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.



Lower bound of positive entropies of automorphisms on tori
For fixed $d$ the best known lower bound is due to Dobrowolski, who showed that if $h(A)>0$ then $$h(A) > \log\Big[1+\frac{1}{1200}\Bigl(\frac{\log\log d}{\log d}\Bigr)^3\Bigr].$$


For this and much more, see Chris Smyth's article *The Mahler measure of algebraic numbers: a survey*, arXiv:math/0701397v2 [math.NT]. 


