$00000001$ is its own inverse in the Rijndael field, because polynomial multiplication by itself gets $00000001$ and is unchanged by the modulo operation. This will always be true in any ring, as noted in a comment above.
However, the specification of the Rijndael S-box is that after taking the inverse of the number in the Galois field, you have to multiply the result by a matrix then add (XOR) a constant: the "affine transformation". That is what makes
0x01 map to
Note that for the purposes of the Rijndael S-box, zero's inverse in the field is considered to be zero. The affine transformation is applied to zero, getting the expected