title: Dynamics of the functions [Formula: see text] with the real parameter.
abstract: In this paper, the dynamics of the functions [Formula: see text] with the real parameter is studied. We say that a real parameter [Formula: see text] belongs to the set [Formula: see text] for a positive integer n if [Formula: see text] has an attracting cycle of n-order. We prove that the Fatou set [Formula: see text] is a completely invariant attracting basin for every parameter [Formula: see text]. Further, regarding the set [Formula: see text] for [Formula: see text], we prove the following results: (1) There exists [Formula: see text] such that [Formula: see text]. (2) For every positive integer [Formula: see text], the set [Formula: see text] is non-empty. (3) For every prime number [Formula: see text], the set [Formula: see text] has at least two components.
authors.affiliation: School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006 People's Republic of China.
authors.forename: Xiaocheng
authors.initials: X
authors.lastname: Deng
authors.affiliation: School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006 People's Republic of China.
authors.forename: Fanning
authors.initials: F
authors.lastname: Meng
authors.affiliation: School of Economic and Management, Guangzhou University of Chinese Medicine, Guangzhou, 510006 People's Republic of China.
authors.forename: Jianming
authors.initials: J
authors.lastname: Lin
authors.affiliation: School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006 People's Republic of China.
authors.forename: Wenjun
authors.initials: W
authors.lastname: Yuan
keywords: Critical value; Fatou set; Julia set; Periodic point