Regularity and Irregularity of Fiber Dimensions of Non-Autonomous Holomorphic Dynamical Systems

Mariusz Urbanski
University of North Texas

This talk will concern non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide A necessary and sufficient condition for holomorphic stability will be provided and discussed. It leads to H\"older continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the $l^\infty$-topology on the parameter space. On the other hand, in drastic contradisctinction to the deterministic case, for some particular non-autonomous family of rational (functions, the Hausdorff and packing dimension functions are not differentiable at any point and these dimensions do not coincide on an open dense set of the parameter space still with respect to the $l^\infty$-topology.

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