A deep understanding of change

Marco Martens is a professor at the Stony Brook University, New York. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, Uppsala University.

A dynamical system is a mathematical model that describes how the state of a system evolves over time. The state of the system could be anything from the position and speed of a pendulum to the current weather conditions or a snapshot of the entire universe. Often, its evolution over time is extremely sensitive to initial conditions, and the system is then described as chaotic. A typical chaotic system is weather, which cannot be predicted for long periods of time, because even a small change in initial values can have large and unpredictable effects in the long term. 

However, even simple dynamical systems can turn out to be astonishingly complex. Some one-dimensional dynamical systems are now well understood, thanks to the development of renormalisation theory, one of the more powerful techniques used in the field in recent decades. The aim of the planned project is to use renormalisation theory to deepen the understanding of two-dimensional dynamical systems. 

Crucial properties of the one-dimensional systems are that they exhibit rigidity, as well as a certain universality. However, the renormalisation of two-dimensional dissipative systems has opened up a source of surprising phenomena. For example, Marco Martens and others have discovered that rigidity does not persist in two dimensions, and they have found new probabilistic rigidity and universality phenomena. The natural next step in the analysis, and the main goal of the proposed research, is to rigorously construct a renormalisation fixed point in the case of a conservative two-dimensional system.