  The propagation of an electromagnetic wave in a one-dimensional fractal
object, the Cantor set, is studied. The transfer matrix of the wave amplitude
is formulated and its renormalization transformation is analyzed. The focus is
on resonant states in the Cantor set. In Cantor sets of higher generations,
some of the resonant states closely approach the real axis of the wave number,
leaving between them a wide region free of resonant states. As a result, wide
regions of nearly total reflection appear with sharp peaks of the transmission
coefficient beside them. It is also revealed that the electromagnetic wave is
strongly enhanced and localized in the cavity of the Cantor set near the
resonant frequency. The enhancement factor of the wave amplitude at the
resonant frequency is approximately $6/|\eta_\mathrm{r}|$, where
$\eta_\mathrm{r}$ is the imaginary part of the corresponding resonant
eigenvalue. For example, a resonant state of the lifetime
$\tau_\mathrm{r}=4.3$ms and of the enhancement factor $M=7.8\times10^7$ is
found at the resonant frequency $\omega_\mathrm{r}=367$GHz for the Cantor set
of the fourth generation of length L=10cm made of a medium of the dielectric
constant $\epsilon=10$.
