Tertiary butyl methyl ether on treatment with HI forms 

The S_N1 reaction mechanism proceeds step-by-step, starting with the formation of the carbocation through the elimination of the leaving group. The nucleophile then attacks the carbocation. Finally, the protonated nucleophile is deprotonated to produce the desired product.

The S_N1 process was pursued because the stable tertiary carbocation formed. Iodine is joined to the tertiary C atom, and the methyl group is joined to the -OH group.