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−f(x)), if at least one of these

integrals is ﬁnite. In this case f is called µ-summable. If

|f| dµ < +∞ we

say that f is µ-integrable. Let A

∈ A; f is said to be µ-integrable on A if the

function f χ

is µ-integrable on X.

13.2

Analytical mechanics

549

We set

f dµ =

f χ

dµ.

(13.6)

The space of µ-integrable functions on X is denoted by L

(X,

A, µ).

Consider 0 < p < +

∞. The space of functions f such that |f|

is µ-integrable

on X is denoted by L

(X,

A, µ).

A particular and well-known case is that of the functions on R

which are

Lebesgue integrable.

Remark 13.2

Assume that X is a compact metric space and

A is the σ-algebra of Borel

sets X. In this case one can deﬁne the support of a measure µ as the smallest

compact set K

⊂ X such that µ(A) = 0 for all A ⊂ X \ K. Moreover, it is

possible to endow

M(X) with a topological structure by deﬁning at every point

∈ M(X) a basis of neighbourhoods

ϕ,ε

(µ) :=

∈ M(X) |

ϕ dν

−

ϕ dµ

≤ ε ,

(13.7)

where ε > 0 and ϕ : X

→ R is continuous. In this topology a sequence of

measures (µ

)

∈N

⊆ M(X) converges to µ ∈ M(X) if for every ϕ : X → R

continuous we have

ϕ dµ

→

ϕ dµ.

(13.8)

Remark 13.3

In what follows we always assume that X is totally σ-ﬁnite, and hence that X =

∞

i

=1

, where the sets A

∈ A have measure µ(A

) < +

∞ for every i ∈ N.

efinition 13.8 Let X be a set, and A be a σ-algebra of subsets of X. If µ, ν :

A → [0, +∞] are two measures, we say that µ is absolutely continuous with respect

to ν if for every A

∈ A such that ν(A) = 0 we have µ(A) = 0. If µ is not absolutely

continuous with respect to ν then it is said to be singular (with respect to ν).

An important characterisation of measures which are absolutely continuous

with respect to another measure is given by the following theorem. The proof,

that goes beyond the scope of this limited introduction, can be found in the

book of Rudin, already cited.

heorem 13.1 (Radon–Nikodym) A measure µ : A → [0, +∞] is absolutely

continuous with respect to another measure ν :

A → [0, +∞] if and only if there

exists a function ρ : X

→ R, integrable with respect to ν on every subset A ∈ A

such that ν(A) < +

∞, and such that for every A ∈ A we have

µ(A) =

A

ρ dν.

(13.9)

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Analytical mechanics

13.3

The function ρ is unique (if we identify any two functions which only diﬀer

on a set of ν measure zero), and it is called the Radon–Nikodym derivative of

µ with respect to ν, or density of µ with respect to ν, and it is denoted by

ρ = dµ/dν.

Remark 13.4

We have that g

∈ L

(X,

A, µ) if and only if gρ ∈ L

(X,

A, ν). In this case

g dµ =

gρ dν.

(13.10)

13.3

Measurable dynamical systems

The objects of study of ergodic theory are the dynamical systems that preserve

a measure, in a sense that we now make precise.

efinition 13.9 Let (X, A, µ) be a measure space. A transformation S : X → X

is said to be measurable if for every A

∈ A, we have S

−1

(A)

∈ A.

A measurable transformation is non-singular if µ(S

−1

(A)) = 0 for all A

∈ A

such that µ(A) = 0.

Obviously S

−1

(A) =

{x ∈ X | S(x) ∈ A} and S is not necessarily invertible.

For example, if X is a topological space,

A is the σ-algebra of Borel sets and

S is a homeomorphism, then S is measurable and non-singular if and only if the

inverse map is measurable and non-singular.

efinition 13.10 Let (X, A, µ) be a measure space. A measurable non-singular

transformation S : X

→ X preserves the measure (i.e. the measure µ is invariant

with respect to the transformation S) if for every A

∈ A, we have µ(S

−1

(A)) =

µ(A).

If S is invertible with a measurable non-singular inverse and if it preserves the

measure, then clearly µ(S

−1

(A)) = µ(A) = µ(S(A)),

∀A ∈ A.

If however S is not invertible, the following simple example highlights the

need to use the condition µ(S

−1

(A)) = µ(A) in the previous deﬁnition. Choose

X = (0, 1) and the σ-algebra

A of Borel sets on (0, 1); the transformation

S(x) = 2x (mod 1) preserves the Lebesgue measure λ, while if we take an

interval (a, b)

⊂ (0, 1) then λ (S (a, b)) = 2λ ((a, b)).

Remark 13.5

Let f be µ-integrable and assume that S preserves the measure µ. Then

f (x) dµ =

f (S(x)) ˙dµ.

Conversely, if this property holds for every f : X

→ R continuous, then S

preserves the measure µ.

efinition 13.11 A measurable dynamical system (X, A, µ, S) is constituted by

a probability space (X,

A, µ) and by a transformation S : X → X which preserves

13.3

Analytical mechanics

551

the measure µ. The orbit of a point x

∈ X is the inﬁnite sequence of points

x, S(x), S

(x) = S(S(x)), . . . , S

(x) = S(S

(x)), . . . obtained by iterating S.

Remark 13.6

The recurrence theorem of Poincar´

e (Theorem 8.4) can be extended without

diﬃculty to the case of measurable dynamical systems (X,

A, µ, S). We state it,

and leave the proof as an exercise: for every A

∈ A the subset A

of all points

∈ A such that S

(x)

∈ A for inﬁnitely many values of n ∈ N belongs to A and

µ(A) = µ(A

).

A ‘topological’ version of the recurrence theorem of Poincar´

e is presented in

Problem 13.15.

A particularly interesting case arises when X is a subset of R

(or, more

generally, of a diﬀerentiable Riemannian manifold M of dimension l) and µ

is a

probability measure which is absolutely continuous with respect to the Lebesgue

measure

dµ

(x) = ρ(x) dx

(13.11)

(or dµ

= ρ dV

, where dV

=

det(g

) d

x is the volume element associated

with the metric g on the manifold M ). The deﬁnition of a measure that is

invariant with respect to a non-singular transformation S : X

→ X is therefore

equivalent to

−1

(A)

ρ(x) dx =

ρ(x) dx,

(13.12)

for every A

∈ A. A very important problem in ergodic theory is the problem of

determining all measures that are invariant for a given transformation. A case

when this is possible is given by the following systems.

Let X = [0, 1], and S : X

→ X be non-singular. Assume that S is piecewise

monotone and of class

1

, and hence that there exists a ﬁnite or countable

decomposition of the interval [0, 1] into intervals [a

, a

], i

∈ I, on which S is

monotone (and

1

in the interior). On each of these subintervals the inverse S

−1

of S is well deﬁned. Let A = [0, x]. Equation (13.12) becomes

ρ(s) ds =

∈

−1

([0,x])

ρ(s) ds,

from which, by diﬀerentiating with respect to x, we obtain

ρ(x) =

∈

ρ(S

−1

(x))

|S (S

−1

(x))

(13.13)

where

indicates the subset of

I corresponding to the indices i such that

−1

(x) =

∅. Equation (13.13) is therefore a condition (necessary and suﬃcient)

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Analytical mechanics

13.3

for the density ρ for a measure that is absolutely continuous with respect to the

Lebesgue measure to be invariant with respect to S.

