ACMS Abstracts: Fall 2013
Rob Sturman (Leeds)
Lecture 1: The ergodic hierarchy & uniform hyperbolicity
Ergodic theory provides a hierarchy of behaviours of increasing complexity, essentially covering dynamics from indecomposable, through mixing, to apparently random. Demonstrating that a (real) system possesses any of these properties is typically difficult, but one important class of system - uniformly hyperbolic systems - make the ergodic hierarchy immediately accessible. Uniform hyperbolicity is a strong condition, and so we also describe its weaker counterpart, non-uniform hyperbolicity. Our chief example in this lecture is the Arnold Cat Map, an example of a hyperbolic toral automorphism.
Lecture 2: Non-uniform hyperbolicity & Pesin theory
The connection between non-uniform hyperbolicity and the ergodic hierarchy is more difficult, but is made possible by the work of Yakov Pesin in 1977, and extensions due to Katok & Strelcyn. Here we illustrate the theory with a linked twist map, a paradigmatic example of non-uniformly hyperbolic system.
Rates of mixing in models of fluid flow
The exponential complexity of chaotic advection might be reasonably assumed to produce exponential rates of mixing. However, experiments suggest that in practice, boundaries slow mixing rates down. We give rigorous results from smooth ergodic theory which establishes polynomial mixing rates for linked twist maps, a class of simple models of bounded flows.
Shilpa Khatri (UNC)