I don't know, but theoretically, if culture is a defense to tourism, if their culture blows up and your toursim declines (someone takes your great works), I think you can go from influential back to popular. That would make sense, but I don't know if it's true. The reason why I think this is true because, on the tourism screen, on the far right, there's either a green arrow that says rising, or rising slowly, or a red arrow pointing down that says "falling" I think.