Let $T^2$ be a 2-torus and $f:T^2\rightarrow T^2$ a smooth map. Let $f_*:\pi_1(T^2)\rightarrow\pi_1(T^2)$ be the induced map on the fundamental group $\pi_1$. If $f_*$ has no eigenvalue greater than 1, then all balls of $T^2$ have subexponential growth by $ f $. Why?
closed as off-topic by Chris Gerig, Stefan Kohl, Anthony Quas, Igor Belegradek, Fernando Muro Oct 10 '14 at 22:02
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You should check out the paper of R. Bowen: ENTROPY FOR GROUP ENDOMORPHISMS AND HOMOGENEOUS SPACES (Transactions of the AMS, 1971)