Let $M$ be a closed smooth connected manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:

(a) Every self-map $M \to M$ has degree $0,\pm 1$. (This is usually called  *inflexible*.) 

(b) Every smooth $S^1$-action on $M$ is trivial.

**My question is whether any of (a) or (b) implies the other or, if not, what concrete examples of manifolds are that have one of these properties, but not the other.**

Here are some thoughts:

If $M = S^k \times N$ then of course $M$ has neither of both properties.

If $M$ has positive simplicial volume (aka *Gromov norm*), like it happens for negatively curved manifolds, and in particular surfaces of hgher genus, then $M$ has both properties (a) and (b). For (a), that's rather obvious, for (b) it is a theorem due to Yano, published in 1982. 

It is a theorem due to Gromov that simply-connected manifolds always have simplicial volume $0$, but it is known that there are inflexible simply connected manifolds in any big enough dimension, see http://arxiv.org/pdf/1109.0960.pdf. I am not able at all to check whether these manifolds admit non-trivial $S^1$ ations, though... 

Very naively, it seems that if $M$ has a nontrivial $S^1$-action, we can use the orbit circles to wrap $M$ around itself with a positive degree, but it is not at all clear to me how this intuitive idea can be made precise. Maybe it is also a fallacy to think about $S^1$-actions in general by having pictures in mind where $M$ is a sphere or torus.