The main purpose of this paper is to introduce the rotation number for non-autonomous and ITERATED FUNCTION SYSTEMS. First, we define ITERATED FUNCTION SYSTEMS and the lift of these types of SYSTEMS on the unit circle. In the following, we define the rotation number and investigate the conditions of existence and uniqueness of this number for our SYSTEMS. Then, the notions rotational entropy and rotational shadowing have been considered and some basic properties of these notions in the ITERATED FUNCTION and non-autonomous SYSTEMS have been investigated.

In this paper, the new concept of non-autonomous ITERATED FUNCTION system is introduced and also shown that non-autonomous ITERATED FUNCTION system IFS(f_(1, ∞ )^0, f_(1, ∞ )^1) is topologically transitive for the metric space of X whenever the system has average shadowing property and its minimal points on X are dense. Moreover, such a system is topologically transitive, whenever, there is a point like z∈ U for each open and invariant set U from X so that N(z, U) has a positive upper density. It is also shown that topological transitivity is result of properties of shadowing and chain transitivity. The relation between average shadowing property, topological transitivity and chaotic non-autonomous ITERATED FUNCTION system is studied. Moreover, it is also demonstrated that the first two conditions for the definition of chaos results the third condition. The topological mixing of such a system is obtained from shadowing property and chain mixing. Finally, we evaluated that the dynamical system (X, f) has Li-York e chaos under special conditions.

This study was conducted during summer and winter of 2018- 2019 in the agricultural research field of Shahid Chamran University. Experimental design was split- plot based on RCBD with three replications. The main plot was the type of agricultural system in three levels including conventional (Conv), organic (Org) and sustainable (Sust) (integrated between Conv and Org) and sup- plot was the type of pre- cultivated crop in sequence with wheat including cultivation of mung bean (M- W), corn (C- W), sesame (S- W) and fallow (F- W). Yield quantity (yield and its component) and quality (grain protein), an estimate of photosynthesis matter transfer index of wheat and soil organic carbon (SOC) after one double-cropping were measured. The result showed that the highest (545.04 g/m2) and the lowest (409.28 g/m2) seed yields were obtained in Conv and Org respectively. In contract, with the changing type of system from Conv to Org, grain protein was increased significantly (from 8.3 to 9.6 %). In addition, the highest (535.47 g/m2) yield of wheat was obtained from M- W double cropping. On the other hands the highest remobilization and current photosynthesis matter were obtained in the organic agricultural system with M- W and conventional with M- W double cropping. The situation of SOC showed that the highest (33.18 mg/g) SOC was obtained in the organic agricultural system with C- W double cropping. The reason for improving SOC in the organic and sustainable agricultural system was application of organic matter (compost and vermicompost) and crop residue management. Totally, from the crop ecology point of view, sustainable agricultural method with a sequence of M- W was the most desirable system.

In this paper we consider shadowing and weak shadowing properties for ITERATED FUNCTION SYSTEMS IFS and give some results on these concepts. At first, a sufficient condition for shadowing property is given and by this result we present two IFS which have the shadowing property. It is proved that every uniformly expanding as well as every uniformly contracting IFS has the weak shadowing property. By an example we show that in IFS’ s shadowing property does not imply weak shadowing property. Finally we have the main result of the paper and prove that the weak shadowing is a generic property in the set of all IFS’ s.

Assuming that $\Lambda$ is a bounded operator on a Hilbert space $H$, this study investigate the structure of the $g$-frames generated by  iterations of $\Lambda$. Specifically, we provide  characterizations of $g$-frames  in the form of $\{\Lambda^n\}_{n=1}^{\infty}$ and describe some conditions under which the sequence $\{\Lambda^n\}_{n=1}^{\infty}$ forms  a $g$-frame for $H$. Additionally, we verify the properties of the operator $\Lambda$ when $\{\Lambda^n\}_{n=1}^{\infty}$ is a $g$-frame for $H$. Moreover, we study the $g$-Riesz bases and dual $g$-frames which are generated by iterations. Finally, we discuss the stability of these types of $g$-frames under some perturbations.