Feigenbaum Maps And Substitution Systems

The Feigenbaum's phenomenon is one of the important reserching subjects in the field of dynamical systems. In this paper we are concerned. in the following functional equation which is closely relative to the Feigenbaum's phenomenon:Its C1 solution / is said to be a Feigenbaum map, ifWe have proved that the kneading sequence of / is a fixed point of the following substitution in a symbolic space.where a, is 1 if fi(0) > 0 and 0 otherwise, 1 i p and ap = 1 iff ap = 0.The second important result is constructing a topological semi-conjugacy between the restriction of a Fengenbaum map to it's characteristic set andsubstitution subshift. And we have proved that if / is a Feigenbaum map and a as a fixed point of some substitution is the kneading sequence of /, then the restriction f |c(f) is a factor of the substitution subshift , i.e. thereexists a continuous map H from orb(a, ) onto C(f) such that H o =f H( ) for orb(a, ) .First we define h : orb(a, ) -orb(l,f) by :and the following follow : (i) h is onto .we then let H( ) = lim fni(1). H is well-defined, and the following followwe know that the substitution subshift is minimal, strictly er-godic and has zero topological entropy. Thus we get that the restriction f |c(f) is minimal, strictly ergodic and has zero topological entropy.