Kevin McGoff gave a talk at Orsay on his work on the theory of entropy srtuctures and symbolic extensions. This theory was founded by Mike Boyle and Tomasz Downarowicz, among others.
This theory relates the continuity properties of the measure-theoretic entropy function with the existence of symbolic topological extensions with measure-theoretic entropies as close as possible to those of the initial system. This theory ascribes to any topological dynamical system an order of accumulation. M. Boyle and T. Downarowicz have shown that this is a countable ordinal.
David Burguet and K. McGoff have shown that any countable ordinal can be achieved by some topological dynamics. The proof relies on a realization theorem of T. Downarowicz and S. Serafin.
K. McGoff explained how he was able, by a more precise and direct construction to achieve the same on any prescribed compact manifold. The transformation can be chosen to be homeomorphic if the dimension is 2 or more.