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Modern ergodic theory; from a physics hypothesisto a mathematical theory

Dogan Comez

Department of Physics and Astrophysics, University of North Dakota

October 30, 2015

PreambleMathematical study of ergodicity

Preamble

Origins of Ergodic theory goes back to statistical mechanics;

establishing a connection between the ensembles typically studiedin statistical mechanics and the properties of single systems. Morespecifically, in solving problems of demonstrating the equality ofinfinite time averages and phase averages.

Dogan Comez Modern ergodic theory; from a physics hypothesis to a mathematical theory


Preamble

Origins of Ergodic theory goes back to statistical mechanics;establishing a connection between the ensembles typically studiedin statistical mechanics and the properties of single systems.

Morespecifically, in solving problems of demonstrating the equality ofinfinite time averages and phase averages.



Preamble

Origins of Ergodic theory goes back to statistical mechanics;establishing a connection between the ensembles typically studiedin statistical mechanics and the properties of single systems. Morespecifically, in solving problems of demonstrating the equality ofinfinite time averages and phase averages.



Consider a physical system of N particles confined in a compactphase space X .

The state of a single particle moving in this spacecan be described by the trajectory of a point x = (p, q), wherep, q ∈ RN are position and momenta of all N particles in thesystem.

If the energy of the system is E , then x must lie on the energysurface H(x) = E , where H is the Hamiltonian

dqidt

=∂H

∂pi,

dpidt

= −∂H∂qi

, 1 ≤ i ≤ N.



Consider a physical system of N particles confined in a compactphase space X . The state of a single particle moving in this spacecan be described by the trajectory of a point x = (p, q), wherep, q ∈ RN are position and momenta of all N particles in thesystem.


dqidt

=∂H

∂pi,

dpidt

= −∂H∂qi

, 1 ≤ i ≤ N.




If the energy of the system is E , then x must lie on the energysurface H(x) = E ,

where H is the Hamiltonian

dqidt

=∂H

∂pi,

dpidt

= −∂H∂qi

, 1 ≤ i ≤ N.





dqidt

=∂H

∂pi,

dpidt

= −∂H∂qi

, 1 ≤ i ≤ N.



Given an initial state x, such a system always has a uniquesolution, which determines the state Tt(p, q) = (p(t), q(t)) at anytime t ∈ R.

Hence, we have a one-parameter continuous flow oftransformations τ = {Tt}t∈R on the phase space X ⊂ R2N thatdescribes the evolution of the system.

So, the orbit of a particle x = (p, q) is Ox = {Tt(x)} ⊂ R2N .

By Liouville’s Theorem, τ preserves the normalized Lebesguemeasure on X .



Structures on X :

X ⊂ R2N is a compact manifold inheriting its topologicalstructure from R2N .

X has a measurable structure inherited from R2N ;

namely,(X ,B, µ), where B is the Borel σ-algebra of subsets of X andµ is the normalized Lebesgue measure.

(X ,B, µ, τ) is a measurable as well as topological dynamicalsystem.

If f : X → R denotes a function of a physical quantity, measuredduring an experiment, for any t ≥ 0, f (Ttx) is the value it takesat the instant t, provided that the system is at x when t = 0.



Structures on X :

X ⊂ R2N is a compact manifold inheriting its topologicalstructure from R2N .

X has a measurable structure inherited from R2N ; namely,(X ,B, µ), where B is the Borel σ-algebra of subsets of X andµ is the normalized Lebesgue measure.

(X ,B, µ, τ) is a measurable as well as topological dynamicalsystem.

If f : X → R denotes a function of a physical quantity, measuredduring an experiment, for any t ≥ 0, f (Ttx) is the value it takesat the instant t, provided that the system is at x when t = 0.



Boltzmann: The measurements of the precise values of f (Ttx) isnot possible since it requires knowing the detailed positions andmomenta of all N particles.

Hence, the result of a measurement isactually the time average of f , i.e.,

1

t

∫ t

0f (Ttx)dt.

Since macroscopic interval of time for the measurements isextremely large from the microscopic point of view, one mayactually consider the limit of the time averages:

limt→∞

1

t

∫ t

0f (Ttx)dt.



Boltzmann: The measurements of the precise values of f (Ttx) isnot possible since it requires knowing the detailed positions andmomenta of all N particles. Hence, the result of a measurement isactually the time average of f , i.e.,

1

t

∫ t

0f (Ttx)dt.

