It is proved that to every invariant measure of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for the measure. It follows that the asymptotic average shadowing property implies that every invariant measure has a generic point. The proof is based on the properties of the Besicovitch pseudometric DB which are of independent interest. It is proved among the other things that the set of generic points of ergodic measures is a closed set with respect to DB. It is also showed that the weak specification property implies the average asymptotic shadowing property thus the theory presented generalizes most known results on the existence of generic points for arbitrary invariant measures.
In this paper we present a systematic study of shadowing properties with average error in tracing such as (asymptotic) average shadowing, $\underline{d}$-shadowing, $\overline{d}$-shadowing and almost specification. As the main tools we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center. Next, we show that always (e.g. without assumption that the map is onto) almost specification implies asymptotic average shadowing,…
We look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits, products, factor maps and the induced maps for symmetric products and hyperspaces. The shadowing properties we consider are the following: shadowing, h-shadowing, eventual shadowing, orbital shadowing, strong orbital shadowing, the first and second weak shadowing properties, limit shadowing, s-limit shadowing, orbital limit shadowing and inverse shadowing.
In this paper, we introduce and study notions of average chain transitivity, average chain mixing, total average chain transitivity and almost average shadowing property. We also discuss their interrelations.
In this paper we generalize the notion of asymptotic average shadowing property to parameterized IFS and prove some related theorems on this notion. Specially, this is proved that every uniformly contracting IFS has the asymptotic average shadowing property. As an important result, we show that if a continuous surjective IFS F on a compact metric space X has the asymptotic average shadowing property then F is chain transitive. Moreover, we introduce some examples and investigate the relationship between the original asymptotic average shadowing property and asymptotic average shadowing propert…
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.
In this paper we examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, we demonstrate that if the dynamical system $(X,f)$ has shadowing, then it is recurrent if and only if it is minimal. Furthermore, we show that a uniformly rigid system $(X,f)$ has shadowing if and only if $X$ is totally disconnected and use this to demonstrate the existence of a space $X$ for which no surjective system $(X,f)$ has shadowing. We further refine these results to discuss the dynamics that can occur in spaces with compact spa…
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 order to determine whether a system has shadowing, it is sufficient to check that continuously generated pseudo-orbits can be shadowed.
The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The "gluing" construction used in the analysis of the last property turns out to be the key point of this theory.
This paper is concerned with the study of various notions of shadowing of dynamical systems induced by a sequence of maps, so-called time varying maps, on a metric space. We define and study the shadowing, h-shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties of these dynamical systems. We show that h-shadowing, limit shadowing and s-limit shadowing properties are conjugacy invariant. Also, we investigate the relationships between these notions of shadowing for time varying maps and examine the role that expansivity plays in shadowing properties of such dyn…
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 homeomorphisms with bounded variation.
We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property and the asymptotic average shadowing property in the conservative case.
In the present paper we study the C1-robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, and the third one implies that the flow is Anosov.
This paper is concerned with the study of fuzzy dynamical systems. Let (X;M; *) be a fuzzy metric space in the sense of George and Veeramani. A fuzzy discrete dynamical system is given by any fuzzy continuous self-map defined on X. We introduce the various fuzzy shadowing and fuzzy topological transitivity on a fuzzy discrete dynamical systems. Some relations between this notions have been proved.
We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed invariant subspace. We prove that every chain transitive linear dynamical system with the shadowing property is frequently hypercyclic, and as a corollary obtain that every shadowing hypercyclic linear dynamical system is frequently hypercyclic.
Topology of global attractors for homeomorphisms with the topological shadowing property in $\mathbb{R}^m$
Centro Universitario Regional Este, Universidad de la República（乌拉圭共和国大学东方地区中心）；Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la República（乌拉圭共和国大学工程学院拉斐尔·拉瓜迪亚数学与统计研究所）
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 reduced shadowing equations the rotation number, and show that it depends only on the geometry of the escaping curve. Two notions called the critical shadowing distance and turning shadowing distance are introduced to characterize different dynamical behaviors. We show that a planar closed escaping curve could have shadowing curves of different types includi…
We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.
The average shadowing property is considered for set-valued dynamical systems, generated by parameterized IFS, which are uniformly contracting, or conjugacy, or products of such ones. We also prove that if a continuous surjective IFS F on a compact metric space X has the aver- age shadowing property, then every point x is chain recurrent. Moreover, we introduce some examples and investigate the relationship between the original shadowing property and shadowing property for IFS. For example, this is proved that the Sierpinski IFS has the average shadowing property. Then we shows that there is a…
Weak mixing for dynamical systems with ${\bm{\mathscr M}}_{\bmα}$-shadowing
School of Mathematics and Statistics, Guizhou University of Finance and Economics（贵州财经大学数学与统计学院）；Department of Mathematics, Shandong University, Weihai（山东大学威海数学系）
For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost 1-1 extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\varepsilon>0$ the collection of ergodic measures (supported on almost 1-1 extensions of odometers) with entropy between $c$ and $c…
We prove that when $f$ is a continuous selfmap acting on compact metric space $(X,d)$ which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy. Using this fact we prove that in the class of $C^0$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergant Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak$^*$ topology. Moreover, such points carry zero entropy. In contrast, irregu…
We explore connections among the regional proximal relation, the asymptotic relation and the distal relation for a topological dynamical system with the shadowing property, and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.
