Iterative Roots Of Piecewise Homeomorphic Maps Of Y-Star

In this paper, we discussed mainly the iterative roots of piecewise homeomorphic maps of an Y- star. This paper is composed of three chapters. In the first chapter,we introduce briefly some known results about iterative roots. In the second chapter,we introduce some lemmas and concepts about iterative roots of piecewise homeomorphic maps of y-stars. In the third chapter,we study mainly iterative roots of piecewise homeomorphic maps of an Y-star. In this paper the following results are obtained.Let F be a piecewise homeomorphic map of an Y-star, and N(F) be the number of no homeomorphic points of F ,then1 When N(F) ≠N(F2), F has no n-th iterative root for every natural number n≥ N(F).Furthermore. If N(F) = N(F2) and the characteristic region W of F is contained in some component of y, then2 When W (F(Y) - F(W)) ≠0. If / is an n-th iterative root of F, then N(f) ≤N(F)-n + 2.3 When W (F(Y) - F(W)) ≠0. If F is increasing on W, then F has n-th iterative roots for every natural number n ≥2 . If F is decreasing on W, then F has no n-th iterative root if n > 1 is even and F has n-th iterative roots if and only if F(W) = W or F(W) W ≠0 if n > 1 is odd.If N(F) = N(F2) and the characteristic region W of F has three components J1,J2,J3, then4 When W (F(Y) - F(W))≠0. If / is an n-th iterative root of F, then N(f) ≤N(F) - n + 3.5 When dW n (F(Y) - F(W)) = 0. If F(Ji) Ji for every i {1,2,3}, then F has n-th iterative roots for every natural number n ≥2. If F(J1) J1,F(J2) J3 and F( J3) J2 , then F has no n-th iterative root if n > 1 is even and F has n-th iterative roots if and only if F( J2 J3) = J2 J3 or F( J2 J3) n dW = 0 if n > 1 isodd. If F(J1) J2, F(J2) J3 and F(J3) J1 , then for any natural number k, F has no 3k-th iterative root and F has 3k +1 - iterative roots or 3k+2-iterative roots if and only if F(W) = W or F(W) n dW = 0.