[Solved] Dynamical Quiz

Problem 1.

Let

x a_i T(x) = ai+1 ai

with x [a_i,a_i+1). T : J J where J = [0,1). And 0 = a₀ < a₁ < < a_k = 1. We want to show that T is a measure preserving transformation of (J,L(J),) regardless of choice of {a_i}.

We have that T¹(x) = x(a_i+1a_i)+a_i. Let C be the collection of left-closed, right-open dyadic intervals in [0,1). We saw in Section 2.7 that C is a sufficient semi-ring. For I we write I = [k/2ⁱ,(k + 1)/2ⁱ) for integers k,i with i 0 and k {0,,2ⁱ 1}. Observe that (I) = 1/2ⁱ for all I C. Assume for a fixed I that I [a_i,a_i+1) for some i {0,k}.

Then,

!

1 k k + 1

T (I) = 2i(ai+1 ai) + ai, i (ai+1 ai) + ai

2

T¹(I) is an interval for any I and is hence a measurable set. Moreover,

1 k + 1 k

(T (I)) = i (ai+1 ai) + ai ( i(ai+1 ai) + ai)

2 2

k + 1 k + 1 k k

= 2ⁱ ai+1 2_i ai _2iai+1 + _2iai

(k + 1)(a_i+1 a_i) k(a_i+1 a_i)

=

2i

((k + 1) k)(a_i+1 a_i)

=

=

2i

Observe that T maps some values onto I for each T defined on the intervals [a_i,a_i+1). Hence, there will be k such intervals resulting from I with the same length as the above when applying T¹.

1

If we add this up over all intervals [a_i,a_i+1), we get,

a_k a₀ 1

= = (I) 2i 2i

as required. Hence, we can apply Theorem 3.4.1 in order to assert that T is then a measure-preserving transformation.