Danijela Damjanovic: When symmetries tame chaos

Abstract:
Chaotic dynamical systems have some beautiful features: they often mix well, have positive entropy, and inherently can be as unpredictable as a coin toss (even though they are completely deterministic). Models for such behaviour are prevalent in the large class of algebraic chaotic systems. These systems live on homogeneous spaces, they are simply defined by automorphisms or group multiplication, and due to algebraicity they are amenable to delicate analysis. One extra feature that these algebraic systems have is quite counterintuitive from the dynamical point of view: their chaotic orbit structure is completely preserved under many symmetries. Non-algebraic systems are of course prevalent in nature, and in presence of constraints they also may be equipped with some symmetries. The question I will address in this talk is: to what extent symmetries of general (somewhat) chaotic dynamical systems completely determine these systems? More concretely, do symmetries of an arbitrary chaotic system force the system to live on a homogeneous space and to be effectively an algebraic system? I will present a recent result in this direction where homogeneous structures are built from purely dynamical features of the system, with help of symmetries.