Related papers: Barcode entropy and wrapped Floer homology

In this article, the main goal is to give a dynamical point of view of Floer homology barcodes for Hamiltonian homeomorphisms of surfaces. More specifically, we describe a way to construct barcodes for Hamiltonian homeomorphisms of surfaces…

Let $S^*Q$ be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism $\varphi$ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In…

In this paper we use broken book decompositions to study Reeb flows on closed $3$-manifolds. We show that if the Liouville measure of a nondegenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for…

We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case…

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that:…

We develop a forcing theory of topological entropy for Reeb flows in dimension $3$. A transverse link $L$ in a closed contact $3$-manifold $(Y,\xi)$ is said to force topological entropy if $(Y,\xi)$ admits a Reeb flow with vanishing…

Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such…

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original…

We prove that to any smooth vector field of a closed manifold it can be assigned a nonnegative number called {\em rescaled topological entropy} satisfying the following properties: it is an upper bound for both the topological entropy and…

In this paper, we show that the pointwise finite-dimensional two-parameter persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of interlevel filtered Floer homology, is rectangle-decomposable. This allows for the…

In this expository article, we present the proof of the invariance of the wrapped Floer homology under the subcritical handle attachment. This is proved by Irie. Here, we fix a minor gap in the proof about the choice of a cofinal family of…

We explore the gapless topological phases of a $p$-wave superconductor, probing its rich topologically ordered phases and underlying quantum phenomena. The topological order of the system is characterized by studying its entanglement…

We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguy\^en and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted…

On every closed contact manifold there exist contact forms with volume one whose Reeb flows have arbitrarily small topological entropy. In contrast, for many closed manifolds there is a uniform positive lower bound for the topological…

We study stability properties of the topological entropy of Reeb flows on contact 3-manifolds with respect to the C^0-distance on the space of contact forms. Our main results show that a C^\infty-generic contact form on a closed co-oriented…

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (ie, slow entropy) can be computed directly from the Jordan block structure of the…

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

Evaluating the performance of generative models in image synthesis is a challenging task. Although the Fr\'echet Inception Distance is a widely accepted evaluation metric, it integrates different aspects (e.g., fidelity and diversity) of…

A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and…

We show that, up to connected sums with integer homology $L$-spaces, bordered Floer homology detects handlebodies, as well as whether a mapping class extends over a given handlebody or compression body. Using this, we combine ideas of…