The authors consider the space of inner functions of finite entropy endowed with the topology of stable convergence. They assign to each of such inner functions \(F\) a singular measure \(\lambda(F)\) on the boundary of the unit disc and a singular value measure \(\nu(F)\) on the unit disc. They prove that for an inner function of finite entropy \(F\) the radial limit in the unit disc exists \(\lambda(F)\) a.e. and that the singular value measure \(\nu(F)\) varies continuously in \(F\).
An inner function \(F\) has finite entropy if \(F'\) belongs to the Nevalinna class. So \(F'\) can be decomposed in a product \(BSO\), where \(B\) is a Blaschke factor, \(S\) is a singular factor and \(O\) an outer factor. The singular measure \(\lambda(F)\) is the one corresponding to the singular factor \(S\). In this decomposition the Blaschke factor \(B\) encodes the critical points of \(F\) and the singular factor \(S\) encodes the boundary critical structure. Alternatively, one can encode all the critical structure of \(F\) as a measure \(\mu_F\) on the closed unit disc defined as\[\mu_F = \sum (1 - |c|) \cdot \delta_c + \lambda(F),\]where the sum is extended over the set \(\mathrm{crit}(F)\) of the critical points of \(F\). Instead of the measure \(\mu_F\) one can use the measure \(\nu_F\) which is the pushforward of \(\mu_F\) with respect to the radial extension of \(F\) to the unit circle, that is proved to exist; that is\[\nu_F = F_*(\mu_F) = \sum (1 - |c|) \cdot \delta_{F(c)} + F_*(\sigma(F')),\]with the sum extended over \(c \in \mathrm{crit}(F)\). In the space of the inner functions with finite entropy one introduces the topology of the stable convergence stating that the convergence \(F_n \to F\) is stable if and only if\[\mu_{F_n} \to \mu_F.\]The continuity of the singular value measure \(\nu(F)\) with respect to \(F\) is given by the following result:

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For almost every point \(\zeta\) on the unit circle with respect to the singular measure \(\lambda(F)\), the radial limit \(\lim_{r\to1} F(r\zeta)\) exists and lies in the open unit disc. If the convergence \(F_n \to F\) is stable then the measures \(\nu_{F_n}\) converge weakly to \(\nu_F\).

The authors say that a function \(F\) has a thick limit \(L\) at a point \(\zeta\) in the boundary of the unit disc \(D\) if the set \(\{ z \in D : | F(z) - L | < \delta \}\) has a connected component that is thick at \(\zeta\) (this means that it is the image of \(D\) by a conformal mapping with non-zero angular derivative at \(\zeta\)). They prove:

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For almost every point \(\zeta\) in the boundary of \(D\) with respect to the singular measure \(\lambda(F)\), \(F\) has a thick limit at \(\zeta\).

A central notion in the context of this paper is that of a component of an inner function, introduced by Pommerenke. Suppose \(V\) is a Jordan domain compactly contained in the unit disc and \(U\) is a connected component of the preimage \(F^{-1}(V)\). Then \(U\) is a Jordan domain and one can define\[F_U = \psi^{-1} \circ F \circ \phi,\]where \(\phi, \psi\) are Riemann maps from \(D\) to \(U\) and \(V\) respectively. It follows that \(F_U\) is an inner function and if \(F\) has finite energy the same is true for \(F_U\). The final result compares the singular measures \(\lambda(F)\) and \(\lambda(F_U)\) or better \(\lambda(F)\) and the pushforward of \(\lambda(F_U)\) by \(\phi\).

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Let \(Z\) denote the set of points on the unit circle where \(U\) is thick. Then,\[\phi^*\lambda(F_U) = |(\phi^{-1})'(\zeta)| d\lambda(F)_|Z,\]where \((\phi^{-1})'(\zeta)\) is interpreted as the inverse of the angular derivative of \(\phi\) at \(\phi^{-1}(\zeta)\), a point on the unit circle.

Reviewer: Julià Cufí (Bellaterra)

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