About me

I am from Jiangxi Province, China. My Chinese name is 张道飞.

This is my fourth year studying for a PhD in the department of mathematics at Warwick. I am a member of the research group of Prof. Mark Pollicott. I am funded by Mark's ERC research project, and my research topic is mixing rates of hyperbolic flows and their compact group extensions.

Previously, I got my bachelor's degree from Jiaxing University in 2017, and then I became a graduate student of Prof. Dawei YangLink opens in a new window in Suzhou University and I got my master's degree in 2020.

Contact

Email: Daofei dot Zhang at warwick dot ac dot uk

A brief description of my research topic

Many years ago, Bowen, Ruelle and Sinai, etc., established a theory of hyperbolic dynamical systems. Hyperbolic systems have many interesting properties, one of which is the rate of mixing (also known as the decay of correlation). For continuous-time hyperbolic systems (i.e. hyperbolic flows), unlike discrete-time hyperbolic systems (i.e. hyperbolic diffeomorphisms) are automatically exponentially mixing with respect to equilibrium states, estimating their rate of mixing is more difficult. In this research direction, Dolgopyat made an outstanding contribution. He developed techniques and methods to study mixing rates (e.g. rapid mixing, exponential mixing) of hyperbolic flows.

Beyond uniform hyperbolicity, compact group extensions of hyperbolic systems have attracted much attention because they provide one of the simplest examples of partially hyperbolic systems. For compact group extensions of hyperbolic diffeomorphisms, Dolgopyat developed techniques and methods to study rapid mixing. He also developed techniques and methods to study exponential mixing of compact group extensions of expanding maps.

My topic is to study mixing rates of compact group extensions of hyperbolic flows. The most important example of this type of system is frame flows on compact n-dimensional manifolds (n>2) with negative sectional curvatures. To do this, I will develop Dolgopyat's techniques and methods to this type of system. More specifically, I will study rapid mixing of compact group extensions of hyperbolic flow. The ultimate goal is to show that reasonable frame flows are rapidly mixing. For exponential mixing, I will focus on torus extensions of Anosov flows and expanding semi-flows. I will first study torus extensions of expanding semi-flows since they are symbolic models of torus extensions of Anosov flows. The objective of this phase is to demonstrate that reasonably constructed torus extensions of expanding semi-flows exhibit exponential mixing behavior. Subsequently, we can employ the concepts established in this phase to address the broader category of torus extensions associated with Anosov flows. The overarching aim is to establish the exponential mixing property for the frame flow on a compact 3-dimensional manifold of 1/4-pinched negative curvature.