I don’t have the Handbook near me at the moment so I can’t look at the example you mention, but at https://vaughnclimenhaga.wordpress.com/2013/11/24/fubini-foiled/ I describe a construction of Milnor’s (see the Intelligencer article linked to there) that is my go-to example for an elementary instance of this phenomenon. It’s similar to Katok’s original example but not identical and avoids things like entropy.