  We establish the continuous functoriality of wrapped Fukaya categories with
respect to Liouville automorphisms, yielding a way to probe the homotopy type
of the automorphism group of a Liouville sector. These methods prove Liouville
and monotone cases of a conjecture of Teleman from the 2014 ICM. In the case of
a cotangent bundle, we show that the Abouzaid equivalence between the wrapped
category and the infinity-category of local systems intertwines our action with
the action of diffeomorphisms of the zero section. In particular, our methods
yield a typically non-trivial map from the rational homotopy groups of
Liouville automorphisms to the rational string topology algebra of the zero
section.
