Number Theory: An Introduction to Mathematics

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)|≤μ(B\(C∩σ−nC))+μ((C∩σ−nC)\B)< 2 ε.

Thus

0 ≤μ(B)−μ(B)^2 =μ(B)−μ(C∩σ−nC)+μ(C∩σ−nC)−μ(B)^2

< 2 ε+μ(C)^2 −μ(B)^2 < 4 ε.

Sinceεis arbitrary, we conclude thatμ(B)=μ(B)^2. Henceμ(B)=0or1,andσis
ergodic.

Similarly, ifYis the set of all infinite sequencesy=(y 1 ,y 2 ,y 3 ,...)withyi∈A
for everyi∈N, then theone-sided Bernoulli shift B+p 1 ,...,pr,i.e.themapτ:Y→Y
defined byτy=y′,wherey′i=yi+ 1 for everyi∈N, is a measure-preserving transfor-
mation of the analogously constructed probability space (Y,B,μ). It should be noted
that, althoughτY=Y,τis not invertible. In the same way as for the two-sided shift,
it may be shown that the one-sided Bernoulli shiftB+p 1 ,...,pris always ergodic.

(iv) An example of some historical interest is the ‘continued fraction’ orGaussmap.
LetX=[0,1] be the unit interval andT:X→Xthe map defined (in the notation of
§1) by

Tξ={ξ−^1 } ifξ∈( 0 , 1 ),

=0ifξ=0or1.

ThusTacts as the shift operator on the continued fraction expansion ofξ:if

ξ=[0;a 1 ,a 2 ,...]=

1

a 1 +

1

a 2 +···

,

thenTξ=[0;a 2 ,a 3 ,...]. (In the terminology of Chapter IV, the complete quotients
ofξareξn+ 1 = 1 /Tnξ.)
It is not difficult to show thatTis a measure-preserving transformation of the prob-
ability space (X,B,μ), whereBis the family of Borel subsets ofX=[0,1] andμ
is the ‘Gauss’ measure defined by

μ(B)=(log 2)−^1

∫

B

( 1 +x)−^1 dx.

It may further be shown thatTis ergodic. Hence, by Birkhoff’s ergodic theorem, iff
is an integrable function on the intervalXthen, for almost allξ∈X,

nlim→∞n−^1

n∑− 1

k= 0

f(Tkξ)=(log 2)−^1

∫

X

f(x)( 1 +x)−^1 dx.