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In this paper, we generalize the notion of expansivity, C-W expansivity and N-expansivity for impulsive dynamical systems and we prove C-W expansivity implies positive [Formula: see text]-topological entropy. Also, we define new version of topological entropy for impulsive semiflows and we can bound this topological entropy of expansive impulsive semiflows from lower by the exponential growth rate of periodic orbits.

In this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

Let Λ be a random compact invariant (with respect to a random map) set with a uniform tangent bundle splitting. We show that if there is a random non-uniformly hyperbolic variable x0 on Λ and the space of limit points of time averages of random Dirac measures on the orbit of x0 coincides with the space of invariant probability measures supported on Λ then, Λ is a random uniformly hyperbolic set.

This paper introduces the notion of robust hyperbolicity along periodic orbits homoclinically related to p (RNUHP) for conservative diffeomorphisms. It is proved that if f ∈ Diff1+ m (M) is RNUHP, then f is Anosov. It is also shown that f admits a dominated splitting, provided that f is expansive conservative stable.

In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, [Formula: see text]-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

In this paper, we study transitivity on the semigroup actions. Indeed, we introduce two criteria, “entropy minimality” and “nonuniformly expanding property”, to obtain transitivity.

In this paper, we introduce the definitions of shadowing and average shadowing properties for iterated function systems and give some examples characterizing these definitions. We prove that an iterated function system has the shadowing property if and only if the step skew product corresponding to the iterated function system has the shadowing property. Also, we study some notions such as transitivity, chain transitivity, chain mixing and mixing for iterated function systems.

In this short paper we show that generically if f is tame and non uniformly hyperbolic on the its periodic points, then averge shadowing property and hyperbolicity are equivalent.

In this paper we show that if X is a compact metric space and φ : ℝ × X → X is a flow and φ is weak uniformly almost periodic with positively asymptotic average shadowing property, then φ is strongly ergodic. As well as corollary we show that if φ : ℝ × X → X is a Lyapunov stable flow with the asymptotic average shadowing property then φ is not distal.

In this paper we show that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous method and and some of the -stable diffeomorphisms have weak inverse shadowing property with respect to classes . In addition we study relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class .

In this short paper we show that if homeomorphism f on a compact metric space X has the shadowing property and only one chain component, then f is ∑-chaotic. As a corollary we show that generically f is ∑-chaotic.

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied.

In this paper we introduce the notions of strict persistence and weakly strict persistence which are stronger than those of persistence and weak persistence, respectively, and study their relations with shadowing property. In particular, we show that the weakly strict persistence and the weak inverse shadowing property are locally generic in Z(M).

Let M be a generalized homogeneous compact space, and let Z(M ) denotes the space of homeomorphisms of M with the C 0 topol- ogy. In this paper, we show that if the interior of the set of weak stable homeomorphisms on M is not empty then for any open subset W of Z(M ) containing only weak stable homeomorphisms the orbital shadow- ing property is generic in W.

We prove that if for a continuous map f on a compact metric space X , the chain recurrent set, R ( f ) has more than one chain component, then f does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f on a compact metric space X has the asymptotic average shadowing property and if A is an attractor for f , then A is the single attractor for f and we have A = R ( f ) . We also study diffeomorphisms with asymptotic average shadowing property and prove that if M is a compact manifold which is not finite with dim M = 2 , then the C 1 interior of the set of all C 1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω -stable diffeomorphisms.

Let X be a divergence-free vector field on a three-dimensional compact connected Riemannian manifold. In this paper, we show that if X is in the C 1 -interior of the set of divergence-free vector fields which satisfy the average shadowing property then X is Anosov. We also obtain similar result for asymptotic average shadowing property.

In this paper, we prove that every non-uniformly expanding transitive group (or semigroup) action of C1+α conformal local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure.