Apple essentially owns the mobile application market — and the tablet application market, if you're inclined to separate that out. Apple's position in those markets is at least as strong as Google's in search. (Yes, they make more money from hardware, but remember we're discounting that under the premise that market influence is the decisive competitive factor.)
Apple also has disproportionate influence even in markets where it is not a monopoly.
It's not clear that those markets have ever been truly competitive or relevant in the first place. To someone like the FTC or the Justice Department, mobile apps are a nascent market, and they're not going to touch it with a 10-foot pole.
As for Apple's skill in managing media exposure and marketing, that's certainly not an area in which they can hope to have any kind of monopoly.
Apple also has disproportionate influence even in markets where it is not a monopoly.