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Abstract : There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). We will discuss several natural decompositions that arise in the study of rational maps, such as Pilgrim's canonical decomposition and Levy decomposition (by Bartholdi and Dudko). I will also introduce a new decomposition of rational maps based on the topology of their Julia sets (obtained jointly with Dima Dudko and Dierk Schleicher). At the end of the talk, we will briefly consider connections of this novel decomposition to geometric group theory and self-similar groups.

Keywords : rational dynamics; Julia sets; invariant graphs; Thurston maps; obstructions; decomposition theory

MSC Codes :
20E08 - Groups acting on trees
20F67 - Hyperbolic groups and nonpositively curved groups
37B10 - Symbolic dynamics
37B40 - Topological entropy
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20 - Combinatorics and topology in relation with holomorphic dynamical systems

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https://videos.cirm-math.fr/2021-09-24_Hlushchanka.mp4

Information on the Event

Event Title : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe
Dates : 20/09/2021 - 24/09/2021
Event Year : 2021
Event URL : https://conferences.cirm-math.fr/2546.html

Citation Data

DOI : 10.24350/CIRM.V.19812403
Cite this video as: (2021). Decomposition results in rational dynamics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19812403
URI : http://dx.doi.org/10.24350/CIRM.V.19812403

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Bibliography

BARTHOLDI, Laurent et DUDKO, Dzmitry. Algorithmic aspects of branched coverings IV/V. Expanding maps. Transactions of the American Mathematical Society, 2018, vol. 370, no 11, p. 7679-7714. - http://dx.doi.org/10.1090/tran/7199

DOUADY, Adrien et HUBBARD, John H. A proof of Thurston's topological characterization of rational functions. Acta Mathematica, 1993, vol. 171, no 2, p. 263-297. - https://projecteuclid.org/journalArticle/Download?urlid=10.1007%2FBF02392534

DUDKO, Dzmitry et HLUSHCHANKA, Mikhail et SCHLEICHER, Dierk. A canonical decomposition of postcritically finite rational maps. Preprint. - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtaWtoYWlsaGx1c2hjaGFua2F8Z3g6MzJjMzdiNjI0MmY5YzdlYg

PILGRIM, Kevin M. Canonical Thurston obstructions. Advances in Mathematics, 2001, vol. 158, no 2, p. 154-168. - https://doi.org/10.1006/aima.2000.1971

SELINGER, Nikita. Thurston's pullback map on the augmented Teichmüller space and applications. Inventiones mathematicae, 2012, vol. 189, no 1, p. 111-142. - http://dx.doi.org/10.1007/s00222-011-0362-3

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