Translation equivalent in free groups

Let F_n be a free group of rank n?2. Two elements g,h in F_n are said to be translation equivalent in F_n if the cyclic length of ?(g) equals the cyclic lenght of ?(h) for every automorphism ? of F_n. In this thesis, let F(a,b) be the free group generated by {a,b} and let w(a,b) be an arbitrary word in F(a,b), we show that w(g,h) and w(h,g) are translation equivalent in F_n whenever g,h?F_n are translation equivalent in F_n. Besides let F_2 be a free group of rank 2, we show that there is an algorithm that decides whether or not, for given two element g,h of F_2, g and h are translation equivalent in F_2.