On the set where the iterates of an entire function are bounded

TL;DR

This paper demonstrates that for transcendental entire functions, the set of points with bounded orbits under iteration can have arbitrarily small positive Hausdorff dimension, highlighting complex dynamical behavior.

Contribution

It establishes that the bounded orbit set's Hausdorff dimension can be made arbitrarily small for transcendental entire functions, revealing new insights into their dynamical complexity.

Findings

Bounded orbit set can have arbitrarily small positive Hausdorff dimension.

The result applies specifically to transcendental entire functions.

This advances understanding of the size and structure of dynamical sets.

Abstract

We show that for a transcendental entire function the set of points whose orbit under iteration is bounded can have arbitrarily small positive Hausdorff dimension.