  In this paper, we investigate the upper capacity entropy and packing entropy
for (transitively)-saturated sets for systems with weaker version of
specification. In particular, we show that the packing entropy of
(transitively)-saturated sets satisfies a variant version of
(transitively)-saturated property, while the upper capacity entropy of
(transitively)-saturated sets always carries full entropy.
  We apply these saturated property results to study the topological structure
on $S^{op}_{f}$, the set of initial values of $f$-optimal orbits arising from
uniformly hyperbolic systems, together with the observation $f$ being both real
and matrix valued. We show that $S_{f}^{op}$ contains two subsets $S_{f}^{MR}$
(and $S_{f}^{MR}$), which are set of initial values of an measure (and
topologically) recurrent $f$-optimal orbit. Moreover, the structure of
$S_{f}^{MR}$ and $S_{f}^{TR}$ exhibit differently from measurable and
topological viewpoints, with respect to the observation $f$ admitting different
regularity.
