It is a conjecture of Colin and Honda that the number of Reeb periodic orbits
of universally tight contact structures on hyperbolic manifolds grows
exponentially with the period, and they speculate further that the growth rate
of contact homology is polynomial on non-hyperbolic geometries. Along the line
of the conjecture, for manifolds with a hyperbolic component that fibers on the
circle, we prove that there are infinitely many non-isomorphic contact
structures for which the number of Reeb periodic orbits of any non-degenerate
Reeb vector field grows exponentially. Our result hinges on the exponential
growth of contact homology which we derive as well. We also compute contact
homology in some non-hyperbolic cases that exhibit polynomial growth, namely
those of universally tight contact structures non-transverse to the fibers on a
circle bundle
