Some problems in Milnor's textbook and its solutions (1)

Today, I upload some problems and its solution which is written in J.Milnor's Textbook (Dynamics in One Complex Variable), Problem 6-(a)-4 and 6-(b).

The title of Problem 6-(a) is The derivative of a torus map, Problem 6-(b) is Periodic points of torus maps. 

+ The content in page 1 is just "why the degree of f  is equal to the square of absolute value of a." , If you need not the proof of above conclusion, i recommend to you skip page 1.

( f is holomorphic map, Torus into itself, f=az+c (mod A) )

Problem 6-(a)-4

Show that the closure of every orbit under f is either a finite set, a finite union of parallel circles or the entire torus T.

The idea to prove this problem is, separating case according to c is rational + rational or irrational + rational (rational + irrational) , irrational + irrational.

we can write honestly the orbit of an arbitrary element z of this torus T, we can find the regularity, how to move the point z by n-fold iteration of f.

Problem 6-(b)

The first part is very simple. To prove the second part, we need to prove some properties of holomorphic map in Torus, but i using this fact non-proof. (This content is written in problem 6-(a)-3.)

The idea of the second part is, separating the entire torus T into small torus. if we can the entire torus 'uniformly separate to small torus', we can obtain the conclusion. (choosing the enough large n..)

I'm sorry if there is anything wrong in my proof...

Thank you to read my post.