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Then for any f ∈ L2µ , 1 N N −1 n=0 UTn f −→ PT f. 2 Lµ Proof. Let B = {UT g − g | g ∈ L2µ }. We claim that B ⊥ = I. If UT f = f, then f, UT g − g = UT f, UT g − f, g = 0, so f ∈ B ⊥ . If f ∈ B⊥ then UT g, f = g, f for all g ∈ L2µ , so UT∗ f = f. 9), so f = UT f . 5 The Mean Ergodic Theorem with h ∈ B. We claim that 1 N N −1 n=0 39 UTn h −→ 0. 11) as N → ∞. All we know is that h ∈ B, so let (gi ) be a sequence in L2µ with the property that hi = UT gi − gi → h as i → ∞. Then for any i 1, 1 N N −1 n=0 UTn h 2 1 N N −1 n=0 UTn (h − hi ) + 2 1 N N −1 n=0 UTn hi .

B) Show that the diagonal embedding δ(r) = (r, r) embeds Z[ 12 ] as a discrete subgroup of R× Q2 , and that X2 ∼ = R× Q2 /δ(Z[ 21 ]) ∼ = R× Z2 /δ(Z) as compact abelian groups (see Appendix C for the definition of Qp and Zp ). In particular, the map T2 (which may be thought of as the left shift on X2 , or as the map that doubles in each coordinate) is conjugate to the map (s, r) + δ(Z[ 12 ]) → (2s, 2r) + δ(Z[ 12 ]) on R × Q2 /δ(Z[ 12 ]). The group X2 constructed in this exercise is a simple example of a solenoid.

The first and most important of these is a result due to Poincar´e  published in 1890; he proved this in the context of a natural invariant measure in the “three-body” problem of planetary orbits, before the creation of abstract measure theory(14) . Poincar´e recurrence is the pigeon-hole principle for ergodic theory; indeed on a finite measure space it is exactly the pigeon-hole principle. 11 (Poincar´ e Recurrence). Let T : X → X be a measurepreserving transformation on a probability space (X, B, µ), and let E ⊆ X be a measurable set.