Related papers: Some consequences of shadowing and entropy-expansi…

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in…

The topological entropy of a continuous self-map of a compact metric space can be defined in several distinct ways; when the space is not assumed compact, these definitions can lead to distinct invariants. The original, purely topological…

Let $(X,d)$ be a compact metric space, $f:X\rightarrow X$ be a continuous transformation with the specification property. we consider non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is…

A homeomorphism on a compact metric space is said hyper-expansive if every pair of different compact sets are separated by the homeomorphism in the Hausdorff metric. We characterize such dynamics as those with a finite number of orbits and…

We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the…

In this note we study some properties of topological entropy for noncompact non-metrizable spaces.

We show the existence of Lipschitz-free spaces verifying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of the countably branching diamond graphs. As a…

We show that every positive expansive flow on a compact metric space consists of a finite number of periodic orbits and fixed points.

Open sets and compact saturated sets enjoy a perfect formal symmetry, at least for classes of spaces such as Stone spaces or spectral spaces. For larger classes of spaces, a perfect symmetry may not be available, although strong signs of it…

We prove that a homeomorphism f of a compact metric space X satisfies the L-shadowing property if and only if its induced hyperspace homeomorphism also satisfies the L-shadowing property. In the proof, assuming only the L-shadowing…

We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible…

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact…

It is well-known that a metric space $(X, d)$ is complete iff the set $X$ is closed in every metric superspace of $(X, d)$. For a given pseudometric space $(Y, \rho)$, we describe the maximal class $\mathbf{CEC}(Y, \rho)$ of superspaces of…

Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an…

In this article we extend the notion of expansivity from topological dynamics to automorphisms of commutative rings with identity. We show that a ring admits a 0-expansive automorphism if and only if it is a finite product of local rings.…

For any continuous self-map of a compact metric space, we provide sufficient conditions under which the infinite direct product of the map is $\omega$-chaotic. We also apply the result to obtain some examples of unusual $\omega$-chaotic…

A map $f$ on a compact metric space is expansive if and only if $f^n$ is expansive. We study the exponential rate of decay of the expansive constant of $f^n$. A major result is that this rate times box dimension bounds topological entropy.

We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every…

We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($\kappa $) condition for some fixed $\kappa >0$ and…

For any continuous self-map of a compact metric space, we prove a saturation of distributionally scrambled Mycielski sets under a type of shadowing and the chain transitivity.