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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.

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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].

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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 $SL(d,\mathbb Z)$.)

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