Iteration in Tracts

Waterman, James (2020). Iteration in Tracts. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0001121c

Abstract

This thesis is focused on the iteration of transcendental entire and meromorphic functions within a particular domain of the complex plane called a direct tract. Many new results are given on the rates of escape and the dimension of points of bounded orbit in a direct tract, as well as the geometry of direct tracts.

First, we study iterates of points within one of these tracts. The points that escape to infinity are of particular interest in the iteration of entire functions due both to their simple definition and, in contrast to rational functions, the interesting phenomena and structures they exhibit. We expand on work of Rippon and Stallard to show that in many cases there exist points that escape to infinity within a direct tract as slowly as desired. In order to accomplish this, we develop several tools based both on the expansion of the hyperbolic metric and estimates on the function value in these direct tracts.

Next, we relate the geometry of a direct tract to how well behaved the entire function is inside this direct tract. We consider a particular type of direct tract, called a logarithmic tract. Many results are known for functions with this type of direct tract, so the ability to identify them from their geometric properties is important. In particular, we give new descriptions of when a direct tract is a logarithmic tract or contains logarithmic tracts.

Finally, we show that, for functions with a specific restriction on the geometry of the direct tract, the Hausdorff dimension of the set of points with bounded orbit in the Julia set is strictly greater than one. To do this, we prove new results related to Wiman–Valiron theory.

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