  A positive contactomorphism of a contact manifold $M$ is the end point of a
contact isotopy on $M$ that is always positively transverse to the contact
structure. Assume that $M$ contains a Legendrian sphere $\Lambda$, and that
$(M,\Lambda)$ is fillable by a Liouville domain $(W,\omega)$ with exact
Lagrangian $L$ such that $\omega|_{\pi_2(W,L)}=0$. We show that if the
exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is
positive, then every positive contactomorphism of $M$ has positive topological
entropy. This result generalizes the result of Alves and Meiwes from Reeb flows
to positive contactomorphisms, and it yields many examples of contact manifolds
on which every positive contactomorphism has positive topological entropy,
among them the exotic contact spheres found by Alves and Meiwes.
  A main step in the proof is to show that wrapped Floer homology is isomorphic
to the positive part of Lagrangian Rabinowitz-Floer homology.
