Related papers: Shadowing, chain transitive sets with nonempty int…
We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…
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…
A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by…
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 $G_{\delta}$ set.…
In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some…
We study shadowing-type properties for set-valued dynamical systems. In particular, we investigate the periodic shadowing property and its relationship with expansivity and chain transitivity. We establish that for positively expansive…
Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz…
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…
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…
Continuous and discrete time systems possessing strange non-chaotic attractors are under investigation. It is demonstrated that unpredictable trajectories exist in the dynamics. A recent numerical technique, the sequential test, is utilized…
We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if a set is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a union of…
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.,…
In this paper, we study stochastic stability of a dynamical system with shadowing property, which evolves under small random perturbation. We prove that time averages along the pseudo-trajectory converge with respect to stationary measure…
This paper investigates the shadowing properties in semi-hyperbolic systems. We introduce three classes of shadowing properties defined on families of manifolds, and prove that a semi-hyperbolic family possesses the $L^p$ bi-shadowing…
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,…
We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space…
We present a new result about the shadowing of nontransversal chain of heteroclinic connections based on the idea of dropping dimensions. We illustrate this new mechanism with several examples. As an application we discuss this mechanism in…
This work contains the results from a comprehensive study of a new class of attractors. The attractors in this class are characterized by strong local instability, but they are not uniformly hyperbolic. Rigorous results on their dynamical,…
For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting).…
It has been recently shown that, under appropriate hypotheses, the $\omega$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the…