A continuous map f from the unit closed interval into itself is called a p-order Feigenbaum's map if f^P(λx)=λf(x), f(0)=1 and f|λ,1| is univallecular. In this paper, some characterizations of p order Feigenbaum's maps are discussed and the existence for both types of such maps is proven.
Consider the continuous map f: X → X and the continuous map (f-) of K(X)into itself induced by f, where X is a metric space and K(X) the space of all non-empty compact subsets of X endowed with the Hausdorff metric. According to the questions whether the chaoticity of f implies the chaoticity of (f-) posed by Román-Flores and when the chaoticity of f implies the chaoticity of (f-)posed by Fedeli,we investigate the relations between f and (f-) in the related dynamical properties such as transitivity, weakly mixing and mixing etc.And by using the obtained results,we give the satisfied answers to Román-Flores's question and Fedeli's question.
LIAO Gongfu, WANG Lidong & ZHANG Yucheng Institute of Mathematics, Jilin University, Changchun 130012, China
