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13 - Fundamental Properties of Meromorphic Dynamical Systems
from Part III - Topological Dynamics of Meromorphic Functions
Janina Kotus, Warsaw University of Technology, Mariusz Urbański, University of North Texas
Book:

Meromorphic Dynamics

Published online:

20 April 2023

Print publication:

04 May 2023, pp 3-66
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Summary

In this chapter, we provide a relatively short and condensed account of the topological dynamics of all meromorphic functions with an emphasis on Fatou domains, including Baker domains that are exclusive for transcendental functions and do not occur for rational functions. We also carry out a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates. In particular, we study at length asymptotic values and their relations to transcendental tracts. The results of this analysis will be used very frequently to study the topological structure of connected components (and their boundaries) of Fatou sets in this part of the book and a countless number of times when we move on to dealing with elliptic functions.

On regularity properties of a surface growth model
Part of
Jan Burczak, Wojciech S. Ożański, Gregory Seregin
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 6 / December 2021

Published online by Cambridge University Press:

09 March 2021, pp. 1869-1892

Print publication:

December 2021
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We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

ON COLLINEAR CLOSED ONE-FORMS
Part of
- Differential topology
IRINA GELBUKH
Journal:

Bulletin of the Australian Mathematical Society / Volume 84 / Issue 2 / October 2011

Published online by Cambridge University Press:

16 June 2011, pp. 322-336

Print publication:

October 2011
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We study one-forms with zero wedge-product, which we call collinear, and their foliations. We characterise the set of forms that define a given foliation, with special attention to closed forms and forms with small singular sets. We apply the notion of collinearity to give a criterion for the existence of a compact leaf and to study homological properties of compact leaves.

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