Chapter 25
Ergodicity
This lecture explains what it means for a process to be ergodic
or metrically transitive, gives a few characterizes of these proper
ties (especially for AMS processes), and deduces some consequences.
The most important one is that sample averages have deterministic
limits.
25.1
Ergodicity and Metric Transitivity
Definition 300
A dynamical system
Ξ
,
X
, μ, T
is
ergodic
, or
an ergodic system
or
an ergodic process
when
μ
(
C
) = 0
or
μ
(
C
) = 1
for every
T
invariant set
C
.
μ
is called a
T
ergodic measure
, and
T
is called a
μ
ergodic transformation, or
just an
ergodic measure
and
ergodic transformation
, respectively.
Remark:
Most authorities require a
μ
ergodic transformation to also be
measurepreserving for
μ
. But (Corollary 54) measurepreserving transforma
tions are necessarily stationary, and we want to minimize our stationarity as
sumptions. So what most books call “ergodic”, we have to qualify as “stationary
and ergodic”. (Conversely, when other people talk about processes being “sta
tionary and ergodic”, they mean “stationary with only one ergodic component”;
but of that, more later.
Definition 301
A dynamical system is
metrically transitive
,
metrically inde
composable
, or
irreducible
when, for any two sets
A, B
∈
X
, if
μ
(
A
)
, μ
(
B
)
>
0
,
there exists an
n
such that
μ
(
T

n
A
∩
B
)
>
0
.
Remark:
In dynamical systems theory, metric transitivity is contrasted with
topological
transitivity:
T
is topologically transitive on a domain
D
if for any
two open sets
U, V
⊆
D
, the images of
U
and
V
remain in
D
, and there is
an
n
such that
T
n
U
∩
V
=
∅
.
(See, e.g., Devaney (1992).)
The “metric”
in “metric transitivity” refers not to a distance function, but to the fact that
a measure is involved.
Under certain conditions, metric transitivity in fact
167
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CHAPTER 25.
ERGODICITY
168
implies topological transitivity: e.g., if
D
is a subset of a Euclidean space and
μ
has a positive density with respect to Lebesgue measure. The converse is not
generally true, however: there are systems which are transitive topologically but
not metrically.
A dynamical system is
chaotic
if it is topologically transitive, and it contains
dense periodic orbits (Banks
et al.
, 1992). The two facts together imply that a
trajectory can start out arbitrarily close to a periodic orbit, and so remain near
it for some time, only to eventually find itself arbitrarily close to a
di
ff
erent
periodic orbit.
This is the source of the fabled “sensitive dependence on ini
tial conditions”, which paradoxically manifests itself in the fact that all typical
trajectories look pretty much the same, at least in the long run. Since metric
transitivity generally implies topological transitivity, there is a close connection
between ergodicity and chaos; in fact, most of the wellstudied chaotic systems
are also ergodic (Eckmann and Ruelle, 1985), including the logistic map. How
ever, it is possible to be ergodic without being chaotic:
the onedimensional
rotations with irrational shifts are, because there periodic orbits do not exist,
and
a fortiori
are not dense.
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 Spring '06
 Schalizi
 Markov chain, Ergodic theory, invariant set, invariant sets, ergodic invariant measures

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