(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)

(* For problem 6 *)

tentmap[x_] := Piecewise[{{3 x, x <= 1/2}, {3 - 3 x, x > 1/2}}]

(* This defines the tent map *)

Reduce[0 <= tentmap[x] <= 1]

(* Reduces the inequality 0 <= tentmap[x] <= 1 to as simple a form as possible.  In this case, Mathematica will return a list of inequalities separated by the "or" operator || *)
(* In general, Reduce[] will reduce an expression to as simple a form as Mathematica can find. *)

Reduce[0 <= Nest[tentmap, x, 2] <= 1]
(* This finds the intervals that are mapped inside [0,1] by two iterations of the tent map. *)
(* Nest[f,x,n] will compute the nth iterate f^n[x]. *)



(* For problem 7 *)

q[x_] := x^2 - 6
r1[x_] := Nest[q, x, 8]
list1 := NSolve[r1[x] == x, x]

(* This asks Mathematica to numerically compute the 256 points of order dividing 8 for q *)

r1[x] - x /. list1

(* This evaluates r1[x] - x where x is replaced, successively, by the values in list1 *)
(* If the computations are accurate, the results should be close to zero *)


r2[x_] := (Nest[q, x, 8] - x)/(Nest[q, x, 4] - x)
list2 = NSolve[r2[x] == 0, x]


(* This asks Mathematica to numerically compute the 240 points of order exactly 8 for q *)
(* This computation may take some time! *)

r1[x] - x /. list2

(* This evaluates r1[x] - x where x is replaced, successively, by the values in list2 *)
(* If the computations are accurate, the results should be close to zero *)