ESAM Student Presents at 2019 SIAM Conference

On May 23, 2019, ESAM student Thomas Lynn presented his work, “Characterizing Fractal Mixing with Ergodic Subsets,” at the 2019 Society for Industrial and Applied Mathematics (SIAM) Conference on Applications of Dynamical Systems. His abstract reads:

Mixing by cutting-and-shuffling can be mathematically described by the dynamics of piecewise isometries, higher dimensional representations of one-dimensional interval exchange transformations. The mixed state of a two-dimensional domain is highly dependent on the structure created by iterated cutting lines in the domain, which often create a fat fractal structure. We investigate dynamics within this fat fractal to explain subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets within the structure of cutting lines. The size and shape of the cutting line fractal, used in the past to correlate with mixing, only indicate where mixing is possible, not where it is actually occurring. A new approach, based on examining the variation in a measured ‘density’ of cutting lines, easily assesses mixing within the fractal since ergodic sets necessarily share the same density. Since the fractal is generated by a one dimensional set of cutting lines, every two dimensional orbit in the fractal can be projected in a one-to-one fashion to a one dimensional set where measurements on the ‘connectivity,’ and therefore the mixing, of the fractal can be easily evaluated.