Since macroscopic interval of time for the measurements isextremely large from the microscopic point of view, one mayactually consider the limit of the time averages:

limt→∞

1

t

∫ t

0f (Ttx)dt.



Claim: (Boltzmann) Such a system left to itself will pass throughall the points of the phase space; hence, the phase space iscompletely filled by the orbit of a single particle.

Then, it follows that the time average should coincide with theaverage value of f over X :

∫X f (x)dµ (space average of f ).

Thus, one can hypothesize that

limt→∞

1

t

∫ t

0f (Ttx)dt =

∫Xf (x)dµ.

This is the ergodic hypothesis of Boltzmann.



The ergodic hypothesis was instrumental in laying the foundationof statistical mechanics, but it was also controversial.

Doubts were raised about verifiability of this hypothesis for manysystems (Rosenthal and Pancherel (1913), Landau (1930’s) andseveral others). Some sources for such historical arguments:

J. van. Leth, Ergodic Theory, Interpretations of Probabilityand the Foundations of Statistical Mechanics, Studies in theHistory of the Philosophy of Modern Physics, 32, 581-594,(2001)

A. Patrascioiu,The ergodic hypothesis: a complicated problemin Mathematics and Physics, Los Alamos Science, (Specialissue), 263-279, (1987)

L. Sklar, Physics and Chance: Philosophical issues in thefoundations of statistical mechanics, Cambridge UniversityPress, (1993)



The ergodic hypothesis was instrumental in laying the foundationof statistical mechanics, but it was also controversial.Doubts were raised about verifiability of this hypothesis for manysystems (Rosenthal and Pancherel (1913), Landau (1930’s) andseveral others).

Some sources for such historical arguments:






The ergodic hypothesis was instrumental in laying the foundationof statistical mechanics, but it was also controversial.Doubts were raised about verifiability of this hypothesis for manysystems (Rosenthal and Pancherel (1913), Landau (1930’s) andseveral others). Some sources for such historical arguments:






The arguments pro and con are still being brought up:

C.R. de Olivera and T. Werlang, Ergodic hypothesis inclassical statistical mechanics, Rev. Brasileira de Ensino deFisica, 29, 189-201, (2007)

L. Markus and K.R. Meyer, Generic Hamiltonian dynamicalsystems are neither integrable nor ergodic, Mem. AMS, 144,(1974)

D.B. Malament and S.L. Zabell, Why Gibbs phase averageswork-the role of ergodic theory, Philosophy of Science, 47,339-349, (1980)

P.B.M. Vranas, Epsilon-ergodicity and the success ofequilibrium statistical machanics, Philosophy of Science, 65,688-708, (1998)

An interesting talk by CalTech physicist Sean Carroll in YouTube.



Problems with the original formulation of the hypothesis:

1. f must be integrable on X (this is usually the case)

2. Deeper problem: It’s impossible that ”the orbit of a singlepoint in the phase space visits every point in the space.”Reason: A curve is the continuous image of an interval in Rand any curve in Rn is one dimensional. So, a continuouscurve cannot fill a space with dimension greater than one.

P. Ehrenfest (quasi-ergodic hypothesis, 1911):

”The orbit of a single point comes arbitrarily close to anypoint in the phase space.”

The orbit of a point is dense in the phase space. This is areasonable assumption, which is accepted as the actual andworkable hypothesis by adherents of the theory and manymathematicians.



The ergodic hypothesis also created lots of interest amongmathematicians of early 20th century. For mathematicians, mainproblems of interest were:

1. Which dynamical systems satisfy the ergodic hypothesis?

2. In a dynamical system, can we always expect the time averagebe equal to space average?

3. What is the structure of dynamical systems satisfying theergodicity?



Set-up

For the rest of the talk (X ,B, µ) is a probability space.

Will consider a discrete group of transformations: τ = {T n}n∈Z,where T : X → X is be a measure preserving transformation (i.e.,µ(E ) = µ(T−1E ) for all E ∈ B).

If f : X → R is an integrable function, then the time averages andthe space average of f take the forms

1

n

n−1∑k=0

f (T kx), and

∫Xf (x)dµ(x), respectively.

Consequently, the ergodic hypothesis of Boltzmann becomes

limn→∞

1

n

n−1∑k=0

f (T kx) =

∫Xf (x)dµ(x).



