On Mix-Norms and the Rate of Decay of Correlations

Explore a 22-minute lecture on mix-norms and the rate of decay of correlations presented by Jean-Luc Thiffeault from the University of Wisconsin-Madison. Delve into the quantitative notions of mixing, including the decay of correlations and mix-norms, and their relationship to measuring mixing intensity. Discover how correlations can be uniformly dominated by mix-norms and investigate whether they can decay asymptotically faster. Learn about the construction of observables with correlations that approach the decay rate of mix-norms, and understand the conditions under which they achieve the same rate. Examine the concepts of q-recurrence and q-transience in mixed functions, and explore various examples to illustrate these ideas. The lecture, part of the Transport and Mixing in Complex and Turbulent Flows 2021 series at the Institute for Pure & Applied Mathematics (IPAM), covers topics such as evolving scalar fields, Mie-norms, different notions of decay rates, the baker's map, sine flow, and the construction of test functions to answer fundamental questions about mixing processes.

Intro
An evolving scalar field
Mie-norms and correlations
Different nations of the rate of decay of correlations
Different nations: sketch
Example: baker's map and q-transience (cont'd)
Example: baker-like action and g-recurrence (cont'd)
Sine flow example (cont'd)
Results: independence on
Results: constructing a test function 1
Results: Answering the original question
Conclusion