Example 13.4 (Ulam and von Neumann 1947)

Consider X = [0, 1], S(x) = 4x(1

− x). The probability measure dµ(x) =

dx/π

x(1

− x) is invariant.

Indeed, to every point x

∈ X there correspond two preimages S

−1

(x) =

−

√

− x) ∈ [0, 1/2] and S

−1

2

(x) =

(1 +

√

− x) ∈ [1/2, 1]. Therefore,

equation (13.13) becomes

x(1

− x)

−1

(x)(1

− S

−1

(x))

|4 − 8S

−1

(x)

−1

(x)(1

− S

−1

(x))

|4 − 8S

−1

(x)

which is immediately veriﬁed.

Example 13.5 : the p-adic transformation

Consider X = [0, 1], and p

∈ N, S(x) = px (mod 1), and hence S(x) = px − m

if m/p

≤ x < (m + 1)/p, m = 0, . . . , p − 1, S(1) = 1. The p-adic transformation

preserves the Lebesgue measure.

Example 13.6 : the Gauss transformation

Consider X = [0, 1), S(x) = 1/x

− [1/x] if x =

/ 0, S(0) = 0, where [

·] denotes the

integer part of a number. The probability measure dµ(x) = dx/(1 + x) log 2 is

invariant. Indeed, S is invertible on the intervals [1/(n + 1), 1/n] , n

∈ N, with

inverse S

−1

(x) = 1/(n + x), and

∞

1 + S

−1

(x)

|S (S

−1

(x))

∞

1 + 1/(n + x)

(n + x)

∞

(n + x + 1)(n + x)

∞

n + x

−

n + x + 1

x + 1

Example 13.7 : the ‘baker’s transformation’

If X = [0, 1]

× [0, 1], then

S(x, y) =

⎧

⎪

⎨

⎪

⎩

2x,

if 0

≤ x <

− 1,

y +

≤ x ≤ 1

13.3

Analytical mechanics

553

(Fig. 13.1) preserves the Lebesgue measure. From a geometrical point of view,

S transforms the square [0, 1]

× [0, 1] in the rectangle [0, 2] × [0,

2

], cuts out the

right half of this rectangle and translates it on top of the left half.

Example 13.8

Let X be a Riemannian manifold, and S : X

→ X be an isometry. The measure

g

is invariant with respect to S.

Example 13.9

Let X = R

, and S : X

→ X be a completely canonical transformation. By the

Liouville theorem, the Lebesgue measure is invariant with respect to S.

Example 13.10 : ‘Arnol’d’s cat’ (Arnol’d and Avez 1967)

If X = T

, then S(x

, x

) = (x

+ x

, x

+ 2x

) (mod 1) preserves the Lebesgue

measure.

Assume that X is a Riemannian manifold and that S : X

→ X is a diﬀeo-

morphism of M such that,

∀ x ∈ X, |det(DS(x))| < 1. Then it can be shown that

there exists an attractor

Ω

⊂ X and a basin of attraction U, i.e. a neighbourhood

Ω

such that S(U )

⊂ U and ∩

≥0

(U ) =

Ω

. In addition, the volume of

Ω

(with respect to the volume form induced by the Riemannian structure on X)

vanishes and all the probability measures that are invariant for S have support

contained in the attractor

Ω

.

Some obvious examples are X = R, S(x) = x/2,

Ω

{0}, U = R, where the

only invariant measure is the Dirac measure δ

(x) at the point x = 0:

(A) =

if 0 /

∈ A,

if 0

∈ A.

2

a

a

a

b

b

b

4

S

S

a

a

a

b

b

b

Fig. 13.1

554

Analytical mechanics

13.4

In general

(A) =

if y /

∈ A,

if y

∈ A.

Another example is given by X = R

, where S(x, y) is the ﬂow at time t = 1

of the following system of ordinary diﬀerential equations:

x = x(1

− x

− y

)

− y,

y = y(1

− x

− y

) + x.

Introducing polar coordinates x = r cos θ, y = r sin θ it is immediate to verify

that ˙r = r(1

− r

2

), ˙

θ = 1, and therefore the circle r = 1 is an attractor. Note

that ˙r > 0 if 0 < r < 1, while ˙r < 0 if 1 < r < +

∞.

In this case

Ω

= S

, U = R

\ {0} and the invariant measure is dµ(r, θ) =

r

=1

dθ/2π. The support of this measure is precisely the limit cycle S

{r = 1}.

13.4

Ergodicity and frequency of visits

Consider a measurable dynamical system (X,

A, µ, S). The ﬁrst fundamental

notion associated with such a system is its ‘statistics’, which is the frequency

with which the orbit

}

∈N

of a point x

∈ X visits a prescribed measurable

set A

∈ A.

To this end, we deﬁne for every n

∈ N the number of visits T (x, A, n) of A

by the orbit of x:

T (x, A, n) :=

−1

x),

(13.14)

where χ

indicates the characteristic function of the set A.

efinition 13.12 We call the frequency of visits ν(x, A) of the set A by the

orbit of x the limit (when it exists)

ν(x, A) =

lim

n

→+∞

T (x, A, n).

(13.15)

The ﬁrst important result for the study of the orbit statistics of a dynamical

system is the existence of the frequency of visits for µ-almost every initial

condition.

13.4

Analytical mechanics

555

T

heorem 13.2 For µ-almost every x ∈ X the frequency of visits ν(x, A) exists.

Proof

Let n

∈ N be ﬁxed, and set

ν(x, A, n) :=

n

T (x, A, n)

(average frequency after n steps),

ν(x, A) := lim sup

→+∞

ν(x, A, n).

Obviously 0

≤ ν(x, A) ≤ 1 and ∀ k ∈ N we have ν(S

x, A) = ν(x, A).

Analogous properties hold for ν(x, A) := lim inf

→+∞

ν(x, A, n).

We want to prove that for µ-almost every x

∈ X, we have

ν(x, A) = ν(x, A).

To this end, introduce ε > 0 and the function

(x, ε) := min

{n ∈ N such that ν(x, A, n) ≥ ν(x, A) − ε}

so that ν(x, A, τ

(x, ε))

≥ ν(x, A)−ε. Suppose that there exists M > 0 such that

(x, ε)

≤ M for every x ∈ X. In this case we can decompose the orbit of x up

to time n, i.e. the ﬁnite sequence (S

0 ≤ j < n

, in parts on each of which the

average frequency of visits of A is at least ν(x, A)

−ε. Indeed, consider the points

0

= x, x

= S

,ε

)

, x

= S

,ε

)

= S

,ε

)+τ

,ε

)

, and so on.

We then have

T (x

, A, τ

, ε)) = τ

, ε)ν(x

, A, τ

, ε))

≥ τ

, ε)(ν(x

, A)

− ε)

= τ

, ε)(ν(x, A)

− ε),

where we have used the previous remark ν(S

x, A) = ν(x, A)

∀ k ∈ N.

For ﬁxed n > M , proceed in this way until a point x

= S

x, with τ

−1

, ε) < n and τ

+ τ

, ε)

≥ n, is reached. We then have

T (x, A, n) =

−1

T (x

, A, τ

, ε)) + T (x

, A, n

− τ

)

≥

⎛

⎝

−1

, ε)

⎞

⎠ (ν(x, A) − ε) = τ

(ν(x, A)

− ε).

