Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions: • X is a divergence-free vector field satisfying the shadowing property. • X is a divergence-free vector field satisfying the Lipschitz shadowing property. • X is an expansive divergence-free vector field. • X has no singularities a...

In this paper, we show the followings: (i) If a volume preserving diffeomorphism f belongs to the C-interior of the set of all volume preserving diffeomorphims having the ergodic shadowing property then it is transitive Anosov. Moreover, (ii) if a C-generic volume-preserving diffeomorphism f has the ergodic shadowing property then it is transitive Anosov. M.S.C. 2010: 37C50, 37D20.

We study the properties of $ \Phi $-irregular sets (sets points for which Birkhoff average diverges) in dynamical systems with shadowing property. estimate topological entropy set terms on chain recurrent classes and prove that full are typical. also consider $-level whose is a specified interval), relating they carry some ergodic measures. Finally, we problem large deviations considering level...

We prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in [7]. In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic or...