Related papers: Shadowing, asymptotic shadowing and s-limit shadow…

We extend the single-perturbation approach (developed in our earlier publications for the case of a single map) to the analysis of the shadowing property for semigroups of endomorphisms. Our approach allows to give a constructive…

We explore the characteristics of shadows for a general class of spherically symmetric, static spacetimes, which may arise in general relativity or in modified theories of gravity. The chosen line element involves a sum (with constant but…

We introduce a new dynamical system model called the shadowing problem, where a shadower chases after an escaper by always staring at and keeping the distance from him. When the escaper runs along a planar closed curve, we associate to the…

For any continuous self-map of a compact metric space, we extend a partition of each chain component with respect to a chain proximal relation to a $G_\delta$-partition of the phase space. Under the assumption of s-limit shadowing, we use…

For topological dynamical systems defined by continuous self-maps of compact metric spaces, we consider the contractive shadowing property, i.e., the Lipschitz shadowing property such that the Lipschitz constant is less than 1. We prove…

We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing…

We point out a basic dichotomy between the shadowing and Lipschitz shadowing properties for one-sided shift spaces in two infinite-alphabet frameworks: the classical product-topology model $X\subseteq A^{\mathbb{N}}$ and the compact…

We deal with dynamical systems on complex lattices possessing chains of non-transversal heteroclinic connections between several periodic orbits. The systems we consider are inspired by the so-called \emph{toy model systems} (TMS) used to…

We consider low-dimensional systems with the shadowing property. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon$-transitive class, and in contrast we…

For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some…

Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the…

We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate for this class of spaces, that in…

We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates…

We discuss the dynamics of $n$-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every $n\in\mathbb{N}$, we exhibit an $n$-expansive homeomorphism, which is not $(n-1)$-expansive, has the shadowing…

We discuss whether classical examples of dynamical systems satisfying the shadowing property also satisfy the shadowing property for the induced map on the hyperspace of continua, obtaining both positive and negative results. We prove that…

We develop a new metric and inverse-limit framework for Deaconu-Renault systems arising from local homeomorphisms between open subsets of locally compact zero-dimensional spaces. Our starting point is the Paterson-type compactification of…

We propose a novel unifying approach to study the shadowing property for a broad class of dynamical systems (in particular, discontinuous and non-invertible) under a variety of perturbations. In distinction to known constructions, our…

This article is about the shadowing property of homeomorphisms on compact metric spaces and the map associating a point of the space to each pseudo-orbit, called 'shadowing map'. Based on some particular dynamical properties, as…

In this paper, we develop the basic theory of the shadowing property for local homeomorphisms of metric locally compact spaces, with a focus on applications to edge shift spaces connected with C*-algebra theory. For the local homeomorphism…

We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control…