Antiholomorphic Dynamical Systems and Real Slices

• Applicant: Professor Dr. Dierk Schleicher
• Subject Area: Mathematics
• Term: from 2013 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 237518971

• Final Report Year: 2017

Final Report Abstract

We like to think that this project satisfied the expectations we had when developing it, and in fact exceeded them in a number of most interesting directions. Antiholomorphic dynamics had been recognized quite early by John Milnor as a conceptual feature of maps respecting real symmetries (specifically, for the dynamics of real cubic polynomials), and this motivated some early work as early as the 1990’s. However, the connections to “mainstream” holomorphic dynamics were recognized by several people only as a result of this research project. So perhaps a particularly convincing way to express the success of this project is to say that it lead to collaborations with John Milnor in the context of his ongoing study of antipode-preserving rational maps, with Mikhail Lyubich and Nick Makarov (who previously did not have much contact with antiholomorphic dynamics), and Alexandre Eremenko and Walter Bergweiler realized that this research has connections to dynamical systems that are of interest beyond the holomorphic dynamics community. We are particularly pleased that the project resolved a number of long-standing conjectures, for instance about discontinuity of straightening and the dynamical difference between embedded “tricorn-like structures” (that initially were thought of as “little tricorns” in the same spirit as the ubiquitous “little Mandelbrot sets”, but our work indicates that they might all be distinct). Consequently, the work led to several publications at good to excellent journals. It might be worth mentioning, as a “publication beyond research journals”, that the DFG decided to include one of the pictures from this project in their research calendar 2014 (December picture), as well as on their new-year greeting cards.

Publications

• Antiholomorphic Dynamics: Topology of Parameter Spaces and Discontinuity of Straightening, PhD Thesis (2015)
  Sabyasachi Mukherjee
  
• Orbit portraits of unicritical antiholomorphic polynomials. Conformal Geometry and Dynamics of the AMS 19 (2015), 35–50
  Sabyasachi Mukherjee
  (See online at https://doi.org/10.1090/S1088-4173-2015-00276-3)
• A rigidity result for some parabolic germs. Indiana University Journal of Mathematics
  Luna Lomonaco and Sabyasachi Mukherjee
  
• Non-landing parameter rays of the multicorns. Inventiones Mathematicae 204 (2016), 869–893
  Hiroyuki Inou and Sabyasachi Mukherjee
  (See online at https://doi.org/10.1007/s00222-015-0627-3)
• Rational parameter rays of multibrot sets. Dynamical Systems, Number Theory and Applications (2016), chapter 3, 49–84, World Scientific
  Dominik Eberlein, Sabyasachi Mukherjee and Dierk Schleicher
  (See online at https://dx.doi.org/10.1142/9789814699877_0003)
• Antiholomorphic perturbations of Weierstrass Zeta functions and Green’s function on tori. Nonlinearity
  Konstantin Bogdanov, Khudoyor Mamayusupov, Sabyasachi Mukherjee, and Dierk Schleicher
  
• On Multicorns and Unicorns II: bifurcations in spaces of antiholomorphic polynomials. Ergodic Theory and Dynamical Systems 37 (2017), 859–899
  Sabyasachi Mukherjee, Shizuo Nakane and Dierk Schleicher
  (See online at https://doi.org/10.1017/etds.2015.65)
• Parabolic arcs of the multicorns: Real-analyticity of Hausdor↵ dimension, and singularities of Pern (1) curves. Discrete and Continuous Dynamical Systems-A 37 (2017), 2565–2588
  Sabyasachi Mukherjee
  (See online at https://doi.org/10.3934/dcds.2017110)