| Feigenbaum phenomenon is an important research topic in the field of dynamical systems. This paper discusses the following function closely related to Feigenbaum phenomenon:A C 1solution of this equation is called a Feigenbaum's mapping , if it satisfies the following conditions:(1) f is even , i.e. f ( ? x ) = f ( x ), ? x∈[ ? 1,1];(2) If x∈(0,|λ|) ,then f′( x) < 0;(3) If x∈[|λ|,1] ,then f′( x) < ? 1.This paper introduces a type of p-order Feigenbaum's mapping fand the concept of its characteristic set C ( f ). By proving the topological conjugacy between the restricted mapping f|C(f)and p -adic system , we indicate that f|C(f) is minimal , uniquely ergodic and has zero topological entropy . For a type of 2–order Feigenbaum's maps, it is proved that the characteristic set is a 2-adic attractor . This paper discusses the existence of periodic points of the p- order Feigenbaum's map . For a 2–order Feigenbaum map satisfying certain conditions , we point out that there is only one 2n periodic orbit for each n > 0. We also point out the precise location of the fixed point and 2 periodic points for stefan mapping.The main result of this paper is as follows:Theorem 2.1 Let f be a p-order Feigenbaum's mapping . Then the restricted mapping f|C(f) is topologically conjugate to oτ, whereτis p -adic system.Corollary 2.2 If f is a p-order Feigenbaum's mapping , then the restricted mapping f|C(f) is minimal , uniquely ergodic and has zero topological entropy .Theorem 2.3 Let f be a 2-order Feigenbaum's mapping . Then C ( f ) is a 2-adic attractor of f .Theorem 3.1 Let f be a p-order Feigenbaum's mapping .Then x is the fixed point of f if and only ifλx is a p periodic point of f .Corollary 3.2 Let f be a 2-order Feigenbaum's mapping with. f ' ( x ) < 1 for x∈[λ,μ] (μis the minimum point of f ).Then the periods of the periodic points of f are powers of 2 and f has only one 2 n- periodic orbit for each n≥0.Theorem 3.3 The fixed point of Stefan mapping is 715 , the 2 periodic points are (7/45) and (34/75) .
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