[Solved] Dynamical Homework 11

1.1 Section 5.1

Problem 2.

Let T : X X be a measure-preserving transformation with (X) = 1 and suppose that for every measurable set A the limit

1 n_X1 _i

lim IA(T (x))

n n i=0

exists and equals (A) a.e. We want to show that T is ergodic. Let us fix two measurable sets A,B. Other than on two sets of measure 0 (whose union is measure 0), we have,

1 n_X1 _i

(A)(B) = lim _A IB(T (x)) n n i=0 n n i=0

1 nX1nX1 i j

= lim IA(T (x))IB(T (x))

ⁿ n _i=0 j=0

1 n_X1 _i

= lim IA_B(T (x)) n n i=0

= Problem 3.

1.2 Section 5.2

Problem 2.

Problem 4.

Problem 5.

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