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.][1] *Ann. of Math.* (2) 42 (1941), 874–920.

[![**enter image description here**][2]][2]

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. 


  [1]: https://www-jstor-org.pitt.idm.oclc.org/stable/1968772?origin=crossref
  [2]: https://i.sstatic.net/qVTC7.png