The answer is no and a suitable counterexample in any dimension $n\geq 2$ was constructed by Oxtoby and Ulam. This is Theorem 12 p. 919 in:
J. C. Oxtoby, S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42 (1941), 874–920.
Here, the automorphism means homeomorphism. Since it is rigid translation on cubes, the derivative is $DT^*=I$ on each of the cubes. This theorem says that not only strange homeomorphisms answering the question exist, but every homeomorphism preserving the measure can be uniformly approximated by such homeomorphisms.