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 for some , then on for some ; 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:
(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.