Pseudo-orbit Tracing Property Research Articles

On linear non-uniform cellular automata: Duality and dynamics

For linear non-uniform cellular automata (NUCA) which are global perturbations of CA over an arbitrary universe, we introduce and investigate their dual linear NUCA, which are also endomorphisms over a generally infinite dimensional vector space. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Surprisingly, we show that linear NUCA with finite memory satisfy the shadowing property also known as the pseudo-orbit tracing property. Thus, linear NUCA can be useful for fault-tolerant computation. Applications on the dual surjunctivity are also obtained.

Topological stability for homeomorphisms with global attractor

AbstractWe prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

Topological stability and pseudo-orbit tracing property of Borel measures from the viewpoint of open covers

In the paper, we introduce the concepts of topological stability, expansivity and pseudo-orbit tracing property of Borel measures with respect to a given continuous map on a compact topological space from the viewpoint of open covers and prove that every expansive Borel measure with the pseudo-orbit tracing property is topologically stable. Furthermore, we obtain some properties of topologically stable measures.

Inverse pseudo orbit tracing property for robust diffeomorphisms

Let f : M → M be a diffeomorphism of a compact smooth manifold M. Herein, we demonstrate that (i) if f has the C1 robustly inverse pseudo orbit tracing property on the chain recurrent set CR(f), then CRf is hyperbolic of f and (ii) if f has the C1 robustly inverse pseudo orbit tracing property on a nontrivial transitive set Λ ⊂ M, then Λ is hyperbolic for f.111991 Mathematics Subject Classification. 37C50; 37D20.

Garden of Eden and weakly periodic points for certain expansive actions of groups

AbstractWe present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math.397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys.39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math.308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

Partial hyperbolicity and pseudo orbit tracing properties

It is known that if a closed invariant set Λ of a diffeomorphism f of a compact smooth manifold M is hyperbolic then f has the pseudo orbit tracing property. For a weak notion of hyperbolicity, in [4] if a transitive set of a diffeomorphism f of the three dimensional manifold M has a partially hyperbolic structure then f does not have the pseudo orbit tracing property. From the results, we consider a compact smooth manifold M which the dimension is greater than three. It is a general version of the three dimensional manifold M. More detail, we show that if a transitive diffeomorphism f of a compact smooth manifold M is partially hyperbolic, then f does not have the pseudo orbit tracing property. Moreover, if f is partially hyperbolic on M then f does not have the asymtotic and the ergodic pseudo orbit tracing properties.

Topological Stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

Topological Stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

Flows with ergodic pseudo orbit tracing property

<abstract><p>In the manuscript, we deal with a type of pseudo orbit tracing property and hyperbolicity about a vector field (or a divergence free vector field). We prove that a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ has the robustly ergodic pseudo orbit tracing property then it does not contain any singularities and it is Anosov. Additionally, there is a dense and open set $ \mathcal{R} $ in the set of $ C^1 $ a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ such that given a vector field (or a divergence free vector field) has the ergodic pseudo orbit tracing property then it does not contain singularities and it is Anosov.</p></abstract>

A hierarchy of topological systems with completely positive entropy

We define a hierarchy of systems with topological completely positive entropy in the context of countable amenable continuous group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.

On periodic shadowing, transitivity, chain mixing and expansivity in uniform dynamical systems

In this paper we extend some results on the notions such as Expansive, Pseudo Orbit Tracing Property (P.O.T.P.), Chain Transitive, Periodic Shadowing, Chain Recurrent. We prove that if an expansive dynamical system (X,f) on compact uniform space has P.O.T.P., then it has periodic shadowing. If a continuous self map f on a compact uniform space has finite shadowing, then f has P.O.T.P. We find that a dynamical system (X,f) on compact uniform space has shadowing property if (X,f) has periodic shadowing provided f is expansive. If f is chain mixing on a compact uniform space (X,U), then fn is chain transitive for each n ≥ 1. If f has the periodic shadowing then fn has periodic shadowing for all n > 1. The periodic shadowing is invariant of topological conjugacy provided that the conjugacy and its inverse are Lipschitz.

A spectral decomposition for flows on uniform spaces

We study the behavior of flows on uniform spaces and consider expansivity, pseudo orbit tracing property and chain recurrence. The main result of this article is a generalization of spectral decomposition theorem to flows that are expansive and have the pseudo orbit tracing property on uniform spaces.

