摘要：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 $\varphi$.

邀请人：陈剑宇