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Analytical mechanics

13.4

On the other hand, τ

≥ n − τ

, ε)

≥ n − M, and hence we found that

∀ n > M,

T (x, A, n)

≥ (n − M)(ν(x, A) − ε).

Integrating this inequality over all the set X, since

T (x, A, n) dµ =

−1

x) dµ =

−1

(x) dµ = nµ(A), we ﬁnd

nµ(A)

≥ (n − M)

[ν(x, A)

− ε] dµ;

hence we ﬁnd that

∀ ε > 0,

µ(A)

≥

ν(x, A) dµ

− ε

or µ(A)

≥

ν(x, A) dµ.

It is now possible to repeat this procedure considering ν(x, A) in place of

ν(x, A), deﬁning

A

(x, ε) := min

{n ∈ N such that ν(x, A, n) ≤ ν(x, A) + ε}

and supposing that this is also bounded, to arrive at the conclusion that

ν(x, A) dµ

≥ µ(A) ≥

ν(x, A) ˙dµ.

From this, taking into account that ν and ν are non-negative, it obviously follows

that ν(x, A)

≥ ν(x, A) µ-almost everywhere, and therefore that

ν(x, A) exists µ-almost everywhere.

We still need to consider the case that τ

(or τ

) is not a bounded function.

In this case, remembering the deﬁnition, we choose M > 0 suﬃciently large, so

that we have

µ(

{x ∈ X | τ

(x, ε) > M

}) < ε.

13.4

Analytical mechanics

557

From this choice it follows that

A = A

∪ {x ∈ X | τ

(x, ε) > M

so we have µ(A)

≤ µ(A) + ε. Considering now the number of visits T (x, A, n)

relative to A, setting ν(x, A) as before, the function τ

(x, ε) relative to A is now

bounded. Hence proceeding as above, we arrive at the inequality

µ(A) >

ν(x, A) dµ

− ε,

from which, taking into account that µ(A)

≤ µ(A)+ε and that ν(x, A) ≥ ν(x, A),

we deduce

µ(A)

≥

X

ν(x, A) dµ

− 2ε.

Since ε is arbitrary, µ(A)

≥

ν(x, A) dµ, exactly as in the case that τ

bounded.

The frequency of visits describes the ‘statistics’ of an orbit and can depend

essentially upon it.

Example 13.11 : the square billiard

Consider a point particle of unit mass moving freely in a square of side 2π, and

reﬂected elastically by the walls.

To study the motion, instead of reﬂecting the trajectory when it meets a wall,

we can reﬂect the square with respect to the wall and consider the motion as

undisturbed (note that this argument shows how to extend the motion of the

particle to the case in which the trajectory meets one of the vertices of the

square). In this way each trajectory of the billiard corresponds to a geodesic of

the ﬂat torus (recall the results of Sections I.7 and I.8), and hence we can apply

the results of Section 11.7. In particular we ﬁnd that if α denotes the angle of

incidence (constant) of the trajectory on a wall, then the latter is periodic if tan α

is a rational number, and it is dense on the torus if tan α is irrational. Given any

measurable subset A of the torus T

it is evident that the frequency of visits

of A by one of the billiard’s orbits depends essentially on the initial condition

(s, α). However, it is possible to compute it exactly thanks to Theorem 11.9 and

the following theorem of Lusin (see Rudin 1974, pp. 69–70).

There exists a sequence of continuous functions χ

: T

→ [0, 1] such that

j

→ χ

for j

→ ∞ almost everywhere.

558

Analytical mechanics

13.4

Applying Theorem 11.9 to the sequence χ

we ﬁnd that if tan α is irrational,

we have

ν(s, α, A) = lim

→∞

(s + t cos α, t sin α) dt

= lim

→∞

lim

→∞

(s + t cos α, t sin α) dt

= lim

→∞

2π

, x

) = µ(A),

where µ denotes 1/(2π)

multiplied by the Lebesgue measure on T

Therefore, for almost every initial condition, the frequency of visits of a meas-

urable set A by the corresponding trajectory of the billiard in the square is simply

equal to the measure of A, and is hence independent of the initial condition.

What we have just discussed is a ﬁrst example of an ergodic system.

efinition 13.13 A measurable dynamical system (X, A, µ, S) is called ergodic

if for every choice of A

∈ A it holds that ν(x, A) = µ(A) for µ-almost every

∈ X.

We now turn to the study of ergodic systems and their properties.

We start with a remark: consider A

∈ A and let χ

be its characteristic

function. Since µ(A) =

(x) dµ, the ergodicity is equivalent to the statement

that

∀ A ∈ A and for µ-almost every x ∈ X one has

lim

→∞

−1

x) =

(x) dµ.

(13.16)

If instead of the characteristic function of a set we consider arbitrary integ-

rable functions f

∈ L

1

(X,

A, µ), the following corresponding generalisation of

Theorem 13.2 is called Birkhoﬀ ’s theorem.

heorem 13.3 Let (X, A, µ, S) be a measurable dynamical system, and let f ∈

(X,

A, µ). For µ-almost every x ∈ X the limit

f (x) := lim

→∞

−1

f (S

(13.17)

exists and it is called the time average of f along the orbit of the point

∈ X.

For the proof of this theorem see Gallavotti (1981) and Cornfeld et al. (1982).

We remark however that from the proof of Theorem 4.1 it follows in fact that

the time average exists whenever f is a ﬁnite linear combination of characteristic

functions of measurable sets (hence every time that f is a simple function):

f =

m

k

∈ R, A

∈ A, ∀ j = 1, . . . , m.

(13.18)

13.4

Analytical mechanics

559

Recall that every function f

∈ L

(X,

A, µ) is the limit a.e. of a sequence of simple

functions.

Remark 13.7

It is obvious that ˆ

f (Sx) = ˆ

f (x), and hence that the time average depends

on the orbit and not on the initial point chosen along the orbit. In addition,

since µ is S-invariant, by Remark 13.5 we have

f (x) dµ =

f (Sx) dµ,

from which it follows, by an application of the theorem of Lebesgue on dominated

convergence to (13.17):

µ

:=

f (x) dµ =

ˆ

f (x) dµ.

(13.19)

The quantity f

=

X

f dµ is called the phase average of f (or expectation of

f ) and equation (13.19) implies that f and its time average ˆ

f have the same

expectation value.

The ergodicity of a dynamical system has as the important consequence that

the phase and time averages are equal almost everywhere, as the following theorem

shows (see property 4).

heorem 13.4 Let (X, A, µ, S) be a measurable dynamical system. The following

properties are equivalent.

(1) The system is ergodic.

(2) The system is metrically indecomposable: every invariant set A

∈ A (i.e.

every set such that S

−1

(A) = A) has measure µ(A) either zero or equal to

µ(X) = 1.

(3) If f

∈ L

(X,

A, µ) is invariant (i.e. f ◦ S = f µ-almost everywhere) then f is

constant µ-almost everywhere.

(4) If f

∈ L

(X,

A, µ) then f

= ˆ

f (x) for µ-almost every x

∈ X.

(5)

∀ A, B ∈ A then

lim

n

→+∞

−1

µ(S

−j

(A)

∩ B) = µ(A)µ(B).

