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We show that for an expanding map, the maximizing measures of a generic (openand dense) $C^r$ ($r\in\mathbb{N}$) differentiable functions are supported on asingle periodic orbit. [There is a gap in the discussions. For the $C^{\infty}$approximation of the Lipschitz functions, we can only control the $C^1$derivative, but we can not control the $C^r$ derivatives for $r\geq2$. Elegantapproximation methods might be needed to solve this problem.]

Ergodic maximization problem for expanding maps with differentiable observables