EPSRC DTP PhD studentship: Quasi-symmetric conjugation of a one-dimensional family of double cover circle maps

Location: University of Exeter, Streatham Campus, Exeter, Devon

In this PhD project we will study quasi-symmetric conjugation of a one-dimensional family of double cover circle maps. This is a very important problem in understanding the dynamics of low-dimensional dynamical systems and an area that has been very active over the years. The reason for this is the close connection with the renormalization conjecture and density of hyperbolicity, two of the most important questions in one-dimensional dynamical systems.

In particular, this project would answer the following question. Given a particular two-parameter family of double covers of the circle investigated by M. Misiurewicz and A. Rodrigues, Double standard maps (Commun. in Mathem. Phys. 273, 2007, 37--65.) with one of the parameters fixed, it is known that for the set of the other parameter for which the map is aperiodic this family is conjugated to a map similar to rotation numbers of the circle.

So, given two maps belonging to the double standard family for which we have aperiodic behavior, the objective of this PhD project would be to prove that the conjugacy between these two maps is quasi-symmetric.

The main aim of this project would be to provide a proof of this result using only real methods. Note that Rempe-Gillen and van Strien proved density of hyperbolicity in spaces of real transcendental entire functions, bounded on the real line, whose singular set is finite and real and also for transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, they proved density of hyperbolicity in the Arnol’d family of circle maps and its generalizations using techniques from complex analysis.

The main aim of this PhD project would be the use of techniques from real analysis and would probably lead to the development of new techniques in the field

The student’s role would be to prove theorems related to quasi symmetric conjugacy of a specific one dimensional function with the help and guidance of the supervisor. The student would initially need to study the background material to one-dimensional dynamics and in particular my joint work with Michal Misiurewicz on double standard maps. They would then have to study quasi-symmetric conjugacy both on the real line and on the circle for functions known in the literature.

The student will acquire expertise in the area of Dynamical Systems, in particular in the study of low-dimensional systems. They will also participate in 100 hours of taught courses, either at the University of Exeter or in the MAGIC programme. This will allow the student at the end of the PhD to pursue an academic career in Mathematics with a strong background in Analysis, Topology and Dynamical Systems (pure Mathematics).