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In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C 0 incompressible flows generated by Lipschitz vector fields. We prove that a C 0 -generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non-wandering set. In particular, a C 0 -generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C 0 -generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive. [ABSTRACT FROM AUTHOR]

In this work we consider a dissipative reaction–diffusion equation in a d -dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in [6] and Shadowing theory. [ABSTRACT FROM AUTHOR]

We study the computational complexity of some problems which arise in the dynamics of certain hyperbolic toral automorphisms, using the shadowing lemma. Given p , q ∈ T 2 , saddles of common period n , and rational r > 0 sufficiently small, we study how quickly orbits ( f n ) N ( z ) ​transit from a basin of p , the open ball B ( p , r ) to one of q , B ( q , r ) . A priori, z is of arbitrary precision. We compute a finite precision L , polynomial in the size of an instance, so that if an orbit ( f n ) N ( z ) makes such a transition, so does a reliable L -precision pseudo-orbit, but at the price of possibly using duration N + 1 . If we allow duration N + 2 , this precision depends on n and r but is independent of N . It follows that in an appropriate model of computation such a saddle transition problem is in Oracle NP i.e. in NP up to a restricted oracle, and we show there exist closely related problems which are in NP . [ABSTRACT FROM AUTHOR]

We call that a flow has the orbital shadowing property if for any ε > 0 there is d > 0 such that, for any d -pseudo orbit g ( t ) there exists an orbit Orb ( x ) satisfying d i s t H ( g ( t ) ‾ , Orb ( x ) ‾ ) < ε . In this paper, we show that the C 1 -interior of the set of 3-dimensional flows having the orbital shadowing property is contained in the set of Ω-stable 3-flows. [ABSTRACT FROM AUTHOR]

This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be more efficient and simpler to implement, especially for PDEs. [ABSTRACT FROM AUTHOR]

The intersection shadow theorem of Katona is an important tool in extremal set theory. The original proof is purely combinatorial. The aim of the present paper is to show how it is using linear independence latently. [ABSTRACT FROM AUTHOR]

We discuss the dynamics of n -expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every n ∈ N , we exhibit an n -expansive homeomorphism, which is not ( n − 1 ) -expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of n -expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing n -expansive homeomorphisms that are not expansive. [ABSTRACT FROM AUTHOR]

We consider a hyperbolic ergodic measure of a C 1 flow on a compact manifold. Under the hypothesis that there are no fixed points and that the Oseledec splitting of the normal bundle satisfies a limit domination property, we prove that the measure has a shadowing property. As an application of this result we prove that the measure can be approached on the weak ⁎ topology by measures supported on hyperbolic periodic orbits. [ABSTRACT FROM AUTHOR]

In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

In this paper, we explore a topological system f : M → M with asymptotic average shadowing property and extend Sigmund's results from Bowen's specification case. We show that every non-empty, compact and connected subset V ⊆ M inv ( f ) coincides with some V f ( y ) . Moreover, we show that the set M V = { y ∈ M : V f ( y ) = V } is dense in Δ V = ⋃ ν ∈ V supp ( ν ) ¯ . In particular, if Δ max = ⋃ ν ∈ M inv ( f ) supp ( ν ) ¯ coincides with M , then M max = { y : V f ( y ) = M inv ( f ) } is residual in M . As consequences, we have several corollaries. One is that every invariant measure has generic points. Another is that the set consisting of those points for which the Birkhoff ergodic average does not exist (called irregular set) is either dense in Δ max (residual provided that Δ max = M ) or empty. In particular, we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of [18] with a different method. Remark that for systems with asymptotic average shadowing property, this article studies Birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory. [ABSTRACT FROM AUTHOR]

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type. [ABSTRACT FROM AUTHOR]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

Let n > k > t ≥ j ≥ 1 be integers. Let X be an n -element set, ( X k ) the collection of its k -subsets. A family F ⊂ ( X k ) is called t -intersecting if | F ∩ F ′ | ≥ t for all F , F ′ ∈ F. The j 'th shadow ∂ j F is the collection of all (k − j) -subsets that are contained in some member of F. Estimating | ∂ j F | as a function of | F | is a widely used tool in extremal set theory. A classical result of the second author (Theorem 1.3) provides such a bound for t -intersecting families. It is best possible for | F | = ( 2 k − t k ). Our main result is Theorem 1.4 which gives an asymptotically optimal bound on | ∂ j F | / | F | for | F | slightly larger, e.g., | F | > 3 2 ( 2 k − t k ). We provide further improvements for | F | very large as well. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a compact metric space and be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property. [ABSTRACT FROM AUTHOR]

Abstract: This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property. In particular, we deduce the chaotic behavior of a map with ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property. [Copyright &y& Elsevier]

Abstract: Shadow systems are often used to approximate reaction–diffusion systems when one of the diffusion rates is large. In this paper, we study the global existence and blow-up phenomena for shadow systems. Our results show that even for these fundamental aspects, there are serious discrepancies between the dynamics of the reaction–diffusion systems and that of their corresponding shadow systems. [Copyright &y& Elsevier]

A fundamental result in extremal set theory is Katona's shadow intersection theorem, which extends the Kruskal–Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of k -sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian and Khachatrian by proving tight bounds for families that can be quite small. For example, when k = 3 our result is sharp for all families with n points and at least 3 n − 7 triples. Katona's theorem was extended by Frankl to families with matching number s. We improve Frankl's result by giving tight bounds for large n. [ABSTRACT FROM AUTHOR]

Abstract: We study linear dynamical systems with multidimensional time in Banach spaces. Using Taylor functional calculus we prove that under additional assumptions hyperbolic systems have shadowing property. [Copyright &y& Elsevier]

Abstract: Making use of a dynamical systems notion called shadowing, we prove a stability result for linear functional equations in metric groups. As a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2-divisibility condition. [Copyright &y& Elsevier]

Abstract: Matrices resulting from wavelet transforms have a special “shadow” block structure, that is, their small upper left blocks contain their lower frequency information. Numerical solutions of linear systems with such matrices require special care. We propose shadow block iterative methods for solving linear systems of this type. Convergence analysis for these algorithms are presented. We apply the algorithms to three applications: linear systems arising in the classical regularization with a single parameter for the signal de-blurring problem, multilevel regularization with multiple parameters for the same problem and the Galerkin method of solving differential equations. We also demonstrate the efficiency of these algorithms by numerical examples in these applications. [Copyright &y& Elsevier]

In J. Math. Anal. Appl. 189 (1995) 409–423, Corless and Pilyugin proved that weak shadowing is a C0 generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set. [Copyright &y& Elsevier]