Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 2 10.wxyz/CV72P2 http://www.austms.org.au/Publ/Bulletin/V72P2/ Flavia Colonna 15 February 2006 2006 2 15 283 290 722-5133-Colonna-2005 10.wxyz/C2005V72P2p283 http://www.austms.org.au/Publ/Bulletin/V72P2/722-5133-Colonna/ In this paper, we characterise the analytic functions $\f $ mapping the open unit disk $\D $ into itself whose induced composition operator $C_\f : f\mapsto f\circ \f $ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation $\f =gB$ where $g$ is a non-vanishing analytic function from $\D $ into the closure of $\D $, and $B$ is an infinite Blaschke product whose zeros form a sequence $\{z_n\}$ containing 0 and a subsequence $\{z_{n_{j}}\}$ satisfying the conditions $\bigl |g(z_{n_{j}})\bigr |\to 1$, and $$\lim _{j\to \infty }\prod _{k\ne n_j}\Bigl |\frac {z_{n_j}-z_k}{1-\overline {z_{n_j}}z_k}\Bigr |=1.$$ 30D45, 47B33, 47A30 MR2183409 02246390 I wish to dedicate this article to Professor Maurice Heins for his ninetieth birthday. I owe him a debt of gratitude for his great lectures which deeply stimulated my passion for complex analysis. As a thesis advisor, he was always very patient and generous with his time. J.M. Anderson, J. Clunie and Ch. Pommerenke J. Reine Angew. Math. MR361090 J.M. Anderson, J. Clunie and Ch. Pommerenke; On Bloch functions and normal functions, \textit{J. Reine Angew. Math.} \textbf{279} (1974), pp.~12--37. L. Carleson Amer. J. Math. MR117349 L. Carleson; An interpolation problem for bounded analytic functions, \textit{Amer. J. Math.} \textbf{80} (1958), pp.~921--930. F. Colonna J. London Math. Soc. (2) MR897677 F. Colonna; The Bloch constant of bounded analytic functions, \textit{J. London Math. Soc. (2)} \textbf{36} (1987), pp.~95--101. F. Colonna Rend. Circ. Mat. Palermo (2) MR1029707 F. Colonna; Bloch and normal functions and their relation, \textit{Rend. Circ. Mat. Palermo (2)} \textbf{38} (1989), pp.~161--180. J.B. Garnett Acadademic Press MR628971 J.B. Garnett; \textit{Bounded analytic functions} (Acadademic Press, New York, 1981). M.H. Heins Arch. Math. MR65644 M.H. Heins; A universal Blaschke product, \textit{Arch. Math.} \textbf{6} (1955), pp.~41--44. C. Xiong Bull. Austral. Math. Soc. MR2094297 C. Xiong; Norm of composition operators on the Bloch space, \textit{Bull. Austral. Math. Soc.} \textbf{70} (2004), pp.~293--299.