Set-up

For the rest of the talk (X ,B, µ) is a probability space.Will consider a discrete group of transformations: τ = {T n}n∈Z,where T : X → X is be a measure preserving transformation (i.e.,µ(E ) = µ(T−1E ) for all E ∈ B).

If f : X → R is an integrable function, then the time averages andthe space average of f take the forms

1

n

n−1∑k=0

f (T kx), and

∫Xf (x)dµ(x), respectively.

Consequently, the ergodic hypothesis of Boltzmann becomes

limn→∞

1

n

n−1∑k=0

f (T kx) =

∫Xf (x)dµ(x).



Recurrence

Theorem. (Poincare Recurrence Theorem, 1893) If E ∈ B withµ(E ) > 0, then for almost every x ∈ E there exists k ≥ 1 suchthat T kx ∈ E .

(Almost every point of E returns to E .)

Proof. Let Q be the set of all points of E that do not return to E .Hence Q = E \ (recurrent points).Recurrent points = (E∩T−1E )∪(E∩T−2E )∪· · ·∪(E∩T−nE )∪. . . .Hence, Q = E \ ∪∞n=1T

−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence

Theorem. (Poincare Recurrence Theorem, 1893) If E ∈ B withµ(E ) > 0, then for almost every x ∈ E there exists k ≥ 1 suchthat T kx ∈ E . (Almost every point of E returns to E .)





Recurrence


Proof.

Let Q be the set of all points of E that do not return to E .Hence Q = E \ (recurrent points).Recurrent points = (E∩T−1E )∪(E∩T−2E )∪· · ·∪(E∩T−nE )∪. . . .Hence, Q = E \ ∪∞n=1T




Recurrence


Proof. Let Q be the set of all points of E that do not return to E .

Hence Q = E \ (recurrent points).Recurrent points = (E∩T−1E )∪(E∩T−2E )∪· · ·∪(E∩T−nE )∪. . . .Hence, Q = E \ ∪∞n=1T




Recurrence


Proof. Let Q be the set of all points of E that do not return to E .Hence Q = E \ (recurrent points).

Recurrent points = (E∩T−1E )∪(E∩T−2E )∪· · ·∪(E∩T−nE )∪. . . .Hence, Q = E \ ∪∞n=1T




Recurrence


Proof. Let Q be the set of all points of E that do not return to E .Hence Q = E \ (recurrent points).Recurrent points = (E∩T−1E )∪(E∩T−2E )∪· · ·∪(E∩T−nE )∪. . . .

Hence, Q = E \ ∪∞n=1T−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.

Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence



−nE .

If x ∈ Q, then T nx /∈ Q for all n ≥ 1.Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence



−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.

Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence



−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.

So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence



−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q).

Since µ(X ) = 1, this is possible only ifµ(Q) = 0, equivalently, µ(E ) = µ(recurrent points).



Recurrence



−nE . If x ∈ Q, then T nx /∈ Q for all n ≥ 1.Thus Q ∩ T−nQ = ∅, and also T−mQ ∩ T−nQ = ∅, for anym, n ≥ 1.So, Q, T−1Q, T−2Q, . . . ,T−nQ, . . . are all pairwise disjoint,each having measure µ(Q). Since µ(X ) = 1, this is possible only ifµ(Q) = 0,

equivalently, µ(E ) = µ(recurrent points).



Recurrence






Remark: If x /∈ E , then PRT cannot guarantee return of x to E insome foreseeable future.

Indeed, such a point may never enter E .

Example: Consider S = unit circle with arc-length measure andT : S → S be the rational rotation given by Tz = e iπz . Let E bethe union of the arc from (1, 0) to (0, i) and from (−1, 0) to

(0,−i). Take a point outside E , say z = e i3π4 . Then T nz /∈ E for

all n ≥ 0!

Observe: E is T -invariant, i.e., T−1E = E , and 0 < µ(E ) < 1. So,the remedy is

Definition. A mpt T : X → X is ergodic if E ∈ B withT−1E = E , then µ(E ) = 0 or 1.



Remark: If x /∈ E , then PRT cannot guarantee return of x to E insome foreseeable future. Indeed, such a point may never enter E .



all n ≥ 0!






Example: Consider S = unit circle with arc-length measure andT : S → S be the rational rotation given by Tz = e iπz .

Let E bethe union of the arc from (1, 0) to (0, i) and from (−1, 0) to


all n ≥ 0!







(0,−i).