(13.20)

Proof

(1)

⇒ (2) Suppose that there exists an invariant set A ∈ A with measure

µ(A) > 0. Since A is invariant for every choice of x

∈ A the frequency of visits

of A is precisely ν(x, A) = 1. But since the system is ergodic for µ-almost every

x, then ν(x, A) = µ(A). The hypothesis that µ(A) > 0 then yields µ(A) = 1.

(2)

⇒ (3) If f ∈ L

(X,

A, µ) is invariant, for every choice of γ ∈ R the

set A

{x ∈ X | f(x) ≤ γ} is invariant. Since the system is metrically

indecomposable it follows that either µ(A

) = 0 or µ(A

) = 1. On the other

560

Analytical mechanics

13.4

hand, if γ

1

< γ

clearly A

⊂ A

. Therefore setting γ

= inf

{γ ∈ R | µ(A

) = 1

}

it follows that f (x) = γ

for µ-almost every x.

(3)

⇒ (4) Since the time average ˆ

f is invariant we have that ˆ

f is constant

µ-almost everywhere. From equation (13.19) it then follows that ˆ

f (x) = f

for

µ-almost every x

∈ X.

(4)

⇒ (1) It suﬃces to apply hypothesis (4) to the characteristic function χ

of the set A.

(4)

⇒ (5) Let f = χ

. For µ-a.e. x

∈ X we have

µ(A) =

dµ = ˆ

(x) = lim

→∞

−1

(x)).

By the dominated convergence theorem, we have

µ(A)µ(B) =

lim

→∞

−1

(x))χ

(x) dµ

= lim

→∞

−1

(x))χ

(x) dµ

= lim

→∞

−1

µ(S

−j

(A)

∩ B).

(5)

⇒ (2) Let A be invariant. Setting B = A

we have by (5) that

µ(A)µ(A

) = lim

→∞

−1

µ(S

−j

(A)

∩ A

) = 0

because of the invariance of A. Hence µ(A) = 0 or µ(A

) = 0.

In general a dynamical system has more than just one invariant measure. For

example if it has a periodic orbit

i

}

, x

= S(x

) i = 0, . . . , n

−1, x

= S(x

the measure

µ(x) =

(x)

(13.21)

is invariant, where δ

(x) denotes the Dirac measure at the point y:

(A) =

if y /

∈ A,

if y

∈ A.

(13.22)

Given that a system often has many periodic orbits it follows that it also has

many distinct invariant measures, and (13.21) clearly implies that they are not

absolutely continuous with respect to one another. For ergodic transformations,

the distinct invariant measures are necessarily singular.

heorem 13.5 Assume that (X, A, µ, S) is ergodic and that µ

A → [0, 1]

is another S-invariant probability measure. The following statements are then

13.4

Analytical mechanics

561

equivalent:

(1) µ

/ µ;

(2) µ

is not absolutely continuous with respect to µ;

(3) there exists an invariant set A

∈ A such that µ(A) = 0 and µ

(A) =

/ 0.

Proof

(1)

⇒ (2) If µ

were absolutely continuous with respect to µ, the Radon–

Nykodim derivative dµ

/dµ would be an invariant function in L

(X,

A, µ). Since

the system (X,

A, µ, S) is ergodic it follows that (dµ

/dµ)(x) is constant µ-

almost everywhere and therefore it is necessarily equal to 1 as both µ and µ

are probability measures. It follows that µ

= µ, a contradiction.

(2)

⇒ (3) Since µ

is not absolutely continuous with respect to µ there exists

∈ A such that µ(B) = 0, while µ

(B) =

/ 0. Setting A =

∞

(B), it is

immediate to verify that A

∈ A, µ(A) = 0, µ

(A) =

/ 0.

(3)

⇒ (1) Obvious.

Suppose that X is a compact metric space and

A is the σ-algebra of the Borel

sets of X.

In some exceptional cases, a dynamical system (X,

A, µ, S) can have a unique

invariant measure. In this case the system is called uniquely ergodic. This has

the following motivation.

heorem 13.6 Let (X, A, µ, S) be a uniquely ergodic system. Then the system

is ergodic and for every choice of f : X

→ R continuous and x ∈ X the sequence

(1/n)

n

−1

f (S

(x)) converges uniformly to a constant that is independent of x.

Therefore the time average exists for every x

∈ X and has value

f dµ.

Proof

If the system were metrically decomposable, there would exist an invariant subset

⊂ X such that 0 < µ(A) < 1. The measure dν = χ

(dµ/µ(A)) is an invariant

probability measure distinct from µ: µ(A

) = 1

− µ(A) while ν(A

) = 0. This

contradicts the hypothesis that the system is uniquely ergodic, and hence the

system cannot be metrically decomposable.

Suppose then that there exists a continuous function f : X

→ R for which

the sequence of functions

(1/n)

−1

◦ S

∈N

does not converge uniformly

f dµ (because of ergodicity, this is the limit of the sequence µ-almost

everywhere). There then exist ε > 0, and two sequences (n

)

i

∈N

⊂ N, n

→ ∞

and (x

)

∈N

⊂ X such that for every i ∈ N we have

−1

f (S

))

−

f dµ

≥ ε.

(13.23)

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Analytical mechanics

13.4

Consider the sequence of probability measures on X:

−1

)

(13.24)

By the compactness of the space of probability measures on X (see Problem 1

of Section 13.13 for a proof) there is no loss of generality in assuming that the

sequence ν

converges to a probability measure ν. We show that ν is invariant; to

this end, thanks to Remark 13.5, it is suﬃcient to show that for every continuous

g : X

→ R we have

g(S(x)) dν =

g(x) dν. On the other hand

g(S(x)) dν = lim

→∞ X

g(S(x)) dν

= lim

→∞

−1

g(S

))

= lim

→∞

g(x) dν

−

g(x

) +

g(S

)) .

(13.25)

Since X is compact and g is continuous, the second and third terms of the sum

in (13.25) have limits of zero. It follows that the measure ν is invariant. Recalling

equations (13.23) and (13.24) we have

f dν

−

f dµ = lim

→∞

f dν

−

f dµ

= lim

→∞

−1

f (S

))

−

f dµ

≥ ε,

which shows that ν =

/ µ and contradicts the hypothesis that µ is the only

invariant measure of the system.

Remark 13.8

It is not diﬃcult to prove that if for every continuous function f : X

→ R the

limit lim

→∞

(1/n)

−1

f (S

(x)) exists for every ﬁxed point x, independently

of x, then the system is uniquely ergodic.

Example 13.12

Let ω

∈ R

be such that ω

· k + p = 0 for every p ∈ Z and for every k ∈

\{0}. Consider the measurable dynamical system determined by X = T

is the σ-algebra of Borel sets on T

, χ

∈ T

, dµ(χ) = 1/(2π)

χ is the Haar

measure on T

and Sχ = χ + ω (mod 2πZ

). Theorem 11.9 guarantees that

the time average exists

∀χ ∈ T

and it is independent of the choice of the

initial point χ

∈ T

. Therefore, by the previous remark, the system is uniquely

ergodic.

13.5

Analytical mechanics

563

13.5

Mixing

One of the equivalent characterisations of ergodicity for a measurable dynamical

system (X,

A, µ, S) is the fact that on average the measure of the preimages

−j

(A) of any set A

∈ A is distributed uniformly on the whole support of the

measure µ in the sense described by (13.20) (see Theorem 13.4). However, the

existence of the limit in (13.20) does not guarantee that the limit of the sequence

µ(S

−j

(A)

∩ B) exists, but it guarantees that if this sequence converges, then its

limit is µ(A)µ(B).

