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Suppose you have an DG-algebra $A$, and a DG-bimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here $M^{\otimes_A n}$ is the A-bimodule $M\otimes_A M\otimes_A\cdots\otimes_A M$ (with $n$ terms). Anything, even a counterexample would be interesting..

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I don't know in general, but in Heegaard Floer theory, there are bimodules naturally associated to a mapping class of a surface self-homeomorphism. The rank of $H_*(M^{\otimes_A n})$ grows exponentially iff the underlying mapping class group element is pseudo-Anosov. See our paper at https://arxiv.org/abs/1012.1032. In particular, it's easy to give examples where the growth rate is linear, although I'm sure there are also more elementary constructions.

(There are also other, earlier constructions for the braid group, by Khovanov-Seidel and Khovanov-Thomas. I don't know if they explicitly stated the fact about growth rates, but it follows directly from their results.)

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