unchanged for a given contact manifold and contact one-form, regardless of the choice of filling. Furthermore, we establish that the barcode entropy provides a lower bound for the topological entropy of the Reeb flow under consideration. 

Abstract: We extend our previous work (see item 5) to all compactly supported exact symplectic automorphism on a Weinstein manifold. More precisely, we showed that the categorical entropy of such symplectic automorphisms bounds the topological entropy of it from below. 

Abstract: After Dimitrov-Haiden-Katzarkov-Kontsevich defined the categorical entropy, a question arises: "what is the connection between categorical and topological entropies?". We answer the question in a symplectic setting. More precisely, we fix a type of symplectic automorphism. Then, for a symplectic automorphism of that type, we show that the categorical entropy of it (on the Fukaya category) is smaller than the topological entropy. Also, we answer a question in category theory, which concerns a relation between the localization functor and the categorical entropy. 

Abstract: My previous paper, 'Lefschetz fibrations on cotangent bundles and some plumbings,' constructed diffeomorphic families of Weinstein manifolds. In this paper, we prove that the diffeomorphic familes are exotic, i.e., their members are not symplectomorphic to each other.

Abstract: We constructed a cylinder object for a semifree dg category. By using this, we can formulate a homotopy colimit formula which helps us to apply the result of Ganatra-Pardon-Shende. As an example of the fomula, we computed the wrapped Fuakay category of cotangent bundles of lens spaces. 

Abstract: In this paper, I gave an algorithm for constructing Lefschetz fibrations on cotangent bundles of manifolds and some plumbing spaces. The main idea of the algorithm is to use Weinstein handle decompositions. As an application of the results, we can obtain infinitely many diffeomorphic families of Weinstein manifolds.

Abstract: I constructed an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination as higher-dimensional generalizations of a train track and a geodesic lamination in the theory of surface.