Decomposition Space Theory and the Bing Shrinking Criterion

‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.

Properties of Dynamical Systems on Dendrites and Graphs

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

Cyclic G ‐ Ω -Weak Contraction-Weak Nonexpansive Mappings and Some Fixed Point Theorems in Metric Spaces

This paper introduces novel concepts of joint Y , Z cyclic G ‐ Ω S , T , a b e f -weak contraction and joint Y , Z cyclic G ‐ Ω S , T , a b e f -weak nonexpansive mappings and then proves the existence of a unique common fixed point of such mappings in case of complete and compact metric spaces, respectively, in particular, it proves the existence of a unique fixed point for both cyclic G ‐ Ω S , a b e f -weak contraction and cyclic G ‐ Ω S , a b e f -weak nonexpansive mappings, and hence, it also proves the existence of a unique fixed point for both cyclic Ω S , a b e f -weak contraction and cyclic Ω S , a b e f -weak nonexpansive mappings. The results of this research paper extend and generalize some fixed point theorems previously proved via the attached references.

Metric Versus Topological Receptive Entropy of Semigroup Actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

Ahlfors regularity and fractal dimension of Smale spaces

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on $$\mathbb R^3$$ R 3 and the Grassmannian of all 2-dimensional linear subspaces of $${\mathbb {R}}^4$$ R 4 . Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.

Refinements of topological invariants of flows

<p style='text-indent:20px;'>We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.</p>