It is therefore natural to consider the dynamical systems satisfying the following

deﬁnition.

efinition 13.14 A measurable dynamical system (X, A, µ, S) is mixing if

∀ A, B ∈ A one has

lim

→+∞

µ(S

−n

(A)

∩ B) = µ(A)µ(B).

(13.26)

Since equation (13.26) implies (13.20) every mixing system is ergodic. An

independent veriﬁcation of this fact can be obtained assuming that A is invariant,

in which case from (13.26) it follows that

µ(A)µ(A

) = lim

→∞

µ(S

−n

(A)

∩ A

) = µ(A

∩ A

) = 0,

and therefore either µ(A) = 0 or µ(A

) = 0, and the system is metrically

indecomposable.

The converse is false: the irrational translations on tori (see Example 13.12)

are uniquely ergodic but not a mixing (see Problem 9 of Section 13.12). A

simple example of a mixing dynamical system is given by the so-called ‘baker’s

transformation’ of Example 3.4, as we shall see below (see Problem 2 of Section

13.13).

Just as ergodicity has an equivalent formulation in terms of the behaviour of

the time average of integrable functions, mixing can be characterised by studying

the functions f : X

→ R which are measurable and square integrable.

efinition 13.15 Let the measurable dynamical system (X, A, µ, S) be given.

The linear operator U

: L

(X,

A, µ) → L

(X,

A, S) deﬁned by

f = f

◦ S

(13.27)

is called Koopman’s operator.

Recalling the deﬁnition of the scalar product of two functions f, g

∈ L

(X,

A, µ):

f, g :=

f g dµ,

(13.28)

Recall that in a probability space (

X, A, µ, S), two sets (or events) A, B ∈ A are

independent if

µ(A ∩ B) = µ(A)µ(B).

564

Analytical mechanics

13.5

it is immediate to verify that since S preserves the measure µ, U

is an isometry:

S

f, U

g = f, g ,

∀ f, g ∈ L

(X,

A, µ).

(13.29)

heorem 13.7 A necessary and suﬃcient condition for the measurable dynamical

system (X,

A, µ, S) to be mixing is that

lim

n

→∞

f, g = f, 1

1, g

(13.30)

for every f, g

∈ L

2

(X,

A, µ).

Remark 13.9

The quantity

f, g

− f, 1 1, g =

◦ S

g dµ

−

f dµ

g dµ

is also called the correlation between f and g at time n. Theorem 13.7 therefore

states that a system is mixing if and only if the correlation between any two

functions tends to zero as n

→ ∞.

Proof of Theorem 13.7

It is immediate to verify that (13.30) implies that the system is mixing; it is

enough to apply it to f = χ

and g = χ

, A, B

∈ A.

Conversely, assuming that (X,

A, µ, S) is mixing, then (13.26) implies that

equation (13.30) holds when f and g are two characteristic functions of sets

belonging to

A. By linearity, we therefore ﬁnd that equation (13.30) is valid

when f and g are two simple functions.

Recall now that simple functions are dense in L

(X,

A, µ) (see Rudin 1974);

hence it follows that

∀ f, g ∈ L

(X,

A, µ) and ∀ ε > 0 there exist two simple

functions f

, g

∈ L

(X,

A, µ) such that

− f

, f

− f

≤ ε

− g

, g

− g

≤ ε

lim

→∞

, g

= f

, 1

1, g

Writing

f, g = U

, g

+ U

f, g

− g

+ U

− f

), g

since U

is an isometry, using the Schwarz inequality

| f, g | ≥ f g one has

| U

f, g

− f, 1 1, g | ≤ | U

, g

− f

, 1

1, g

+ f

− g

+ f

− f

+ ε f + ε g

13.6

Analytical mechanics

565

There then exists a constant c > 0 such that if n is suﬃciently large

| U

f, g

− f, 1 1, g | ≤ cε,

and hence (13.30) follows.

Example 13.13 : linear automorphisms of the torus T

Consider the ﬂat two-dimensional torus with the σ-algebra of Borel sets, and the

Haar measure dµ (χ) = (1/4π

) dχ

dχ

. A linear automorphism of the torus is

given by

S(χ

, χ

) = (aχ

+ bχ

, cχ

+ dχ

) mod 2πZ

,

(13.31)

where a, b, c, d

∈ Z and |ad − bc| = 1. It is easy to verify that the Haar measure

is S-invariant.

We now prove that if the matrix σ =

b

c

has no eigenvalue with unit

modulus, then the system is mixing.

To this end, we check that (13.30) is satisﬁed by the functions f

(χ) = e

ik·χ

, k

∈

, which form a basis of L

). We want to show therefore that for every pair

k, k

∈ Z

we have

lim

→∞ T

(χ))f

(χ) dµ(χ) =

2

f

χ dµ(χ)

χ dµ(χ).

(13.32)

If k = k = 0 the two sides are constant and equal to 1 for every n

∈ N. It

is not restrictive to assume that k = 0 which yields immediately that the right-

hand side is equal to 0. On the other hand, we have f

(S(χ)) = f

(χ) hence

(χ)) = f

(σ

)

(χ) and since σ has an eigenvalue with absolute value > 1

the norm is

|(σ

)

| → ∞.

It follows that if n is suﬃciently large we necessarily

have (σ

T

)

k = k and as the basis (f

)

k∈Z

is orthonormal, then the left-hand

side of (13.32) vanishes. This concludes the proof that the system is mixing.

13.6

Entropy

Let (X,

A, µ, S) be a measurable dynamical system. Ergodicity and mixing give

two qualitative indications of the degree of randomness (or stochasticity) of the

system. An indication of quantitative type is given by the notion of entropy

which we shall soon introduce.

We start by considering the following situation. Let α be an experiment with

∈ N possible mutually exclusive outcomes A

, . . . , A

(for example the toss of

Since

σ transforms the vectors of Z

into vectors of

and is invertible, no non-zero

vector with integer components can be entirely contained in the eigenspace corresponding to

the eigenvalue less than 1, because this would imply that by iterating

σ a ﬁnite number of

times the vector has norm less than 1, contradicting the hypothesis that it belongs to

2

.

566

Analytical mechanics

13.6

a coin m = 2 or the roll of a die m = 6). Assume that each outcome A

happens

with probability p

∈ [0, 1]:

= 1.

In a probability space (X,

A, µ, S) this situation is described by assigning a

ﬁnite partition of X = A

∪ . . . ∪ A

(mod 0), A

∈ A, A

∩ A

∅ if i =

/ j,

µ(A

) = p

The following deﬁnition describes the properties which must hold for a function

measuring the uncertainty of the prediction of an outcome of the experiment

(equivalently, the information acquired from the execution of the experiment α).

