Entropy of inner functions (Q1182830)

Let \(f\) be an inner function, i.e. an analytic map of the unit disk into itself such that \(| f(z)|=1\) for almost all \(z\) on the unit circle \(T\). Furthermore let \(f(0)=0\). In this interesting paper it is shown that the entropy of \(f\) considered as a (measure-preserving) map of \(T\) onto \(T\) is given by \[ h=\int_ T\log| f'(z)|| dz|\tag{*} \] where \(f'\) is the angular derivative. This is to be understood in the sense that \(h\) is finite if and only if \(f'\) belongs to the Nevanlinna class \(N\). The identity \((*)\) was conjectured by J. L. Fernández and established by N. G. G. Martin for several cases. The proof of the estimate \(\geq\) in \((*)\) is based on the result of Rohlin (on general endomorphisms) that if \(h\) is finite then \(T\) can be partitioned into countably many sets where \(f\) is injective. This allows the application of a result of M. Heins on the angular derivative. The proof of \(\leq\) in \((*)\) is much more difficult. A key step is to establish a certain smoothness property of the \(n\)-th preimage of \(f\) by a careful study of finite Blaschke products.