What is the ruling on taking back one's wife after divorcing her and making a vow of Thihar against her on the same day, and is expiation for Thihar obligatory, knowing that there was a previous expiation?

If it was the first or second divorce, you have the right to take back your wife before her 'iddah (waiting period) ends. Zihar occurs here, and its expiation becomes obligatory upon you before you touch your wife, because a revocably divorced woman is like a wife. However, if it was the third divorce, your wife has been irrevocably separated from you (baynunah kubra), and she is not lawful for you until she marries another man and then is divorced from him after consummation. The validity of zihar from her in this case is a matter of dispute, but the preponderant opinion is its validity, and you are obliged to pay the expiation for zihar if you marry her again.