  The hyperon dynamics in heavy-ion collisions near threshold energy has been
investigated within the quantum molecular dynamics transport model. The isospin
and momentum dependent hyperon-nucleon potential and the threshold energy
correction on the hyperon elementary cross section are included in the model.
It is found that the high-density symmetry energy is dependent on the isospin
ratios $\Sigma^{-}/\Sigma^{+}$ and $\Xi^{-}/\Xi^{0}$, in particular in the
domain of high kinetic energies. The isospin diffusion in heavy-ion collisions
influences the neutron/proton ratio in the high-density region. The
$\Sigma^{-}/\Sigma^{+}$ ratio depends on the stiffness of symmetry energy, in
particular at the beam energy below the threshold value (E$_{th}$=1.58 GeV),
i.e., the kinetic energy spectra of the single ratios, excitation functions and
energy spectra of the double ratios in the isotopic reactions of $^{108}$Sn +
$^{112}$Sn, $^{112}$Sn + $^{112}$Sn, $^{124}$Sn + $^{124}$Sn and $^{132}$Sn +
$^{124}$Sn. The double strangeness ratio $\Xi^{-}/\Xi^{0}$ weakly depends on
the symmetry energy because of the hyperon-hyperon collision mainly
contributing the $\Xi$ production below the threshold energy (E$_{th}$ = 3.72
GeV).
