In this paper, a braid is regarded as a dynamical system of points in the plane. The states of this dynamical system are given by Delaunay triangulations. This construction makes it possible to define an abstract groupoid $\overset{abc}{\mathcal{G}^{4}_{n+3}}$, which gives a representation of the colored braid groupoid $\text{ColB}(n)$. We define homomorphisms ${f}_{n+3}:\overset{abc}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{Q})$ and ${f}'_{n+3}:\overset{abc}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{C})$, and describe an algorithm for computing the resulting invariants.

Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. We show first that, for this factor representation alone, both assumptions are unnecessary: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type. In particular, the mere existence of such a symbolic inverse-limit representation is not specific to shadowing. The main contribution of the paper is to identify the additional stability which shadowing provides in the metric case. We prove that every compact metric system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type whose bonding maps are surjective. Hence the inverse sequence satisfies the Mittag-Leffler condition, and the corresponding zero-dimensional extension still has shadowing. This strengthens the metric representation theorem of Good and Meddaugh and completes their characterization in terms of ALP factors of Mittag-Leffler inverse sequences of shifts of finite type. Finally, for arbitrary compact Hausdorff spaces, we show that every compact shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps. In this sense, compact shadowing systems are generated from shifts of finite type by applying, at most three times, the two shadowing-preserving operations of taking Mittag-Leffler inverse limits and passing to ALP factors.

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.