# Lower bound of positive entropies of automorphisms on tori

Let $$A$$ be an automorphism on tori $$\mathbb{T}^d$$. It is well known that the topological entropy $$h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \}$$ where $$\lambda$$ goes through all eigenvalue of $$A$$ with multiplicity.

Consider the case when $$h(A)>0$$. I would like to ask what are the lower bounds $$\inf_{A\in SL(d,\mathbb{Z}),h(A)>0} h(A)$$ and $$\inf_{A\in SL(d,\mathbb{Z}),h(A)>0, d\ge 2} h(A).$$ Thanks.

For fixed $$d$$ the best known lower bound is due to Dobrowolski, who showed that if $$h(A)>0$$ then $$h(A) > \log\Big[1+\frac{1}{1200}\Bigl(\frac{\log\log d}{\log d}\Bigr)^3\Bigr].$$
We have $$h(A)=\log\mathcal M(p)$$, the Mahler measure of the characteristic polynomial of $$M$$, where $$M$$ is the matrix which specifies $$A$$. So if we assume Lehmer's conjecture (which remains unproven), then the answer is $$\approx\log 1.17628$$. (I presume you allow negative determinants, i.e., it's $$GL(d,\mathbb Z)$$ rather than $$SL(d,\mathbb Z)$$.)