Fundamental Statistical Mechanics

10. Lyapunov exponents for a Map
11. Baker's transformation is Ergodic
11.1 Proof
We now have all the tools needed to prove that the baker's transformation is ergodic. In fact it is
possible to prove much stronger properties of the baker's transformation - it is a Bernoulli
process - which implies that it is mixing. A Bernoulli process is one in which it is possible to
establish some kind of isomorphism between the process and a random Markov process. This is
exactly what we did when we showed that the baker's transformation can be mapped onto a
Bernoulli shifts. Here we will give the proof that the transformation is ergodic since this proof is
simple and very illustrative of methods often employed in more complicated cases.
Consider an infinitesimal neighbourhood of a point (x, y). The vertical line through (x, y) is the
stable manifold of that point, and the future images of nearby points on this line approach the
future images of (x, y) as they travel together. On the horizontal line, the unstable manifold, the
future images of points move away from the future images of (x, y). Under time reversal the role
of the x and y directions are interchanged, the stable manifold becomes the unstable manifold
and vice versa. The invariant measure on the unit square is dµ = dxdy = d x ′ d y ′ = d µ ′ .
Now define forward and backward time averages as
B + (Γ) = lim
n→∞
1
n
B − 1
(Γ) = lim
n→∞ n
n−1
∑
j = 0
n−1
∑
j = 0
B(b j (Γ))
B(b − j (Γ))
Step 1 of the proof is to show that the forward and backward time averages are equal almost
everywhere
B + (Γ) = B − (Γ)
11.2 Baker's transformation and Irreversibility
Statistical Mechanics Page 44