  We investigate the effect of a hyperbolic (or, more generally, isolated as an
invariant set) closed Reeb orbit on the dynamics of a Reeb flow on the
$(2n-1)$-dimensional standard contact sphere, extending two results previously
known for Hamiltonian diffeomorphisms to the Reeb setting. In particular, we
show that under very mild dynamical convexity type assumptions, the presence of
one hyperbolic closed orbit implies the existence of infinitely many simple
closed Reeb orbits. The second main result of the paper is a higher-dimensional
Reeb analogue of the Le Calvez-Yoccoz theorem, asserting that no closed orbit
of a non-degenerate dynamically convex Reeb pseudo-rotation is locally maximal,
i.e., isolated as an invariant set. The key new ingredient of the proofs is a
Reeb variant of the crossing energy theorem.
