This analysis reveals monotonicity properties in hyperbolic projections of holomorphic functions, suggesting a new framework for understanding dynamics.
We consider hyperbolic projections of orbits of holomorphic self-maps of the unit disc, onto curves landing on the unit circle with a given angle. We show that under certain, necessary, assumptions, the projections exhibit monotonicity properties akin to those present in continuous dynamics. Our techniques are purely hyperbolic-geometric in nature and provide the general framework for analysing projections of arbitrary sequences onto curves.
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Christodoulou et al. (2025) studied this question.
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