CHAPTER 9. The Tent Map

9.1 The Tent Mapping

The tent mapping is defined by
		  
(Subscript[T, μ])(x)=μ x, 0≤ x ≤ 1/2, (Subscript[T, μ])(x)=μ (1-x), 1/2≤ x ≤ 1. 
						  
In Chapter 9 we investigated this map for different values of  μ.  In Exercises 9.2  3(iii) the polynomial  (Subscript[p, k])(μ)=(μ^(k-1))-(μ^(k-2))-(μ^(k-3))+(μ^(k-4))- (μ^(k-5))+…+μ-1, for
k>3 odd, turns out to be important. In fact a period k orbit (for k odd) first appears as the largest root of the equation (Subscript[p, k])(μ)=0.

Exercises 9.1.

1.     Use Mathematica to show that  (Subscript[p, k])((Sqrt[2]))=1-(Sqrt[2]),  for k=5  and k=7. Deduce that this is true for every value of k≠3, odd.  Explain why this shows that (Subscript[T, (Sqrt[2])]) has no points of odd period (see also Section 2.8).