Characterisation of the Isometric Composition Operators on the Bloch Space

In this paper, we characterise the analytic functions φ mapping the open unit disk ∆ into itself whose induced composition operator Cφ : f 7→ f ◦ φ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation φ = gB where g is a non-vanishing analytic function from ∆ into the closure of ∆, and B is an infinite Blaschke product whose zeros form a sequence {zn} containing 0 and a subsequence {znj} satisfying the conditions ∣∣g(znj )∣∣→ 1, and lim j→∞ ∏