Most of topological dynamics studies systems of the form where is a continuous self-map and is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., where is a closed subset of and such that .
J. Auslander asked about the obstructions for a topological system to have a symbolic extension, i.e., a symbolic system and a continuous surjection commuting with the dynamics: . There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?
M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.
T. Downarowicz and A. Maas showed that interval maps also always have symbolic extensions for .
Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.