Marcin Kulczycki

On almost specification and average shadowing properties

In this paper we study relations between almost specification property, asymptotic average shadowing property and average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples that compactness is a necessary condition for these implications to hold. As a consequence of our methodology we also obtain a proof that limit shadowing in chain transitive systems implies shadowing.

Exploring the asymptotic average shadowing property

The main aim of this paper is to relate a recently introduced notion of the asymptotic average shadowing property (AASP) with other notions known from topological dynamics. For a compact metric space X and a continuous map we prove the following facts: 1. If f has the AASP and the minimal points of f are dense in X then f is totally transitive. 2. If f is a surjection and has the specification property then f has the AASP. 3. If f is c-expansive and has the shadowing property then f is mixing if and only if f has the AASP.

Noncontinuous maps and Devaney’s chaos

Vu Dong Tô has proven in [1] that for any mapping f: X → X, where X is a metric space that is not precompact, the third condition in the Devaney’s definition of chaos follows from the first two even if f is not assumed to be continuous. This paper completes this result by analysing the precompact case. We show that if X is either finite or perfect one can always find a map f: X → X that satisfies the first two conditions of Devaney’s chaos but not the third. Additionally, if X is neither finite nor perfect there is no f: X → X that would satisfy the first two conditions of Devaney’s chaos at the same time.

Liapunov stability and adding machines revisited

We give a revised proof of a theorem originally due to Buescu and Stewart (1995). The new version of this theorem dispenses with the assumption that the space X is locally compact. The theorem is as follows: let X be a locally connected metric space, let f,:,X→X be a continuous map, and let A⊂X be a Liapunov stable compact transitive set that has infinitely many connected components. Then the map induced by f on the space of connected components of A is topologically conjugate to an adding machine.

A class of continua that are not attractors of any IFS

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

COUPLED-EXPANDING MAPS AND MATRIX SHIFTS

For an irreducible transition matrix A of size m × m, which is not a permutation, a map f : X → X is said to be strictly A-coupled-expanding if there are nonempty sets V1,…, Vm ⊂ X such that the distance between any two of them is positive and f(Vi) ⊃ Vj holds whenever aij = 1. This paper presents two theorems that give sufficient conditions for a strictly A-coupled-expanding map to be chaotic on part of its domain in the sense of, respectively, Auslander and Yorke and Devaney. These results improve on the work of Zhang and Shi [2010]. An example is provided to illustrate that the class of maps the new theorems apply to is significantly wider.

Shadowing vs. distality for actions of ℝ n

This article introduces the notion of weakly parametrized (wp) shadowing for actions of groups ℤ m  × ℝ n , where m, n ≥ 0 and m + n > 0. The possibility of coexistence of distality and shadowing for actions of ℝ n is discussed. It is proven that an equicontinuous action of ℝ n on a compact connected space possessing wp-shadowing is actually minimal. Moreover, distal real flows (ℝ-actions) on one-dimensional compact metric spaces are characterized as constant-one suspensions over adding machines.

Entropy of Subordinate Shift Spaces

Abstract We introduce a new family of shift spaces—the subordinate shifts. Using subordinate shifts, we prove in an elementary way that for every nonnegative real number t there is a shift space with entropy t.

A Unified Approach to Theories of Shadowing

This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing.It is proven that there exists a metric space X such that the sets of maps with many types of general approximation properties (including the classic shadowing, the Lp-shadowing, limit shadowing, and the s-limit shadowing) are not dense in C(X), S(X), and H(X) (the space of continuous self-maps of X, continuous surjections of X onto itself, and self-homeomorphisms of X) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in C(M), S(M), and H(M). Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the Lp-shadowing, and the s-limit shadowing) are dense in the space of continuous self-maps of the Cantor set.A condition is given that guarantees transfer of general approximation property from a map on X to the map induced by it on the hyperspace of X. It is also proven that the transfer in the opposite direction always takes place.