Introduction to Complex Trees
Channel: Bernat Espigulé
Date: 2019-11-13

Talk about my research project on complex trees given at the Illustrating Mathematics Special Interest Seminar, ICERM 10-29-2019 Brown University.
Complex Trees: Structural Stability of Connected Self-similar Sets 
Bernat Espigulé, Universitat de Barcelona

The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets which includes Cantor sets, Koch curves, Lévy curves, Sierpinski gaskets, Rauzy fractals, plane-filling curves, and fractal dendrites. We note a fundamental dichotomy for n-ary complex trees that allows us to study topological changes in regions where one-parameter families of connected self-similar sets are defined. Moreover, we show how to obtain these families from systems of equations encoded by tip-to-tip equivalence relations. The parameter space maps that we introduce to study these families are related to a set introduced in Bill Thurston’s last paper, though in many ways easier to understand.

Preprint: https://arxiv.org/abs/1902.11282
Slides: https://www.complextrees.com/ICERM_IM19_SIS_Espigule_28Oct
Publications: https://scholar.google.com/citations?user=FNt6DKMAAAAJ 
Artwork: http://gallery.bridgesmathart.org/exhibitions/2019-icerm-illustrating-mathematics/geonat
ICERM's project: https://im.icerm.brown.edu/portfolio/math-cartography/