Lyapunov optimizing measures and periodic measures for $C^2$ expanding maps

We prove that there exists an open and dense subset $\mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.

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