maryam rabii

In this paper we consider the family fa(x) = axd(x − 1) + x whena < 0 is a real number and d ≥ 2 is an even integer. The function fa has aunique positive critical point. By decreasing the parameter a, the behavior ofthe orbit of this critical point changes. In this paper we consider two cases. Inthe first case the orbit of the positive critical point converges to 0 and in thesecond case the positive critical point is mapped to a repelling periodic pointof period 2. In each case we give a recursive formula to determine the numberof the periodic points of fa. Also, in each case we introduce an invariant seton which fa is chaotic. We employ conjugacy map and symbolic dynamics inour investigations.

We consider the real polynomials of degree d + 1 with a fixed point of multiplicity d ≥ 2. Such polynomials are conjugate to fa,d(x) = axd(x − 1) + x, a ∈ R \ {0}. In this family, the point 0 is always a non-hyperbolic fixed point. We prove that for given d, d′, and a, where d and d′ are positive even numbers and a belongs to a special subset of R−, there is a′ < 0 such that fa,d is topologically conjugate to fa′ ,d′ . Then we extend the properties that we have studied in case d = 2 to this family for every even d > 2.

In this paper we consider the dynamics of the real polynomials of degree d 1 with a fixed point of multiplicity d ≥ 2. Such polynomials are conjugate to fa,d(x) = axd(x−1), a ∈ R\{0}, d ∈ N. Our aim is to study the dynamics fa,d in some special cases.

The treatise entitled “Arithmetic in astronomy”, which is edited and commented upon in this paper, is part of an old manuscript collection preserved in the National Library of Bosnia and Herzegovina. At the end of this treatise there are some phrases in Arabic which show that the treatise was originally part of a Persian book written by ʿAlī al-Qūshchī. This manuscript is, in fact, an abridged form of the second chapter of Qūshchī’s Book prepared by Naṣrullāh al-Naṣīr bin Naṣīr nicknamed as Wāqif al-Khalkhālī and copied by Ḥājī Uthmān bin Ḥajī ʿUmar. Following the well-known practice of arithmetic books, the treatise contains a breaf discussion of topics such as addition, substraction, multiplication, division and extractactin of the second root in sexagesimal system. Some astronomical applications are also mentioned.

The aim of this paper is to present a proof of the hyperbolicity of the family $f_c(x)=c(x-frac{x^3}{3}), |c|>3$, on an its invariant subset of $mathbb{R}$.