Let

∆

(m)

be the (m

− 1)-dimensional standard symplex of R

, given by

∆

(m)

, . . . , x

)

∈ R

| x

∈ [0, 1],

i

=1

= 1

efinition 13.16 A family of continuous functions H

(m)

∆

(m)

→ [0, +∞],

where m

∈ N, is called an entropy if the following properties hold:

(1) symmetry:

∀ i, j ∈ {1, . . . , m} we have

(m)

, . . . , p

) = H(p

, . . . , p

);

(2) H

(m)

(1, 0, . . . , 0) = 0;

(3) H

(m)

(0, p

, . . . , p

) = H

(m−1)

, . . . , p

∀ m ≥ 2, ∀ (p

, . . . , p

)

∈

∆

(m−1)

;

(4)

∀ (p

, . . . , p

)

∈

∆

(m)

we have H

(m)

, . . . , p

)

≤ H

(m)

(1/m, . . . , 1/m) and

the equality holds if and only if p

= 1/m for every i = 1, . . . , m;

(5) consider (π

, . . . , π

, π

, . . . , π

)

∈

∆

(ml)

; then for every

, . . . , p

)

∈

∆

(m)

we have

(ml)

(π

, . . . , π

, π

, . . . , π

)

= H

(m)

, . . . , p

) +

(l)

, . . . ,

Property (2) expresses the absence of uncertainty of a certain event. Property

(3) means that no information is gained by impossible outcomes and (4) means

that the maximal uncertainty is attained when all outcomes are equally probable.

Property (5) describes the behaviour of the entropy when distinct experiments

are compared. Let β be another experiment with possible outcomes B

, . . . , B

(i.e. another partition of (X,

A, µ, S)). Let π

be the probability of A

and B

together. The probability of B

conditional on the fact that the outcome of α

is A

is prob (B

| A

) = π

(= µ(A

∩ B

)). Clearly the uncertainty in the

prediction of the outcome of the experiment β when the outcome of α is A

measured by H

(l)

(π

, . . . , π

). From this fact stems the requirement that

(5) be satisﬁed. In the following, we use the simpler notation H(p

, . . . , p

heorem 13.8 The function

H(p

, . . . , p

) =

−

log p

(13.33)

13.6

Analytical mechanics

567

(with the convention 0 log 0 = 0) is, up to a constant positive multiplier, the only

function satisfying (1)–(5).

Proof (see Khinchin 1957, pp. 10–13).

Let H(p

1

, . . . , p

) be an entropy function, and for any m set K(m) =

H(1/m, . . . , 1/m). We show ﬁrst of all that K(m) = +c log m, where c is a

positive constant.

Properties (3) and (4) imply that K is a non-decreasing function. Indeed,

K(m) = H

, . . . ,

≤ H

m + 1

, . . . ,

m + 1

= K(m + 1).

Consider now any two positive integers m and l. The property (5) applied to

the case π

≡ 1/ml, p

≡ 1/m yields

K(lm) = K(m) +

K(l) = K(m) + K(l),

from which it follows that

K(l

m

) = mK(l).

Given any three integers r, n, l let m be such that l

≤ r

≤ l

, i.e.

≤

log r

log l

≤

We know that

mK(l) = K(l

)

≤ K(r

) = nK(r)

≤ K(l

) = (m + 1)K(l),

from which it follows that

n

≤

K(r)

K(l)

≤

i.e.

K(r)

K(l)

−

log r

log l

≤

Because of the arbitrariness of n we deduce that K(r)/ log r = K(l)/ log l and

therefore K(m) = c log m, c > 0.

Assume now that p

, . . . , p

are rational numbers. Setting the least common

multiple of the denominators equal to s, we have p

= r

/s, with

= s. In

addition to the experiment α with outcomes A

, . . . , A

with respective probabil-

ities p

1

, . . . , p

we consider an experiment β constituted by s outcomes B

, . . . , B

divided into m groups, each containing, respectively, r

, . . . , r

outcomes.

We now set π

= p

= 1/s, i = 1, . . . , m, j = 1, . . . , r

Given any outcome A

of α, we therefore have that the outcome β is the

outcome of an experiment with r

equally probable outcomes, and hence

, . . . ,

= c log r

568

Analytical mechanics

13.6

and

, . . . ,

= c

log r

= c

log p

+ c log s.

On the other hand, H(π

, . . . , π

) = c log s and by property (5) we have

H(p

, . . . , p

) = H(π

, . . . , π

m

)

−

, . . . ,

−c

log p

The continuity of H ensures that the formula (13.33), proved so far when p

∈ Q,

is also valid when p

is a real number.

Remark 13.10

H can be characterised as the (

−1/N) × logarithm of the probability of a ‘typical’

outcome of the experiment α repeated N times. Indeed, if N is large, repeating

the experiment α N times one expects to observe each outcome A

approximately

N times (this is a formulation of the so-called law of large numbers).

The probability of a typical outcome containing p

N times A

, p

N times A

,

etc. is therefore

. . . p

From this it follows precisely that

H(p

, . . . , p

) =

−

log p

. . . p

−

log p

Remark 13.11

The maximum value of H is attained when p

= 1/m, i = 1, . . . , m (as required

by property (4)) and has value H (1/m, . . . , 1/m) = log m.

We now consider how to extend the notion of entropy to measurable dynamical

systems (X,

A, µ, S).

We introduce some notation. If α and β are two partitions of

A, the joined

partition α

∨ β of α and β is deﬁned by the subsets {A ∩ B, A ∈ α, B ∈ β}. If

1

, . . . , α

are partitions, we write

i

=1

for the joined partition of α

, . . . , α

.

If S is measurable and non-singular, and α is a partition, S

−1

α is the partition

deﬁned by the subsets

−1

A, A

∈ α}. Finally, we say that a partition β is ﬁner

than α, which we denote by α < β, if

∀ B ∈ β there exists A ∈ α such that B ⊂ A.

Obviously, the joined partitions are ﬁner than the starting ones. The entropy

H(α) of a partition α =

1

, . . . , A

} is given by H(α) = −

i

=1

µ(A

) log µ(A

13.6

Analytical mechanics

569

D

efinition 13.17 Let (X, A, µ, S) be a measurable dynamical system, and let α

be a partition. The entropy of S relative to the partition α is deﬁned by

h(S, α) := lim

→∞

1

−1

−i

(13.34)

The entropy of S is

h(S) := sup

{h(S, α), α is a ﬁnite partition of X}.

(13.35)

Remark 13.12

It is possible to prove, exploiting the strict convexity of the function x log x

on R

+

, that the limit (13.34) exists. Indeed, the sequence (1/n)H

∨

−1

−i

is monotone non-increasing and non-negative. Hence h(S, α)

≥ 0 for every α.

Remark 13.13

The entropy of a partition α measures the quantity of information acquired by

observing the system using an instrument that distinguishes between the points

of X with the resolution given by the sets of the partition

, . . . , A

} = α.

For x

∈ X, consider the orbit of x up to time n − 1:

x, Sx, S

x, . . . , S

−1

Since α is a partition of X, the points S

x, 0

≤ i ≤ n − 1, belong to precisely

one of the sets of the partition α: setting x

= x, x

= S

x, we have x

∈ A

with k

∈ {1, . . . , m} for every i = 0, . . . , n − 1.

∨

n

−1

−i

measures the quantity of information deduced from the know-

ledge of the distribution with respect to the partition α of a segment of ‘duration’

n of the orbit. Therefore (1/n)H

∨

−1

−i

α is the average quantity of inform-

ation per unit of time and h(S, α) is the the quantity of information acquired

(asymptotically) at each iteration of the dynamical system, knowing how the

orbit of a point is distributed with respect to the partition α.

This remark is made rigorous by the following theorem. The proof can be

found in Ma˜

ne (1987).

heorem 13.9 (Shannon–Breiman–McMillan) Let (X, A, µ, S) be a measurable

ergodic dynamical system, and α be a ﬁnite partition of X. Given x

∈ X let

(x) be the element of

−1

−i

α which contains x. Then for µ-almost every

∈ X we have

h(S, α) = lim

→∞

−

log µ(α

(x)).

