Examples are given of holomorphic self-maps of the unit ball on C2 which induce unbounded composition operators on the Hardy space H2. In particular, an example is given which is one-to-one on the closed ball. Also, a valence condition on the boundary of this ball is given which is sufficient for unboundedness of the induced composition operator. 1. Introduction. Let Bn be the open unit ball in C™ and let H2 = H2(Bn) be the Hardy space on Bn. If 1, there are many examples (see (1, 2)) which show that C may map an arc on dBn to a point on dBn. The main result of this note is the construction (Theorem 2) of a mapping $: B2 —> B2 which is holomorphic and one-to-one on B2 and such that C® is unbounded on H2. $ is in fact a polynomial mapping. B. MacCluer and J. Shapiro show in (4, Theorem 6.4) that if : Bn —► Bn is one-to-one and if the derivative of _1 is bounded on (Bn), then Cj, is bounded on i/2 (see also (1, Theorem 2)). Our example shows that even for one-to-one mappings, some additional hypothesis on <h must be imposed to guarantee that Cj, is bounded. Example 4 is also related to the above theorem. In Theorem 1 we give a valence condition on <f) which is sufficient for unboundedness of C<i,. All of our results rely on the following Carleson measure criterion for boundedness of C<j,.