Number Theory: An Introduction to Mathematics

5 Recurrence 483

Any translationTaof the torusRd/Zdhas entropy zero, whereas the endomor-
phismRAofRd/Zdhas entropy

∑

i:|λi|> 1

log|λi|,

whereλ 1 ,...,λdare the eigenvalues of the matrixAand the summation is over those
of them which lie outside the unit circle.
The two-sided Bernoulli shiftBp 1 ,...,prhas entropy

−

∑r

j= 1

pjlogpj,

and the entropy of the one-sided Bernoulli shiftB+p 1 ,...,pris given by the same formula.
It follows thatB 1 / 2 , 1 / 2 is not isomorphic toB 1 / 3 , 1 / 3 , 1 / 3 , since the first has entropy
log 2 and the second has entropy log 3. Ornstein (1970) established the remarkable re-
sult that two-sided Bernoulli shifts are completely classified by their entropy:Bp 1 ,...,pr
is isomorphic toBq 1 ,...,qsif and only if

−

∑r

j= 1

pjlogpj=−

∑s

k= 1

qklogqk.

This is no longer true for one-sided Bernoulli shifts. Walters (1973) has shown that
B+p 1 ,...,pris isomorphic toBq+ 1 ,...,qsif and only ifr=sandq 1 ,...,qsis a permutation
ofp 1 ,...,pr.
The Gauss mapTx={x−^1 }has entropyπ^2 /6 log 2. Although it is mixing, it is
not isomorphic to a Bernoulli shift.
Katznelson (1971) showed that any ergodic automorphism of the torusRd/Zdis
isomorphic to a two-sided Bernoulli shift, and Lind (1977) has extended this result to
ergodic automorphisms of any compact abelian group.
Ornstein and Weiss (1973) showed that, ifφtis the geodesic flow on a smooth
(of class C^3 ) compact two-dimensional Riemannian manifold whose curvature at each
point is negative, thenφtis isomorphic to a two-sided Bernoulli shift for everyt>0.
Although, as Hilbert showed, a compact surface of negative curvature cannot be imbed-
ded inR^3 , the geodesic flow on a surface of negative curvature can be realized as the
motion of a particle constrained to move on a surface inR^3 subject to centres of at-
traction and repulsion in the ambient space.The isomorphism with a Bernoulli shift
shows that a deterministic mechanical system can generate a random process. Thus
philosophical objections to ‘Laplacian determinism’ or to ‘God playing dice’ do not
seem to have much point.

5 Recurrence.................................................

It was shown by Poincar ́e (1890) that the paths of a Hamiltonian system of differential
equations almost always return to any neighbourhood, however small, of their initial