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 approach is local: it is based on the gluing property which takes into account the shadowing under a single (not necessarily small) perturbation.
We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with "jumps" along the central foliation. The proof is based on the Tikhonov-Shauder fixed point theorem.
In this paper, we explore a topological system $f:M\rightarrow M$ with average shadowing property. We extend Sigmund's results and show that every non-empty, compact and connected subset $V\subseteq\mathcal {M}_{inv}(f)$ coincides with $V_f(y)$, where $\mathcal {M}_{inv}(f)$ denotes the space of invariant Borel probability measures on M, and $V_f(y)$ denotes the accumulation set of time average of Dirac measures supported at the orbit of $y$. We also show that the set $M_{V}=\{y\in M\,\,|\,\,V_{f}(y)=V\}$ is dense in $Δ_{V}=\bigcup_{ν\in V}supp(ν)$. In particular, if $Δ_{max}=\bigcup_{ν\in\mat…
We study the properties of $Φ$-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $Φ$-irregular set in terms of entropy on chain recurrent classes and prove that $Φ$-irregular sets of full entropy are typical. We also consider $Φ$-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measur…
For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\ < J\ > = \underset{n\rightarrow\infty}{\lim} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares sensitivity method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $d\ < J\ > \over ds$ from the solution. This paper proves that as the size of the least squares problem approaches infinity, the computed approximation converges to the true derivative.
We consider inverse periodic shadowing properties of discrete dynamical systems generated by diffeomorphisms of closed smooth manifolds. We show that the $C^1$-interior of the set of all diffeomorphisms having so-called inverse periodic shadowing property coincides with the set of $Ω$-stable diffeomorphisms. The equivalence of Lipschitz inverse periodic shadowing property and hyperbolicity of the closure of all periodic points is proved. Besides, we prove that the set of all diffeomorphisms that have Lipschitz inverse periodic shadowing property and whose periodic points are dense in the nonwa…
We study various weaker forms of inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called Ergodic Inverse Shadowing property (Birhhoff averages of continuous functions along the exact trajectory and the approximating one are close). We demonstrate that this property implies continuity of the set of invariant measures in Hausdorff metrics. We show that the class of systems with Ergodic Inverse Shadowing is quite broad, it includes all diffeomorphisms with hyperbolic nonwandering sets. Secondly, we study the so-called Individual In…
Let $\mathfrak{X}^{1}(M)$ be the space of $C^{1}$-vector fields on $M$ endowed with the $C^{1}$-topology and let $Λ$ be an isolated set for a $X\in\mathfrak{X}^{1}(M)$. In this paper, we directly prove that every $X\in\mathfrak{X}^{1}(M)$ having the (asymptotic) average shadowing property in $Λ$ has no proper attractor in $Λ$. Our proof is a direct version of the results by Gu and Ribeiro. We also show that every $X\in\mathfrak{X}^{1}(M)$ having the (two-sided) limit shadowing property with a gap in $Λ$ is topologically transitive and has the shadowing property in $Λ$.
A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expa…
The orbital shadowing property (OSP) of discrete dynamical systems on smooth closed manifolds is considered. Nondensity of OSP with respect to the C^1-topology is proved. The proof uses the method of skew products developed by Yu.S. Ilyashenko and A.S. Gorodetski.
In this paper, we study continuum-wise expansive non-autonomous discrete dynamical systems. We discuss various properties of such non-autonomous systems. We further obtain results for cw-expansive non-autonomous systems with shadowing property and obtain an important equivalence. \keywords{Non-autonomous dynamical systems, continuum-wise expansive, shadowing property
We find sufficient conditions for commutative non-autonomous systems on certain metric spaces to be topologically stable. In particular, we prove that (i) Every mean equicontinuous, mean expansive system with strong average shadowing property is topologically stable. (ii) Every equicontinuous, recurrently expansive system with almost shadowing property is topologically stable. (iii) Every equicontinuous, expansive system with shadowing property is topologically stable.
Let $(X,G,Φ)$ 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, fix some finite subset $S\subset G$. We prove that if $X$ is totally disconnected, then $Φ$ has $S$-shadowing property if and only if $(X,G,Φ)$ is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric space (may be not totally disconnected), we prove that if $Φ$ has $S$-sha…
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 together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant m…