Take a point outside E , say z = e i3π4 . Then T nz /∈ E for

all n ≥ 0!








all n ≥ 0!





Ergodic transformations

The definition of ergodicity guarantees that any x ∈ X enters agiven set of positive measure at some point in the future. Also it isnothing but measurable version of the quasi-ergodic hypothesis.

Question: Do ergodic dynamical systems exist?

Answer: Yes! I’ll provide three examples.

Example 1. (Irrational rotation) Let X = S be the unit circle asabove. Let T : S → S be defined by Tz = eα2πiz , whereα ∈ (0, 1) is an irrational number. This is a very rigid system, inthe sense that, the orbit of every point is dense in S .



Example 2. (Bernoulli shifts)

Lets toss a coin and record whatcomes up as H or T. Continuing for a very long time, we end upwith a sequence of H’s and T’s. Without any loss of generality,assume that these are infinite sequences. Hence, the collection ofall such experiments is the sequence space X = {H,T}N. Let theprobability of getting an H be P(H) = p and the probability ofgetting T be P(T ) = 1− p. Let’s define a probability measure µon X . First define it on cylinder sets [x] formed by long finitesequences: if x = (x1, x2, . . . , xn), xi ∈ {H,T}, letµ([x]) =

∏ni=1 P(xi ). By Kolmogorov’s extension theorem extend µ

to whole of X . Hence (X , µ) is a probability space. Defineσ : X → X by

σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),

i.e., the shift. Then σ is measure preserving and ergodic.



Example 2. (Bernoulli shifts) Lets toss a coin and record whatcomes up as H or T.

Continuing for a very long time, we end upwith a sequence of H’s and T’s. Without any loss of generality,assume that these are infinite sequences. Hence, the collection ofall such experiments is the sequence space X = {H,T}N. Let theprobability of getting an H be P(H) = p and the probability ofgetting T be P(T ) = 1− p. Let’s define a probability measure µon X . First define it on cylinder sets [x] formed by long finitesequences: if x = (x1, x2, . . . , xn), xi ∈ {H,T}, letµ([x]) =



σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),




Example 2. (Bernoulli shifts) Lets toss a coin and record whatcomes up as H or T. Continuing for a very long time, we end upwith a sequence of H’s and T’s. Without any loss of generality,assume that these are infinite sequences. Hence, the collection ofall such experiments is the sequence space X = {H,T}N.

Let theprobability of getting an H be P(H) = p and the probability ofgetting T be P(T ) = 1− p. Let’s define a probability measure µon X . First define it on cylinder sets [x] formed by long finitesequences: if x = (x1, x2, . . . , xn), xi ∈ {H,T}, letµ([x]) =



σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),




Example 2. (Bernoulli shifts) Lets toss a coin and record whatcomes up as H or T. Continuing for a very long time, we end upwith a sequence of H’s and T’s. Without any loss of generality,assume that these are infinite sequences. Hence, the collection ofall such experiments is the sequence space X = {H,T}N. Let theprobability of getting an H be P(H) = p and the probability ofgetting T be P(T ) = 1− p.

Let’s define a probability measure µon X . First define it on cylinder sets [x] formed by long finitesequences: if x = (x1, x2, . . . , xn), xi ∈ {H,T}, letµ([x]) =



σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),




Example 2. (Bernoulli shifts) Lets toss a coin and record whatcomes up as H or T. Continuing for a very long time, we end upwith a sequence of H’s and T’s. Without any loss of generality,assume that these are infinite sequences. Hence, the collection ofall such experiments is the sequence space X = {H,T}N. Let theprobability of getting an H be P(H) = p and the probability ofgetting T be P(T ) = 1− p. Let’s define a probability measure µon X . First define it on cylinder sets [x] formed by long finitesequences: if x = (x1, x2, . . . , xn), xi ∈ {H,T}, letµ([x]) =

∏ni=1 P(xi ).

By Kolmogorov’s extension theorem extend µto whole of X . Hence (X , µ) is a probability space. Defineσ : X → X by

σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),






to whole of X . Hence (X , µ) is a probability space.

Defineσ : X → X by

σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),







σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),

i.e., the shift.

Then σ is measure preserving and ergodic.






σ(x1, x2, . . . , xn, . . . ) = (x2, . . . , xn+1, . . . ),




Irrational rotation and Bernoulli shift represent the two extremes ofthe spectrum of ergodic transformations.