(13.36)

570

Analytical mechanics

13.6

An interpretation of the Shannon–Breiman–McMillan theorem is the following.

For an ergodic system there exists a number h such that

∀ ε > 0, if α is a

suﬃciently ﬁne partition of X, then there exists a positive integer N such that

for every n

≥ N there exists a subset X

of X of measure µ(X

) > 1

−ε made of

approximately e

elements of

−1

−i

α, each of measure approximately e

−nh

If X is a compact metric space and

A is the σ-algebra of Borel sets X, Brin

and Katok (1983) have given an interesting topological version of the Shannon–

Breiman–McMillan theorem. Let B(x, ε) be the ball of centre x

∈ X and radius

ε. Assume that S : X

→ X is continuous and preserves the probability measure

µ :

A → [0, 1]. Consider

B(x, ε, n) :=

{y ∈ X | d(S

x, S

≤ ε for every i = 0, . . . , n − 1},

i.e. B(x, ε, n) is the set of points y

∈ X whose orbit remains at a distance less

than ε from that of x for at least n

− 1 iterations. It is possible to prove the

following.

heorem 13.10 (Brin–Katok) Assume that (X, A, µ, S) is ergodic. For µ-almost

every x

∈ X we have

sup

ε>

0

lim sup

→∞

−

log µ(B(x, ε, n)) = h(S).

(13.37)

An interesting corollary of the previous theorem is that the entropy of the

translations over tori T

is zero. Indeed, in this case d(Sx, Sy) = d(x, y) and

therefore

∀ n ∈ N and ∀ ε > 0 we have B(x, ε, n) = B(x, ε), from which it

follows that h(S) = 0.

The same is true, more generally, if S is an isometry of the metric space (X, d).

The notion of entropy allows one to distinguish between systems in terms of

the ‘predictability’ of their observables. When the entropy is positive, at least

part of the observables cannot be computed from the knowledge of the past

history.

Chaotic systems are therefore the systems that have positive entropy. Taking

into account the Brin–Katok theorem and the recurrence theorem of Poincar´

e, one

sees how in chaotic systems the orbits are subject to two constraints, apparently

contradicting each other. On the one hand, almost every orbit is recurrent and in

the future will pass inﬁnitely many times near the starting-point. On the other

hand, the probability that two orbits remain close for a given time interval n

decays exponentially as n grows.

Since two orbits, that were originally close to each other, must return inﬁnitely

many times near the starting-point, they must be entirely uncorrelated if the

entropy is positive, and hence they must go far and come back in diﬀerent times.

This complexity of motions is called chaos, and it clearly shows how diﬃcult

(or impossible) it is to compute the future values of an observable (corresponding

to a function f : X

→ R) simply from the knowledge of its past history.

13.7

Analytical mechanics

571

13.7

Computation of the entropy. Bernoulli schemes. Isomorphism

of dynamical systems

In the deﬁnition of entropy h(S) of a measurable dynamical system, it is necessary

to compute the supremum of h(S, α) as α varies among all ﬁnite partitions of

X. This seems to exclude the practical possibility of computing h(S). In reality,

this is not the case, and one can proceed in a much simpler way.

In this section we identify a partition α with the σ-algebra generated by α, and

∞

−i

α with the smallest σ-algebra containing all the partitions

−1

−i

for every n

∈ N.

Recall that two σ-algebras

A and B are equal (mod 0), denoted by A = B (mod

0), if

∀ A ∈ A there exists B ∈ B such that µ(A

∆

B) = 0, and vice versa.

The discovery of Kolmogorov and Sinai, which makes possible the computation

of the entropy overcoming the need to compute the supremum in (13.35), is that

it suﬃces to consider the ﬁnite partitions α that generate the σ-algebra

A, and

hence such that

+∞

−∞

−i

α =

A (mod 0) if S is invertible, or

∞

−i

α =

A (mod

0) if S is not invertible. Indeed one can prove the following.

heorem 13.11 (Kolmogorov–Sinai) If α is a partition of X generating the

σ-algebra

A, the entropy of the measurable dynamical system (X, A, µ, S) is

given by

h(S) = h(S, α).

(13.38)

The proof of this theorem does not present special diﬃculties but it is tedious

and will be omitted (see Ma˜

ne 1987).

Among the measurable dynamical systems for which it is possible to compute

the entropy, the Bernoulli schemes, which we now introduce, constitute the

fundamental example of systems with strong stochastic properties.

Consider the space X of inﬁnite sequences x = (x

)

i

∈N

, where the variable x

can only take a ﬁnite number of values which, for simplicity, we assume to be

the integers

{0, . . . , N − 1} (we sometimes use the notation Z

to denote the

integers

{0, 1, . . . , N − 1}). The space of sequences X is often denoted by Z

N

.

When we want to model an inﬁnite sequence of outcomes of the toss of a coin

(or the roll of a die) we ﬁx N = 2 (respectively N = 6) and each possible value

of x

is equally probable.

Consider on X the transformation S : X

→ X deﬁned by

(S(x))

i

= x

∀ i ∈ N,

(13.39)

usually known as a shift.

If instead of one-sided sequences (

)

i∈N

∈ Z

the space

X is made of two-sided

(doubly inﬁnite) sequences (

i

)

i∈Z

∈ Z

we have a so-called bilateral Bernoulli scheme. All

considerations to be developed trivially extend to the case of bilateral Bernoulli schemes.

572

Analytical mechanics

13.7

We proceed as in Example 13.2, associating with Z

a probability measure and

assigning to the value j

∈ Z

a probability equal to p

> 0, with the condition

−1

= 1.

This choice induces a probability measure on the space of sequences X that

we now describe.

Consider ﬁrst of all the σ-algebras

A on X generated by the cylinders, i.e.

the subsets of X corresponding to sequences for which a ﬁnite number of values

is ﬁxed. Given k

≥ 1 elements j

, . . . , j

∈ Z

, not necessarily distinct, and k

distinct positions i

1

< i

2

< . . . < i

∈ N, the corresponding cylinder is

C = C

, . . . , j

, . . . , i

=

{x ∈ X | x

= j

, x

= j

, . . . , x

k

= j

(13.40)

Therefore all sequences in X which take the prescribed values in the positions

corresponding to the indices i

, . . . , i

belong to C.

We therefore deﬁne the measure µ on

A by prescribing its value on cylinders:

C

j

, . . . , j

i

1

, . . . , i

= p

. . . p

(13.41)

Note that in (13.41) the positions i

, . . . , i

do not play any role. Hence it is

immediate to deduce that if C is a cylinder, then µ(S

−1

(C)) = µ(C), and recalling

that the σ-algebra

A is generated by cylinders we conclude that (X, A, µ, S) is

a measurable dynamical system (hence that S preserves the measure µ). This

system is known as a Bernoulli scheme with probability (p

, . . . , p

−1

) and it is

denoted by SB(p

, . . . , p

−1

We leave as an exercise the veriﬁcation that a Bernoulli scheme is mixing (see

Problem 10 of Section 13.12) but we show that the entropy of SB(p

, . . . , p

−1

)

−

−1

log p

The partition α into the cylinders

=0,...,N −1

generates the σ-algebra

A. Indeed we have

∨ S

−1

α =

, j

=0,...,N −1

∨ S

−1

∨ S

−2

α =

= 0, . . . , N − 1

= 0, 1, 2

13.7

Analytical mechanics

573

and so on. The corresponding entropies are (use (13.41)):

H(α) =

−

−1

log p

H(α

∨ S

−1

α) =

−

−1

log p

−

−1

log p

)

−1

−

−1

log p

)

−1

−2

−1

log p

H(α

∨ S

−1

∨ S

−2

) =

−

log p

−3

−1

log p

and so on. From this it follows that h(S, α) =

−

−1

log p

and thus the

entropy of SB(p

, . . . , p

−1

) also follows by the Kolmogorov–Sinai theorem.

