Analytic mappings of the unit disk which almost preserve hyperbolic area

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AbstractIn this paper, we study analytic self‐maps of the unit disk which distort hyperbolic areas of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, Möbius distortion, the distribution of critical points and Aleksandrov–Clark measures. We also examine the Lyapunov exponents of their Aleksandrov–Clark measures.