Example 3. (Tent system) Let X = [0, 1] with Lebesgue measureand define

T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].

The inverse image of any interval is split into two parts that fallsinto two halves of [0, 1] with total length equal to the length of theoriginal interval. So, T is measure preserving. Every rational pointis either convergent (to 0 or 2/3, its fixed points), or periodic oreventually periodic. Irrational points are “wandering”, i.e.,non-convergent, and their orbits come arbitrarily close to any pointin X . Thus (X , µ,T ) is an ergodic dynamical system.




Example 3. (Tent system)

Let X = [0, 1] with Lebesgue measureand define

T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].






T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].






T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].

The inverse image of any interval is split into two parts that fallsinto two halves of [0, 1] with total length equal to the length of theoriginal interval.

So, T is measure preserving. Every rational pointis either convergent (to 0 or 2/3, its fixed points), or periodic oreventually periodic. Irrational points are “wandering”, i.e.,non-convergent, and their orbits come arbitrarily close to any pointin X . Thus (X , µ,T ) is an ergodic dynamical system.





T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].

The inverse image of any interval is split into two parts that fallsinto two halves of [0, 1] with total length equal to the length of theoriginal interval. So, T is measure preserving.

Every rational pointis either convergent (to 0 or 2/3, its fixed points), or periodic oreventually periodic. Irrational points are “wandering”, i.e.,non-convergent, and their orbits come arbitrarily close to any pointin X . Thus (X , µ,T ) is an ergodic dynamical system.





T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].

The inverse image of any interval is split into two parts that fallsinto two halves of [0, 1] with total length equal to the length of theoriginal interval. So, T is measure preserving. Every rational pointis either convergent (to 0 or 2/3, its fixed points), or periodic oreventually periodic.

Irrational points are “wandering”, i.e.,non-convergent, and their orbits come arbitrarily close to any pointin X . Thus (X , µ,T ) is an ergodic dynamical system.





T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].

The inverse image of any interval is split into two parts that fallsinto two halves of [0, 1] with total length equal to the length of theoriginal interval. So, T is measure preserving. Every rational pointis either convergent (to 0 or 2/3, its fixed points), or periodic oreventually periodic. Irrational points are “wandering”, i.e.,non-convergent, and their orbits come arbitrarily close to any pointin X .

Thus (X , µ,T ) is an ergodic dynamical system.





T (x) =

2x if x ∈ [0,

1

2)

2(1− x) if x ∈ [1

2, 1].




The Ergodic Theorem

Having the definition of ergodicity and some examples, let’s checkif we can expect the time average be equal to space average insuch systems.

Theorem. (Birkhoff’s Ergodic Theorem) Let (X , µ) be aprobability space and T : X → X be a mpt and f ∈ L1(X ). Then

a) limn→∞1n

∑n−1k=0 f (T kx) = f ∗(x) exists for almost every

x ∈ X ,

b) f ∗(Tx) = f ∗(x) a.e. x ∈ X ,

c) If T is ergodic, then f ∗(x) =∫X f (x)dµ.

For ergodic dynamical systems the time average must be equal tothe space average!Note: the assertions may not hold on a set of measure zero. Thismay be a concern for philosophically minded, since there are setsthat are topologically dense in X while having measure zero.



The Ergodic Theorem



a) limn→∞1n


x ∈ X ,

b) f ∗(Tx) = f ∗(x) a.e. x ∈ X ,


For ergodic dynamical systems the time average must be equal tothe space average!

Note: the assertions may not hold on a set of measure zero. Thismay be a concern for philosophically minded, since there are setsthat are topologically dense in X while having measure zero.



The Ergodic Theorem



a) limn→∞1n


x ∈ X ,

b) f ∗(Tx) = f ∗(x) a.e. x ∈ X ,


For ergodic dynamical systems the time average must be equal tothe space average!Note: the assertions may not hold on a set of measure zero. Thismay be a concern for philosophically minded, since there are setsthat are topologically dense in X while having measure zero.



Proof. (Sketch)

The assertion is trivial for invariant functions, i.e.,functions of the form f ◦T = f . Similarly, for functions of the formf = g − g ◦ T , where g is bounded, the averages becometelescopic sum, and hence converges to 0.

Note: L1(X ) = {f = f ◦ T} ⊕ {g − g ◦ T : g ∈ L∞(X )}.

