Delving into the Mandelbrot Set and Its Copies

Science / Mathematics

n the realm of complex dynamics, the Mandelbrot set stands as a captivating and central object, a mathematical masterpiece that unveils the intricate behavior of quadratic polynomials. Its mesmerizing patterns, characterized by self-similarity and infinite complexity, have captivated mathematicians and artists alike. The Mandelbrot set serves as the locus of connectedness for the family of quadratic polynomials, providing a framework for understanding their diverse behaviors. An intriguing aspect is the presence of Mandelbrot copies within the set itself and across numerous other parameter planes. Within the Mandelbrot set, there are two distinct types of copies: primitive copies (with a cusp) and satellite copies (without a cusp). This project is focused on studying satellite copies, specifically whether they exhibit mutual similarity on infinitesimal scales, i.e., whether they are quasiconformally homeomorphic. Interestingly, the Mandelbrot set also seems to emerge in the parameter plane of a family of objects that aren’t even functions; they are holomorphic correspondences. This is significant as they represent intersections between objects that inhabit slightly different realms (quadratic maps and the modular group). The aim is to demonstrate that this ‘apparent’ Mandelbrot is, in fact, a Mandelbrot copy, meaning there is a homeomorphism between this set and the Mandelbrot set.

Amount invested

1st phase: R$ 100,000.00
2nd phase: R$ 1,000,000.00 (R$ 700,000.00 + R$ 300,000.00 optional bonus for the integration and training of people from underrepresented groups in science)
R$ 10,000.00 (maternity grant)
  • Topics
  • Complex dynamics
  • Mandelbrot set
  • Quadratic polynomials