  In this paper we study the growth rate of a version of Legendrian contact
homology, which we call strip Legendrian contact homology, in 3-dimensional
contact manifolds and its relation to the topological entropy of Reeb flows. We
show that: if for a pair of Legendrian knots in a contact 3-manifold $(M,\xi)$
the strip Legendrian contact homology is defined and has exponential
homotopical growth with respect to the action, then every Reeb flow on
$(M,\xi)$ has positive topological entropy. This has the following dynamical
consequence: for all Reeb flows (even degenerate ones) on $(M,\xi)$ the number
of hyperbolic periodic orbits grows exponentially with respect to the period.
We show that for an infinite family of 3-manifolds, infinitely many different
contact structures exist that possess a pair of Legendrian knots for which the
strip Legendrian contact homology has exponential growth rate.
