  We investigate the cutoff phenomenon for Markov processes under information
divergences such as $f$-divergences and R\'enyi divergences. We classify most
common divergences into four types, namely $L^2$-type, $\mathrm{TV}$-type,
separation-type and $\mathrm{KL}$ divergence, in which we prove that the cutoff
phenomenon are equivalent and relate the cutoff time and window among members
within each type. To justify that this classification is natural, we provide
examples in which the family of Markov processes exhibit cutoff in one type but
not in another. We also establish new product conditions in these settings for
the processes to exhibit cutoff, along with new results in non-reversible or
non-normal situations. The proofs rely on a functional analytic approach
towards cutoff.
