Number Theory: An Introduction to Mathematics

466 XI Uniform Distribution and Ergodic Theory Thenμis said to be aprobability measureand the triple(X,B,μ)is said to be a proba ...

3 Birkhoff’s Ergodic Theorem 467 Put f ̄(x)= lim n→∞ n−^1 n∑− 1 k= 0 f(Tkx), f(x)= lim n→∞ n−^1 n∑− 1 k= 0 f(Tkx). Thenf ̄andfar ...

468 XI Uniform Distribution and Ergodic Theory and hence τ(∑x)− 1 k= 0 f ̄M(Tkx)≤ τ(∑x)− 1 k= 0 f ̃(Tkx)+τ(x)ε. To estimate the ...

3 Birkhoff’s Ergodic Theorem 469 since the measure-preserving nature ofTimplies that, for anyg∈L(X,B,μ), ∫ X g(Tx)dμ(x)= ∫ X g(x ...

470 XI Uniform Distribution and Ergodic Theory It should be noticed that the preceding proof simplifies if the functionfis bound ...

3 Birkhoff’s Ergodic Theorem 471 Suppose now that (i) holds and let f∈L(X,B,μ). Then the functionf∗in the statement of Theorem 1 ...

472 XI Uniform Distribution and Ergodic Theory 4 Applications................................................ We now give some e ...

4 Applications 473 (ε) for each function f ∈ L(X,B,λ), limn→∞n−^1 ∑n− 1 k= 0 f(T k ax)= ∫ Xfdλfor almost allx∈X. (ii) Again supp ...

474 XI Uniform Distribution and Ergodic Theory and hence cn=cD− (^1) n for everyn∈Zd. Butcm=0 for some nonzerom∈Zd,sincefis not ...

4 Applications 475 n−^1 ∑n−^1 k= 0 f(Akx)→ ∫ X fdλ= 0. Sincefis not zero a.e., it follows that the set of allxwhich are normal w ...

476 XI Uniform Distribution and Ergodic Theory Brown and Moran (1993) have shown, conversely, that ifA,Barecommuting d×d nonsing ...

4 Applications 477 B=[a−m,...,am]∪···∪[a′−m,...,a′m], we define μm(B)=pa−m···pam+···+pa′−m···pam′. Thenμm(X)=1,μm(B)≥0foreveryB∈ ...

478 XI Uniform Distribution and Ergodic Theory We m a y d e fi n e t h egeneral cylinder set Cia 11 ......iakk,wherei 1 ,...,ika ...

4 Applications 479 since B\(C∩σ−nC)⊆(B\C)∪(B\σ−nC)⊆(B\C)∪σ−n(B\C), and similarly μ((C∩σ−nC)\B)≤ 2 μ(C\B). Hence |μ(B)−μ(C∩σ−nC)| ...

480 XI Uniform Distribution and Ergodic Theory Here it makes no difference if ‘integrable’ and ‘almost all’ refer to the invaria ...

4 Applications 481 Suppose further that there exists aninvariant region X⊆Rd.Thatis,Xis the closure of a bounded connected open ...

482 XI Uniform Distribution and Ergodic Theory Matx, can be given the structure of a Riemannian manifold, theunit tangent bundle ...

5 Recurrence 483 Any translationTaof the torusRd/Zdhas entropy zero, whereas the endomor- phismRAofRd/Zdhas entropy ∑ i:|λi|> ...

484 XI Uniform Distribution and Ergodic Theory points. Poincar ́e’s proof was inevitably incomplete, since at the time measure t ...

5 Recurrence 485 Furstenberg’s approach to this result is not really shorter than Szemeredi’s, but it is much more systematic. I ...