The asymptotic behavior of a dynamical process coming from the development in continuous fractions

Abstract

The aim of this paper is the study of a dynamical process generated by a sequence of maps: $${x_{n + 1}} = {f_n}\left( {{x_n}} \right)$$ where $${f_n}{\rm{ : }}\left( {0,\infty } \right){\rm{ }} \to \left( {0,\infty } \right){\rm{, }}{f_n}{\rm{ }}\left( x \right){\rm{ = }}{{{c_n}} \over {1 + x}}{\rm{ for all }}n{\rm{ }} \in {\rm{ }}N{\rm{ and }}{\left( {{c_n}} \right)_n}$$ is a a sequence of positive numbers. This process is generated similar to continuous fractions development. A continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the inverse of another number, then writing this other number as the sum of its integer part and another inverse, and so on. In a finite continued fraction (or terminated continued fraction), the iteration is terminated after finitely many steps by using an integer in stead of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction. We will study the pre-equilibrium points for this process, the attraction basins and the stability.