Next, supn | 1n∑n−1

k=0 f (T kx)| <∞ a.e. for all f ∈ L1.Recall: (Theorem of Banach) If this supremum is finite a.e., thenthe set of functions for which the convergence holds is a closedsubset of L1(X ). This theorem yields the proof.



Proof. (Sketch) The assertion is trivial for invariant functions, i.e.,functions of the form f ◦T = f .

Similarly, for functions of the formf = g − g ◦ T , where g is bounded, the averages becometelescopic sum, and hence converges to 0.






Proof. (Sketch) The assertion is trivial for invariant functions, i.e.,functions of the form f ◦T = f . Similarly, for functions of the formf = g − g ◦ T , where g is bounded, the averages becometelescopic sum, and hence converges to 0.









k=0 f (T kx)| <∞ a.e. for all f ∈ L1.

Recall: (Theorem of Banach) If this supremum is finite a.e., thenthe set of functions for which the convergence holds is a closedsubset of L1(X ). This theorem yields the proof.






k=0 f (T kx)| <∞ a.e. for all f ∈ L1.Recall: (Theorem of Banach) If this supremum is finite a.e., thenthe set of functions for which the convergence holds is a closedsubset of L1(X ).

This theorem yields the proof.



Operator theory ergodic theorems

Over the years the Ergodic Theorem has been generalized andextended to numerous settings.

Theorem. (Dunford-Schwartz, 1956) Let (X ,F , µ) be aprobability space and T : Lp(X )→ Lp(X ) be a linear operator with‖T‖1 ≤ 1 and ‖T‖∞ ≤ 1. Then, for all f ∈ L1(X ),

limn→∞

1

n

n−1∑k=0

T k f (x) exists for almost every x ∈ X .

There are many other operator theoretical generalizations as well.



Multiparameter ergodic theorems

Assume T : X → X and S : X → X be two mpts, orT , S : Lp(X )→ Lp(X ) be linear contractions. Then, it make senseto consider averages of the form 1

mn

∑m,ni ,j=1 T

iS j f (x).

Theorem. (Zygmund, 1951; Fava, 1972) For all f ∈ L log L(X ),limm,n→∞

1mn

∑m,ni ,j=1 T

iS j f (x), exists a.e. The limit is

(limn→∞1n

∑n−1k=0 T

k f (x))(limn→∞1n

∑n−1k=0 S

k f (x)).

Theorem. (Brunel, 1973) If T and S commute, then for allf ∈ L1(X ), limn

1n2

∑ni ,j=1 T

iS j f (x), exists a.e..





mn

∑m,ni ,j=1 T

iS j f (x).


1mn

∑m,ni ,j=1 T

iS j f (x), exists a.e.

The limit is

(limn→∞1n

∑n−1k=0 T


∑n−1k=0 S

k f (x)).


1n2

∑ni ,j=1 T






mn

∑m,ni ,j=1 T

iS j f (x).


1mn

∑m,ni ,j=1 T

iS j f (x), exists a.e. The limit is

(limn→∞1n

∑n−1k=0 T


∑n−1k=0 S

k f (x)).


1n2

∑ni ,j=1 T




Local ergodic theorems

Can we recover f form the averages 1t

∫ t0 f (Tsx)ds?

Theorem. (Wiener, 1939; Terrell, 1972) Let {Tt} and {St} betwo commuting continuous one-parameter flow of measurepreserving transformations on X such that T0 = S0 = I . Then, forall f ∈ L1(X ),

limt→0+

1

t2

∫ t

0

∫ t

0f (TsSrx)dsdr = f (x) exists a.e.

Extension to commuting positive linear contractions is due toOrnstein (1973) and Akcoglu-del Junco (1975).



Subsequential ergodic theorems

Ergodic theorems along subsequences:

Imagine that we’re ablemake precise measurements at each point T kx along the orbit of apoint in our dynamical system. Suppose actual measurements aremade at times n1, n2, n3, . . . and hence, we ended up with theaverages 1

N

∑N−1k=0 f (T nkx).

Question. Does limN1N

∑N−1k=0 f (T nkx) converge a.e.?

If so, does it converge to the right value?

The answer is affirmative.



Subsequential ergodic theorems

Ergodic theorems along subsequences: Imagine that we’re ablemake precise measurements at each point T kx along the orbit of apoint in our dynamical system. Suppose actual measurements aremade at times n1, n2, n3, . . . and hence, we ended up with theaverages 1

N

∑N−1k=0 f (T nkx).

