Wiman-Valiron discs and the dimension of Julia sets
James Waterman
TL;DR
This paper proves that for certain meromorphic maps, the set of points with bounded orbits in the Julia set has Hausdorff dimension greater than one, using new Wiman-Valiron theory results.
Contribution
It introduces novel results in Wiman-Valiron theory to establish lower bounds on the Hausdorff dimension of bounded orbit sets in Julia sets.
Findings
Hausdorff dimension of bounded orbit set > 1
New Wiman-Valiron theory results
Dimension estimates for Julia sets of meromorphic maps
Abstract
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman-Valiron theory.
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