We examine again the p-adic transformation S of Example 13.5 and consider

the partition α =

{(j/p, (j + 1)/p)}

=0,...,p−1

. Using the fact that

∨

−i

α =

{(j/(p

), (j + 1)/(p

))

}

=0,...,p

n+1

−1

it is not diﬃcult to verify that α is a

generating partition and therefore h(S) = h(S, α).

On the other hand,

n

i

−i

= p

· p

−(n+1)

log p

−(n+1)

−(n + 1) log p,

from which it follows that h(S, α) = log p.

Note that SB (1/p, . . . , 1/p) has the same entropy. It is indeed possible to pass

from one system to the other by a very easy construction.

With every point ξ

∈ (0, 1) we associate the sequence x ∈ Z

deﬁned as

follows: for every i = 0, 1, . . . we set

i

= j

⇔ S

(ξ)

∈

j + 1

(13.42)

Denote by (X,

A, µ, S) and by (X , A , µ , S ), respectively, the two 4-tuples: ((0,1),

σ-algebra of Borel sets of (0,1), Lebesgue measure, p-adic transformation) and

, the σ-algebra generated by the cylinders, the measure corresponding to

SB (1/p, . . . , 1/p), and the shift).

In addition, denote by T : X

→ X the transformation deﬁned in (13.42).

574

Analytical mechanics

13.7

The following facts are of immediate veriﬁcation:

(a) T is measurable;

(b)

∀ A ∈ A , µ(T

−1

A ) = µ (A );

∈ X, T (S(x)) = S (T (x));

(d) T is invertible (mod 0), i.e. there exists a measurable transformation T :

→ X, which preserves the measures (so that ∀ A ∈ A, µ (T

−1

A) = µ(A)),

such that T (T (x)) = x for µ-a.e. x

∈ X and T (T (x )) = x for µ -a.e. x ∈ X .

In general, we have the following.

efinition 13.18 Let (X, A, µ, S), (X , A , µ , S ) be two measurable dynamical

systems. A transformation T : X

→ X satisfying the conditions (a), (b), (c), (d)

is called an isomorphism of measurable dynamical systems and the two systems

are then isomorphic.

Ergodic theory does not distinguish between isomorphic systems: two iso-

morphic systems have the same ‘stochastic’ properties.

It is an exercise to prove the following.

heorem 13.12 Two isomorphic systems have the same entropy. If one system

is mixing, then the other is also mixing. If one system is ergodic, then the other

is also ergodic.

In the particular case of the Bernoulli schemes the equality of entropy is

not only a necessary condition but it is also suﬃcient for two schemes to be

isomorphic.

heorem 13.13 (Ornstein) Two Bernoulli schemes with the same entropy are

isomorphic.

The proof of this result goes beyond the scope of this book. Besides the original

article of Ornstein (1970), see also Cornfeld et al. (1982, section 7, chapter 10).

A consequence of this theorem of Ornstein is that the Bernoulli schemes

are completely classiﬁed (up to isomorphism) by their entropy. The last result

we quote in this section shows how the entropy also classiﬁes the hyperbolic

isomorphisms of tori (see Example 13.13): these are given by matrices σ

∈ GL(l, Z)

with no eigenvalue of absolute value = 1.

heorem 13.14 (Katznelson) Every linear hyperbolic automorphism of T

isomorphic to a Bernoulli scheme.

Due to the theorem of Ornstein the classiﬁcation of the ergodic properties of

the automorphisms of T

is given by the entropy.

It can be proved (see Walters 1982, sections 8.4 and 8.10) that if ν

, . . . , ν

are

the eigenvalues of the automorphism σ then

h(σ) =

{i||ν

|>1}

log

|ν

(13.43)

We conclude with the deﬁnition of Bernoulli systems.

13.8

Analytical mechanics

575

D

efinition 13.19 A measurable dynamical system (X, A, µ, S) is a Bernoulli

system if it is isomorphic to a Bernoulli scheme.

Bernoulli systems exhibit the most signiﬁcant stochastic properties. Their equi-

valence classes up to isomorphism, due to the theorem of Ornstein, are completely

classiﬁed by only one invariant, the entropy.

13.8

Dispersive billiards

Many important models of classical statistical mechanics are systems of point

particles or rigid spheres moving freely except for the eﬀect of elastic collisions,

either with ﬁxed obstacles or among themselves. To study the behaviour of

electron gases in metals, Lorentz introduced in 1905 the following model: a point

particle moves in R

subject only to elastic collisions with a distribution of

inﬁnitely many ﬁxed rigid spheres (see Fig. 13.2).

Another important model is the hard spheres gas: a system of spheres which

move freely in a domain V

⊂ R

interacting through elastic collisions between

them and with the boundary ∂V of the domain (see Fig. 13.3).

In all these cases, the main element of the model is the condition that the

collision be elastic. This is the characterising feature of all dynamical systems of

billiard type.

efinition 13.20 A billiard is a dynamical system constituted by the motion of

a point particle with constant velocity inside a bounded open subset V

⊆ R

with

piecewise smooth boundary (

∞

), and with a ﬁnite number of smooth components

intersecting transversally. The particle is subject to elastic reﬂections when it

collides with ∂V (see Fig. 13.3): the incidence angle is equal to the reﬂection

angle and the energy is conserved.

Fig. 13.2 Lorentz gas.

576

Analytical mechanics

13.8

V

−V

Fig. 13.3 Hard spheres gas.

the ‘stadium’

Fig. 13.4 Examples of plane billiards.

In our short introduction to the study of billiards we shall restrict ourselves to

the plane case, when l = 2 (See Fig. 13.4 for some examples of plane billiards).

This is the only case whose stochastic properties are suﬃciently understood.

Since the absolute value of the velocity is constant, it is possible to describe

the motion using a system with discrete time.

We parametrise ∂V using the natural parameter s and suppose that the length

of ∂V is equal to 2π; we can then characterise x

∈ ∂V by choosing arbitrarily

the origin corresponding to s = 0 and via the application S

→ x(s) (note

that x(s + 2π) = x(s)) (see Fig. 13.5). The elastic collision with ∂V is completely

described by assigning the pair (s, α)

∈ S

× (0, π), where x(s) is the collision

point in ∂V and α is the angle formed by the reﬂected velocity (i.e. the velocity

immediately after the collision) and the unit vector tangent to ∂V .

Consider the phase space X = S

× (0, π) with the σ-algebra A of Borel sets,

and the transformation S : X

→ X which associates with (s, α) the next collision

point and reﬂection angle (s , α ).

roposition 13.1 S preserves the probability measure dµ(s, α) = 1/4π sin α ds dα.

13.8

Analytical mechanics

577

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