Question. Does limN1N

∑N−1k=0 f (T nkx) converge a.e.?

If so, does it converge to the right value?

The answer is affirmative.



A short list of sequences {nk} along which a.e. convergence holds,i.e., limN

1N

∑N−1k=0 f (T nkx) exists a.e.:

• The sequence of square-free integers (in L1)

• The sequence [n3/2] (in Lp, 1 < p <∞)

• The sequence [n log n] (in Lp, 1 < p <∞)

• Return time sequences (in L1)

• Randomly generated sequences of positive density (in L1)

• Randomly generated sequences of zero density (in L2)

• Sequences of squares (in Lp, 1 < p <∞)

• Sequences of primes (in Lp, 1 < p <∞)

Furthermore, the limit along the first three sequences is the rightone, namely, it is the space average!



Convergence along moving averages

What if the measurements are made along a sequence like

v1, v1 + 1, v1 + 2, . . . , v1 + r1,

v2, v2 + 1, v2 + 2, . . . , v2 + r2

. . .

vn, vn + 1, vn + 2, . . . , vn + rn, and so on,

where vn ↑, rn ↑ and vn + rn < vn+1.

Hence, we have averages ofthe form 1

rn

∑rnk=0 f (T vn+kx), (called as moving averages).

If the sequence {(vn, rn)}n satisfies a condition called conecondition, then

limn

1

rn

rn∑k=0

f (T vn+kx) exists a.e. for all f ∈ L1.



Convergence along moving averages

What if the measurements are made along a sequence like

v1, v1 + 1, v1 + 2, . . . , v1 + r1,

v2, v2 + 1, v2 + 2, . . . , v2 + r2

. . .

vn, vn + 1, vn + 2, . . . , vn + rn, and so on,

where vn ↑, rn ↑ and vn + rn < vn+1. Hence, we have averages ofthe form 1

rn

∑rnk=0 f (T vn+kx), (called as moving averages).

If the sequence {(vn, rn)}n satisfies a condition called conecondition, then

limn

1

rn

rn∑k=0

f (T vn+kx) exists a.e. for all f ∈ L1.



Modulated ergodic theorems

What if the measurements are somewhat “tainted”, or modulated?

That is, instead of obtaining the values f (T kx) along the orbit ofthe point x , we would be getting values like ak f (T kx) for somesequence {ak}. Then, we’ll be end up with modulated averages:1n

∑n−1k=0 ak f (T kx).

Question. For which sequences {ak} does limn1n

∑n−1k=0 ak f (T kx)

converge a.e.?A list of modulating sequences {ak} along which a.e. convergenceholds, i.e., limn

1n

∑n−1k=0 ak f (T kx) exists a.e. is:

• ak = λk , where |λ| = 1

• {ak} is a bounded Besicovitch sequence

• {ak} is a sequence having a mean.




What if the measurements are somewhat “tainted”, or modulated?That is, instead of obtaining the values f (T kx) along the orbit ofthe point x , we would be getting values like ak f (T kx) for somesequence {ak}.

Then, we’ll be end up with modulated averages:1n





1n








What if the measurements are somewhat “tainted”, or modulated?That is, instead of obtaining the values f (T kx) along the orbit ofthe point x , we would be getting values like ak f (T kx) for somesequence {ak}. Then, we’ll be end up with modulated averages:1n





1n












converge a.e.?

A list of modulating sequences {ak} along which a.e. convergenceholds, i.e., limn

1n













1n







Fractal geometry connection

An interesting feature of some dynamical systems in connectionwith fractal geometry.

Modified tent map: T : [0, 1]→ [0, 1] defined by

T (x) =

3x if x ∈ [0,

1

2)

3(1− x) if x ∈ [1

2, 1].

Notice that T does not map [0, 1] into itself. If x ∈ ( 13 ,

23 ), then

Tx /∈ [0, 1]. So, J1 = {x ∈ [0, 1] : Tx > 1} = ( 13 ,

23 ). Then,

T ([0, 1]) = [0, 1] \ J1 = [0, 13 ] ∪ [ 2

3 , 1], two closed intervals, both ofwhich are mapped one-to-one and onto [0, 1].




An interesting feature of some dynamical systems in connectionwith fractal geometry.Modified tent map: T : [0, 1]→ [0, 1] defined by

T (x) =

3x if x ∈ [0,

1

2)

3(1− x) if x ∈ [1

2, 1].


