482 XI Uniform Distribution and Ergodic Theory
Matx, can be given the structure of a Riemannian manifold, theunit tangent bundle
T 1 M. EvidentlyT 1 Mis a (2n−1)-dimensional manifold ifMisn-dimensional. There
is a natural measureμonT 1 Msuch that dμ=dvqdωq,wheredvqis the volume
element atqof the Riemannian manifoldMandωqis Lebesgue measure on the unit
sphereSn−^1 in the tangent space toMatx.IfMis compact, then the measureμcan
be normalized so thatμ(T 1 M)=1.
AgeodesiconMis a curveγ ⊆ Msuch that the length of every curve inM
joining a pointx∈γto any sufficiently close pointy∈γis not less than the length
of the arc ofγwhich joinsxandy. Given any point(x,v)∈T 1 M, there is a unique
geodesic passing throughxin the direction ofv. The geodesic flow onT 1 Mis the
flowφt:T 1 M →T 1 Mdefined byφt(x,v)=(xt,vt),wherextis the point ofM
reached fromxafter timetby travelling with unit speed along the geodesic deter-
mined by(x,v)andvtis the unit tangent vector to this geodesic atxt.IfMis compact
then, for every realt,φtis defined and is a measure-preserving transformation of the
corresponding probability space (T 1 M,B,μ).
The geodesics on a compact 2-dimensional manifoldMwhose curvature at each
point is negative were profoundly studied by Hadamard (1898). It was first shown by
E. Hopf (1939) that in this caseφtis ergodic for everyt >0. (We must exclude
t=0, sinceφ 0 is the identity map.) This result has been considerably generalized by
Anosov (1967) and others. In particular, the geodesic flow on a compactn-dimensional
Riemannian manifold is ergodic if at each point the curvature of every 2-dimensional
section is negative.
Although the preceding examples look quite different, some of them are not
‘really’ different, i.e. apart from sets of measure zero. More precisely, if (X 1 ,B 1 ,μ 1 )
and (X 2 ,B 2 ,μ 2 ) are probability spaces with measure-preserving transformations
T 1 :X 1 →X 1 andT 2 :X 2 →X 2 , we say thatT 1 isisomorphictoT 2 if there exist sets
X′ 1 ∈B 1 ,X′ 2 ∈B 2 withμ 1 (X′ 1 )=1,μ 2 (X′ 2 )=1andT 1 X′ 1 ⊆X′ 1 ,T 2 X′ 2 ⊆X′ 2 ,and
a bijective mapφofX 1 ′ontoX′ 2 such that
(i) for anyB 1 ⊆ X 1 ′,B 1 ∈ B 1 if and only ifφ(B 1 )∈ B 2 and thenμ 1 (B 1 )=
μ 2 (φ(B 1 ));
(ii)φ(T 1 x)=T 2 φ(x)for everyx∈X′ 1.
For example, it is easily shown that the Bernoulli shiftBp 1 ,...,pr is isomorphic
to the following transformation of the unit square, equipped with Lebesgue measure.
Divide the square intorvertical strips of widthp 1 ,...,pr; then contract the height of
thei-th strip and expand its width so that it has heightpiand width 1; finally combine
these rectangles to form the unit square again by regarding them as horizontal strips
of heightp 1 ,...,pr.(Forr=2andp 1 =p 2 = 1 /2, this transformation of the unit
square is allegedly used by bakers when kneading dough.)
It is easily shown also that isomorphism is an equivalence relation and that it
preserves ergodicity. However, it is usually quite difficult to show that two measure-
preserving transformations are indeed isomorphic. A period of rapid growth was ini-
tiated with the definition by Kolmogorov (1958), and its practical implementation by
Sinai (1959), of a new numerical isomorphism invariant, theentropyof a measure-
preserving transformation. For the formal definition of entropy we refer to the texts on
ergodic theory cited at the end of the chapter. Here we merely state its value for some
of the preceding examples.