The periodicity of two disconnecting arc spaces

Let be two disconnecting arc and be a continuous map . Suppose has a fixed point and a periodic point of period n for some odd integer .Then for every integer , has a periodic point of period . If have periodic points of period and where are odd integer numbers , then the set of periodic points are infinite . Finally , if the set of periodic points is finite , then there are two integers and with and such that the set of period is .