Degrees of self-maps of products

Every closed oriented manifold $M$ is associated with a set of integers $D(M)$, the set of self-mapping degrees of $M$. In this paper we investigate whether a product $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor $D(N)$ contains $d$. We find sufficient conditions so that $d\notin D(M\times N)$, obtaining, in particular, products that do not admit self-maps of degree $-1$ (strongly chiral), that have finite sets of self-mapping degrees (inflexible) and that do not admit any self-map of degree $dp$ for a prime number $p$. Furthermore we obtain a characterization of odd-dimensional strongly chiral hyperbolic manifolds in terms of self-mapping degrees of their products.