In dynamics there are several conjectures related to entropy. The first is the famous Shub Entropy Conjecture:

Let be a differentiable map (or a diffeomorphism), be the spectral radius of the linear map . Then

.

Moveover find characteristics of maps satisfying in each homotopy class.

I am not sure whether the following special problem is still open (it is proved if is orientable):

If is an Anosov diffeomorphism, then .

There is another entropy conjecture due to Katok. In fact Katok proved that the topological entropy and the Liouville entropy of a geodesic flow on a negatively curved surface agree only if the curvature is constant, that is, the metric is locally symmetric. He conjectured that the same holds in any dimension. One of the best result in this direction is

Besson, Courtois and Gallot, the topological entropy of geodesic flow is minimized only for locally symmetric metrics.