Related papers: Stably weakly shadowing transitive sets and domina…

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 say that a compact invariant set $\Lambda$ of a $C^1$-vector field $X$ on a compact boundaryless Riemannian manifold $M$ is robustly shadowable if it is locally maximal with respect to a neighborhood $U$ of $\Lambda$, and there exists a…

Let $\mathfrak{X}^{1}(M)$ be the space of $C^{1}$-vector fields on $M$ endowed with the $C^{1}$-topology and let $\Lambda$ be an isolated set for a $X\in\mathfrak{X}^{1}(M)$. In this paper, we directly prove that every…

We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.

The article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map $f\times f^2 \times...\times f^m$, where $f\colon X\ra X$ is a topological dynamical system on a…

We study various types of shadowing properties and their implication for C1 generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i)…

We prove that any C1-stably weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E + F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result…

We examine certain phenomena in $C^1$-dynamics from a viewpoint of shadowing and improve a known result on hyperbolic sets. We also review a result on the stability of attractor boundaries from the same viewpoint and derive several…

We propose a weak form of domination, called partially dominated splitting and the main result is that there is a partially dominated splitting over a nonsingular compact invariant set for a flow if, and only if, the associated linear…

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.…

For any continuous self-map of a compact metric space, we prove a saturation of distributionally scrambled Mycielski sets under a type of shadowing and the chain transitivity.

Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond-…

Strong bisimulation for labelled transition systems is one of the most fundamental equivalences in process algebra, and has been generalised to numerous classes of systems that exhibit richer transition behaviour. Nearly all of the ensuing…

The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as…

Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid. WLTSs subsume a wide range of LTSs, providing a general notion of strong (weighted) bisimulation. In this paper we extend…

We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $\Lambda$ a non-trivial $C^1$-robustly transitive set of a vector field $X$, there is a $C^1$-perturbation $Y$ of $X$ such…

It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if…

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\Sigma$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values…

We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of…

We prove that the resolvent set of any, possibly singular, quasi periodic Jacobi operator is characterized as the set of all energies whose associated Jacobi cocycles induce a dominated splitting. This extends a well-known result by R. A.…