The persistent homology of a sampled map: from a viewpoint of quiver representations

This paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations. The direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the underlying map if the sample is sufficiently dense. Compared to existing methods using eigenspace functors, our filtration analysis represents an important advantage that no prior information related to the eigenvalues of the underlying map is required. Some numerical examples are given to demonstrate the effectiveness of our method.

Homological epimorphisms, homotopy epimorphisms and acyclic maps

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

Smale endomorphisms over graph-directed Markov systems

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

Dynamic properties of the dynamical system SFnm(X), SFnm(f))

<p>Let X be a continuum and let n be a positive integer. We consider the hyperspaces F_n(X) and SF_n(X). If m is an integer such that n &gt; m ≥ 1, we consider the quotient space SFⁿ_m(X). For a given map f : X → X, we consider the induced maps F_n(f) : F_n(X) → F_n(X), SF_n(f) : SF_n(X) → SF_n(X) and SFⁿ_m(f) : SFⁿ_m(X) → SFⁿ_m(X). In this paper, we introduce the dynamical system (SFⁿ_m(X), SFⁿ_m (f)) and we investigate some relationships between the dynamical systems (X, f), (F_n(X), F_n(f)), (SF_n(X), SF_n(f)) and (SFⁿ_m(X), SFⁿ_m(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent.</p>

Almost minimal systems and periodicity in hyperspaces

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.