Abstract: The Sullivan dictionary provides a conceptual framework to compare the actions of Kleinian groups and the dynamics of rational maps. Both of these settings generate interesting fractal sets (limit sets of Kleinian groups and Julia sets of rational maps). This dictionary provides a particularly strong correspondence when the dimensions of these sets are considered. Restricting to the geometrically finite setting, in both cases there is a 'critical exponent' which returns the Hausdorff, box, and packing dimensions of the associated fractal, as well as the Hausdorff, packing, and entropy dimensions of the associated conformal measure. We show that, by slightly expanding the family of dimensions considered, a much richer theory emerges. This allows us to draw more nuanced comparisons, and also provide novel discrepancies, between the Kleinian and rational map settings. This is joint work with Liam Stuart.

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