Higher rank dynamics on nilmanifolds
Time: Thu 2025-05-15 14.00
Location: F3 (Flodis), Lindstedtsvägen 26 & 28, Stockholm
Language: English
Subject area: Mathematics
Doctoral student: Sven Sandfeldt , Analys, dynamik, geometri, PDE och talteori
Opponent: Professor David Fisher, Rice University, Houston, Texas, USA
Supervisor: Professor Danijela Damjanović, Analys, dynamik, geometri, PDE och talteori; Associate professor Maria Saprykina, Analys, dynamik, geometri, PDE och talteori
QC 2025-04-24
Abstract
The theory of dynamical systems is the area of mathematics that studies systems that change in time. This theory can be used to study everything from the movement of celestial bodies, unpredictability of weather forecasts, and self-organizing behaviour among fireflies to questions within geometry and number theory. Classically, a dynamical system consists of two parts (i) a phase space, the space of all possible states, and (ii) a law for updating from one state to the next as time passes. When viewing time as discrete, this naturally gives rise to an action of the integers, where an integer n moves a state x to the corresponding state after time n. However, motivated by questions in geometry, number theory, or dynamics itself, it is often useful to study group actions of more exotic groups. The symmetries of a classical system are the collection of all global changes of coordinates that leave the dynamics intact; taking all symmetries together we obtain the symmetry group of the system. This group naturally acts on the phase space so, given a classical system, we obtain an induced action of its symmetry group. A general philosophy is that having a large symmetry group is highly restrictive and in special cases, we even expect to be able to classify all systems with sufficiently many symmetries. This is a rigidity phenomenon, i.e. the a priori weak assumption of having a large symmetry group has the seemingly much stronger conclusion that the system can be completely classified. This thesis consists of four papers that aim to understand the symmetries in certain classes of chaotic dynamical systems, as well as understanding the related question of rigidity of large group actions.
Papers I and II investigate the symmetries of perturbations of a certain class of dynamical systems, partially hyperbolic nilmanifold automorphisms. The main results are complete classifications of all possible symmetries for these perturbations, as well as a classification of those perturbations that admit a large symmetry group. It is shown that for these systems the property of having many symmetries is closely related to the preservation of algebraic structures.
Paper III studies actions of higher rank lattices (e.g., the group of all invertible 3 by 3 matrices with integer entries) on a certain type of manifolds (tori and Heisenberg nilmanifolds). It is shown that if these actions contain an element that is sufficiently chaotic, characterized by being partially hyperbolic, then the action must be of an algebraic origin.
Paper IV studies a class of continuous time classical dynamical systems with a property known as being cohomology free. Conjecturally these can only exist on special types of phase spaces, tori. We prove this conjecture when the phase space is a nilmanifold, a larger class of spaces that contain tori as special cases.