Bifurcation in parameter spaces of holomorphic and anti-holomorphic dynamics and its visualization

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400115
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: Inou Hiroyuki 京都大学, 理学研究科, 講師 (00362434)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 反正則力学系 / 放物型分岐 / 没入型仮想現実 / 自己相似性

Outline of Final Research Achievements

The analogue of the Mandelbrot set in anti-holomorphic dynamics is called the tricorn. We prove that external rays and "umbilical cords" accumulating to the boundaries of hyperbolic components of odd period do not converge to a point. This non-landing property implies that the tricorn is much more complicated than the Mandelbrot set. Especially, the non-landingness of "umbilical cords" implies that the tricorn does not have such a self-similar property as for the Mandelbrot set.
We also construct a fundamental environment for visualization of Julia sets and bifurcation measures in complex two dimensional space with recent VR devices, so that we can easily implement various methods for controls, projections, rendering and so on for experiment.

Free Research Field

複素力学系