Entropy, Homology and the Fundamental Group

Mike Shub (CONICET, IMAS, Universidad de Buenos Aires and CUNY Graduate School)

The topological entropy of a transformation of a topological space to itself is an information theoretic notion of the growth rate of the number of epsilon distinguishable orbits of the transformation. When this number is not zero the trnsformation is chaotic. There are connections between the topological entropy and some of the standard notions in algebraic topology, for example the entropy is greater than or equal to the growth rate of the induced map on the fundamental group of the space. There is a conjecture for C^r maps of compact manifolds (known to be true for C^infinity maps) bounding the entropy below by spectral radius of the induced map on the homology of the manifold (Yomdin's Theorem). I will review some of the known facts and the state of some of the conjectures. No knowledge of dynamics will be assumed. In particular entropy will be formally defined.