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In this paper we investigate the quasi-shadowing property for C1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}−∞+∞ on a random partially hyperbolic set there exists a “center” pseudo orbit {yk}−∞+∞ shadowing it in the sense that yk+1 is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above “center” pseudo orbit {yk}−∞+∞ can be chosen such that yk+1 and the image of yk lie in their common center leaf. [ABSTRACT FROM AUTHOR]

We call that a vector field has the oriented shadowing property if for any $$\varepsilon >0$$ there is $$d>0$$ such that each $$d$$ -pseudo orbit is $$\varepsilon $$ -oriented shadowed by some real orbit. In this paper, we show that the $$C^1$$ -interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the $$\Omega $$ -stability. [ABSTRACT FROM AUTHOR]

We consider shadowing properties for vector fields corresponding to different type of reparametrizations. We give an example of a vector field which has the oriented shadowing properties, but does not have the standard shadowing property. [ABSTRACT FROM AUTHOR]

We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag–Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally, we show that any linear action of a non-abelian free group does not have the shadowing property. [ABSTRACT FROM AUTHOR]