Consider a compact one-dimensional dynamic system. We focus on $S_1$, it does not mean $S_1$ is the only compact one-dimensional system, but it is a typical example. Let $T: S_1 \rightarrow S_1$. If $T$ is a homomorphism, then $T$ stays the order of $S_1$ (by continuity and the zero point theorem). This just means: (may be do a reflexion $latex e^{2\pi i\theta}\to e^{-2\pi i\theta}$). In the homomorphism case, we try to define the rotation number to describe the expanding rate of the dynamic sys…