Rigidity problems, renormalization and critical points of the multiplier maps

• Applicant: Professor Dr. Igors Gorbovickis, Ph.D.
• Subject Area: Mathematics
• Term: from 2021 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 455038303

The first part of this project deals with renormalization theory in one-dimensional dynamics and the rigidity problem for sufficiently smooth circle homeomorphisms with a critical point of finite order. Such maps are called critical circle maps.The question of rigidity in the context of circle homeomorphisms goes back to Poincaré, who showed that every circle homeomorphism with an irrational rotation number is semi-conjugate to a rigid rotation. In the particular case of critical circle maps Yoccoz proved that any two such maps with the same irrational rotation number are topologically (in fact, quasisymmetrically) conjugate. The rigidity problem is the question whether the conjugacy is, in fact, smooth.The rigidity problem for critical circle maps has received a lot of attention in the literature. Currently it is solved in a positive way for maps with the critical point of the form f(x)=x|x|^(α-1) + c in some local coordinates, with an odd integer α, known as the critical exponent. An important step in establishing this result was the proof of hyperbolicity of renormalization for analytic critical circle maps.Renormalization has been one of the central themes in modern low-dimensional dynamics. A key breakthrough in the study of the renormalization was made by D. Sullivan, who introduced methods of holomorphic dynamics and Teichmüller theory into the subject. It is important to mention that historically, rigidity was empirically observed for critical circle maps with an arbitrary critical exponent α > 1, however, since the seminal work of Sullivan, the principal development of the subject happened in the realm of analytic circle maps. Analyticity of the maps at the critical point gives an obvious restriction that α must be an odd integer. The broadly stated goal of the first part of this project is to develop the renormalization theory for analytic one-dimensional dynamical systems with a singular critical point of an arbitrary order α > 1 and study the related rigidity problems. We are also planning to consider the case of circle maps with multiple critical points.In the second part of the project we will focus on problems from one-dimensional holomorphic dynamics, related to the study of critical points and critical values of the multipliers of periodic orbits, viewed as (multiple valued) algebraic functions on the parameter space of complex degree d polynomials. We will particularly consider the case d = 2, where this study might lead to a better understanding of the possible shapes of hyperbolic components in the Mandelbrot set.

• DFG Programme: Research Grants
• International Connection: Canada, USA
• Cooperation Partners: Professorin Tanya Firsova, Ph.D.; Professor Michael Yampolsky, Ph.D.