Some Properties of the Continuous Single-Valley Expansion Solution to the Feigenbaum’s Functional Equation

Abstract

AbstractIn this paper, some fundamental properties of the continuous single-valley expansion solution to Feigenbaum’s functional equation were obtained. For λε (0,1) and p ≥ 2, we will discuss the completeness of the function space which consists of unique continuous single-valley expansion solution (resp. unique continuous non-single-valley expansion solution) to p-order Feigenbaum’s functional equation. Let p,q ≥ 2. It was proved that the system of equations $$ \left\{ \begin{array} {ll} f(x)=\frac{1}{\lambda}f^{p}(\lambda x),f(0)=1,(\lambda \epsilon (0,1) for decision) f(x), x \epsilon[0,1];\\ f(x)=\frac{1}{\lambda}f^{q}(\lambda x),f(0)=1,(\lambda \epsilon (0,1) for decision) f(x), x \epsilon[0,1]. \end{array} \right. $$ does not have continuous single-valley expansion solution. MR(2000) Subject Classification: 39B52.KeywordsFeigenbaum’s mapFunctional equationContinuous single-valley expansion solution