Vector Fields with the Asymptotic Orbital Pseudo-orbit Tracing Property

In this paper, we introduce the notion of the asymptotic orbital pseudo-orbit tracing property for a vector field X of a compact smooth manifold M. We show that if a vector field X has the C1 robust asymptotic orbital pseudo-orbit tracing property, then it is Anosov, and if a C1 generic vector field X has the asymptotic orbital pseudo-orbit tracing property, then it is Anosov. Moreover, we also show our results for divergence-free vector fields and Hamiltonian systems.

Gromov-Hausdorff distances for dynamical systems

We study equivariant Gromov-Hausdorff distances for general actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has the pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport, we investigate equivariant Gromov-Hausdorff convergence for actions of locally compact amenable groups on Wasserstein spaces.

Suspensions of homeomorphisms with the two-sided limit shadowing property

In this paper, we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyse some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it. This gives a lot of examples of flows satisfying this property, however, it enlighten an important difference between the case of flows and homeomorphisms. There are flows satisfying the two-sided limit shadowing property that are not topologically mixing, while homeomorphisms satisfying the two-sided limit shadowing property satisfy even the specification property. It can happen that a suspension flow has the two-sided limit shadowing property but the base homeomorphism does not, though it is proved that it must satisfy a strictly weaker property called two-sided limit shadowing with a gap (as in [B. Carvalho, D. Kwietniak. On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl. 420 (2014), 801–813]). We define a similar notion of two-sided limit shadowing with a gap for flows and prove that these notions are actually equivalent in the case of flows. Finally, we prove that singular suspension flows (in the sense of [Komuro: One-parameter flows with the pseudo orbit tracing property. Mh. Math. 98 (1984) 219–253]) do not satisfy the two-sided limit shadowing property.

On Attractor and Pseudo Orbit Tracing of Uniform Limit Dynamical Systems

ABSTRACTSome dynamical properties of the uniform limit dynamical system have been investigated. It has been found that for a sequences (X, Fn) of dynamical systems having same attractor A and converging uniformly to a dynamical system (X, f), the dynamical system (X, f) has the same attractor A. It has been found that a sequence (X, Fn) of uniformly rigid dynamical systems converging uniformly to a dynamical system (X, f), the dynamical system (X, f) is also uniformly rigid. We have proved that the uniform limit (X, f) of a uniformly convergent sequence of dynamical systems having pseudo orbit tracing property has pseudo orbit tracing property. The expansiveness and the equicontinuous set ϵ: of the uniform limit dynamical system have been investigated.

The conditional variational principle for maps with the pseudo-orbit tracing property

Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X \to X$ is a continuous map. We define $n$-ordered empirical measure of $x \in X$ by \begin{document}$\mathscr{E}_n(x) = \frac{1}{n}\sum\limits_{i = 0}^{n-1}δ_{f^ix},$ \end{document} where $δ_y$ is the Dirac mass at $y$. Denote by $V(x)$ the set of limit measures of the sequence of measures $\mathscr{E}_n(x)$. In this paper, we obtain conditional variational principles for the topological entropy of \begin{document}$\Delta_{sub}(I): = \left\{ {x \in X:V(x)\subset I} \right\},$ \end{document} and \begin{document}$\Delta_{cap}(I): = \left\{ {x \in X:V(x)\cap I≠\emptyset } \right\}.$ \end{document} in a dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.

Pointwise Topological Stability

AbstractWe decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a topologically stable point of a homeomorphism f as a point x such that for any C0-perturbation g of f there is a continuous semiconjugation defined on the g-orbit closure of x which tends to the identity as g tends to f. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.

Pseudo-orbit tracing and algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

Topological stability and pseudo-orbit tracing property of group actions

In this paper we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property, then it is topologicaly stable. This represents a group action version of P. Walter’s stability theorem [Lecture Notes in Math., vol. 668, Springer, 1978, pp. 231–244]. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. In particular, we establish a characterization of subshifts of finite type over finitely generated groups in terms of the pseudo-orbit tracing property.

Shadowable Points for Flows

A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a G δ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. These results extends those presented in Morales (Dyn Syst. 2016;31(3):347–356). We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz attractor does not have shadowable points. We show that in the presence of shadowable points chain transitive flows are transitive and that transitivity is a necessary condition for chain recurrent flows with shadowable points whenever the phase space is connected. Finally, as an application, these results we give concise proofs of some well known theorems establishing that flows with POTP admitting some kind of recurrence are minimal.