We study contact geometry, and focus on the study of periodic orbits of the Reeb vector field. It is a conjecture of Colin and Honda that for universally tight contact structures on hyperbolic manifolds, the number of Reeb periodic orbits grows exponentially with respect to the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic manifolds. Along the lines 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 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. Finally we study consequences on Reeb periodic orbits of a bypass attachment, an elementary change of the contact structure consisting in attachment of half an overtwisted disc along a Legendrian arc. We describe new periodic orbits in terms of Reeb chords of the attachment arc, we compute contact homology of a product neighborhood of convex surfaces after a bypass attachment and we compute contact homology for some contact structures on solid tori.
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