Related papers: Beyond topological hyperbolicity: the L-shadowing …

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…

In this paper, we introduce and analyze several key dynamical properties-namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal-within the framework of uniform transformation semigroups.…

Let $M$ be a smooth compact manifold and $\Lambda$ be a compact invariant set. In this paper we prove that for every robustly transitive set $\Lambda$, $f|_\Lambda$ satisfies a $C^1-$generic-stable shadowable property (resp.,…

We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber,…

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…

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other…

We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$…

We study a special type of shadowing (DSP) of chain transitive continuous self-maps of compact Hausdorff spaces. We prove some basic properties of DSP. As application of DSP, we obtain sufficient conditions for a statistical variant of…

We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose…

We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing…

The $C^1$-structurally stable diffeomorphims of a compact manifold are those that satisfy Axiom A and the strong transversality condition (AS). We generalize the concept of AS from diffeomorphisms to invariant compact subsets. Among other…

For any continuous self-map of a compact metric space, we consider the space of chain components and prove that the s-limit shadowing implies the denseness of chain components with the shadowing property. It gives a partial answer to a…

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…

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a $C^\infty$-flow on a…

This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic…

Let $(X,G,\Phi)$ be a dynamical system, where $X$ is compact Hausdorff space, and $G$ is a countable discrete group. We investigate shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property,…

We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of $\mathbb{T}^2$ is homotopic to the identity and has both a fixed point and a periodic…

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism $L$ of $\mathbb T^2$. It is known that the positive Lyapunov exponent of $f$ with respect to the…

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of…

The notion of stochastic shadowing property is introduced. Relations to stochastic stability and standard shadowing are studied. Using tent map as an example it is proved that, in contrast to what happens for standard shadowing, there are…