  Nondegenerate periodic orbits in three-dimensional Reeb flows can be
classified into three types, positive hyperbolic, negative hyperbolic and
elliptic. As a problem which involves refining the three-dimensional Weinstein
conjecture, D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano proposed
whether every nondegenerate closed contact three manifold has at least one
positive hyperbolic orbit except for lens spaces. In the same paper, they
showed the existence of at least one simple hyperbolic orbit under $b_{1}>0$ by
the isomorphism between ECH and Seiberg-Witten Floer (co)homology, especially,
using the result that if $b_{1}>0$, the odd part of ECH which detects the
existence of a positive hyperbolic orbit does not vanish. But in the case of
$b_{1}=0$, such a way doesn't work. In the present paper, we prove the
existence of a positive hyperbolic orbit in the presence of at least one
elliptic orbit except for some well-known trivial cases under $b_{1}=0$. The
key points in this paper are the volume property with respect to ECH spectrums
and the compactification of the moduli spaces of certain $J$-holomorphic curves
counted by the $U$-map.
