10. Let , be a strictly increasing family of homeomorphisms on the unit circle, be the rotation number of . Poincare observed that if and only if admits some periodic points of period . In this case admits fixed points.

Note that is continuous, and non-decreasing. However, may not be strictly increasing. In fact, if and , then is locked at for a closed interval . More precisely, if for some , then on for some ; if ; while if both happen.

Also oberve that if , then is a singelton. So assuming is not unipotent for each , the function is a Devil’s staircase: it is constant on closed intervals , whose union is dense in .

9. Let be a vector field on , be the flow induced by on . That is, . Then we take a curve , and consider the solutions . There are two ways to take derivative:

(1) .

(2) , which induces the tangent flow of .

Combine these two derivatives together:

This gives rise to an equation

Formally, one can consider the differential equation along a solution :

, . Then is called the linear Poincare map along . Suppose . Then determines if the periodic orbit is hyperbolic or elliptic. Note that the path , contains more information than the above characterization.