Given a topological dynamical system $T:X/to X$, and an continuous observable $/varphi:X/to/mathbb{R}$, we say an orbit $/mathcal{O}_{x_{0}}=/{x_{0},T(x_{0}),/cdots/}$ is an $f$-optimal orbit, if the Birkhoff average $/langle /varphi/rangle(x_{0}):=/lim_{n/to/infty}/frac{1}{n}/varphi(T^{i}(x_{0}))$ exists, and $/langle/varphi/rangle(x_{0})/geq/limsup_{n/to/infty}/frac{1}{n}/varphi(T^{i}(x)),/forall x/in X$, and define by $/mathcal{S}_{op}/subset X$, the set of initial states, which give rise to the optimal orbit.  We will investigate the geometric structure of $/mathcal{S}_{op}$, and see how $/mathcal{S}_{op}$ varies, corresponding to the variances on the hyperbolicity of $T$, and regularity of $/varphi$.

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