We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.
"Growth and order of automorphisms of free groups and free Burnside groups." Geom. Topol. 21 (4) 1969 - 2014, 2017. https://doi.org/10.2140/gt.2017.21.1969