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..
# The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$
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).