Constant slope, entropy, and horseshoes for a map on a tame graph

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Alexandroff Manifolds and Homogeneous Continua

We prove the following result announced by the second and third authors: Any homogeneous, metric ANR-continuum is a -continuum provided dimGX = n ≥ 1 and , where G is a principal ideal domain. This implies that any homogeneous n-dimensional metric ANR-continuum is a Vn-continuum in the sense of Alexandroff. We also prove that any finite-dimensional cyclic in dimension n homogeneous metric continuum X, satisfying for some group G and n ≥ 1, cannot be separated by a compactum K with and dimGK ≤ n – 1. This provides a partial answer to a question of Kallipoliti–Papasoglu as to whether a two-dimensional homogeneous Peano continuum can be separated by arcs.

Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

A Continuous Extension Operator for Convex Metrics

We consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

Absolute Approximate Retracts and AR-Spaces

A subset A of a topological space X is an approximateretract of X if for every neighborhood U of A in X there is a retract R of X such that A ⊂ R ⊂ U. A compactum X is an absolute approximate retract (AAR-space) if whenever X is embedded as a subset of a compactum Z, then X is an approximate retract of Z. These concepts were first defined in [2] where it is shown that every AAR-space is a contractible Peano continuum. In [3] an example is given to show that there exists a contractible LC∞ compactum which is not an AAR-space.