Transfer operators and emergent dynamics in hyperbolic systems

Dynamical systems is a field of mathematics that is concerned with studying phenomena that evolve in time. It has deep connections with other areas of mathematics such as analysis, number theory, probability and geometry. Many interesting dynamical systems are chaotic in nature. This means they exhibit sensitive dependence on initial conditions and their long-term behaviour cannot be predicted by following their orbits. Thus, it is natural to study the behaviour of such systems form a probabilistic point of view. An instrumental tool to infer statistical aspects of a chaotic dynamical system is called the transfer operator. Such an operator describes how distributions change over time under the evolution of the dynamics. More importantly, its spectral data encode remarkable information, such as rates of correlation decay, extremes and rare events, on the statistics of the underlying dynamics.

A fundamental class of chaotic smooth hyperbolic dynamical systems is called Anosov. In the past fifteen years transfer operator techniques produced impressive results on the statistical aspects of smooth, 'idealised' (single site), Anosov systems and made remarkable new connections with other areas of mathematics, namely with semiclassical analysis, a modern topic in mathematical analysis.

However, transfer operator techniques are not yet pioneered for coupled Anosov systems, and hence obviously, not for the more general coupled piecewise hyperbolic systems with singularities. Such coupled systems appear naturally as network models in engineering, physical and biological sciences, and are of paramount importance in studying nonequilibrium thermodynamics. This leaves the fruitful approach of transfer operators, and ergodic theory in general, short on providing statistical insights on the behaviour of such complex systems that are capable of producing emergent dynamics: dynamical quantities, such as escape of mass and heat transfer, that appear as a result of interaction among components in a large system.

In this project we aim to achieve a new state-of-the art in smooth ergodic theory and hyperbolic dynamics by developing novel transfer operator techniques to understand emergent dynamical quantities and macroscopic statistical properties of 'large dynamical systems' whose microscopic dynamics are piecewise hyperbolic systems with singularities.