23 ), then

Tx /∈ [0, 1]. So, J1 = {x ∈ [0, 1] : Tx > 1} = ( 13 ,

23 ). Then,

T ([0, 1]) = [0, 1] \ J1 = [0, 13 ] ∪ [ 2






T (x) =

3x if x ∈ [0,

1

2)

3(1− x) if x ∈ [1

2, 1].


23 ), then

Tx /∈ [0, 1]. So, J1 = {x ∈ [0, 1] : Tx > 1} = ( 13 ,

23 ).

Then,T ([0, 1]) = [0, 1] \ J1 = [0, 1

3 ] ∪ [ 23 , 1], two closed intervals, both of

which are mapped one-to-one and onto [0, 1].





T (x) =

3x if x ∈ [0,

1

2)

3(1− x) if x ∈ [1

2, 1].


23 ), then

Tx /∈ [0, 1]. So, J1 = {x ∈ [0, 1] : Tx > 1} = ( 13 ,

23 ). Then,

T ([0, 1]) = [0, 1] \ J1 = [0, 13 ] ∪ [ 2




Define, J2 ={x ∈ [0, 1] : T 2x > 1

}. Thus,

T 2([0, 1]) = [0, 1] \ (J1 ∪ J2) consists of four closed intervals thatare mapped one-to-one and onto [0, 1].

Continuing on we candefine, In = {x ∈ [0, 1] : T nx ∈ [0, 1]} andJn = {x ∈ In−1 : T nx > 1}. Notice that In = [0, 1] \

⋃nk=1 Jk has

2n disjoint intervals that T maps one-to-one and onto [0, 1].

Taking the set of all such intervals, defineC = {x ∈ [0, 1] : T nx ∈ [0, 1], ∀n ≥ 1} = ∩∞n=1T

n([0, 1]). Since allof these intervals are one-to-one and onto [0, 1], C is mapped toitself. Hence, the pair (C,T ) is a dynamical system.

Furthermore, from the construction, we can see that C =⋂

n≥1 In.This is the Cantor set, a fractal!



Define, J2 ={x ∈ [0, 1] : T 2x > 1

}. Thus,

T 2([0, 1]) = [0, 1] \ (J1 ∪ J2) consists of four closed intervals thatare mapped one-to-one and onto [0, 1]. Continuing on we candefine, In = {x ∈ [0, 1] : T nx ∈ [0, 1]} andJn = {x ∈ In−1 : T nx > 1}.

Notice that In = [0, 1] \⋃n

k=1 Jk has2n disjoint intervals that T maps one-to-one and onto [0, 1].







Define, J2 ={x ∈ [0, 1] : T 2x > 1

}. Thus,

T 2([0, 1]) = [0, 1] \ (J1 ∪ J2) consists of four closed intervals thatare mapped one-to-one and onto [0, 1]. Continuing on we candefine, In = {x ∈ [0, 1] : T nx ∈ [0, 1]} andJn = {x ∈ In−1 : T nx > 1}. Notice that In = [0, 1] \

⋃nk=1 Jk has

2n disjoint intervals that T maps one-to-one and onto [0, 1].







One can, of course, construct the Cantor set via a “static”manner. However, this dynamic construction paves way tointroduce many tools of dynamical systems into study of propertiesof fractals.

Indeed, some fundamental features of fractals (such asbox dimension, Hausdorff dimension, subfractal structure, etc.)have been studied in depth only after the introduction of ergodictheory tools into fractal geometry.



One can, of course, construct the Cantor set via a “static”manner. However, this dynamic construction paves way tointroduce many tools of dynamical systems into study of propertiesof fractals. Indeed, some fundamental features of fractals (such asbox dimension, Hausdorff dimension, subfractal structure, etc.)have been studied in depth only after the introduction of ergodictheory tools into fractal geometry.



THANK YOU!



Some sources used

V.I. Arnold and A. Avez, Ergodic Problems of ClassicalMechanics, W.A. Benjamin, Inc., 1968.

G. Gallavotti, Statistical Mechanics. A Short Treatise,Springer-Verlag, Berlin, 1999.

A.I. Khinchin, 1960, Mathematical Foundations of StatisticalMechanics, Dover, 1960

U. Krengel, Ergodic Theorems, de Gruyter, 1985.

K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.

P. Walters, An Introduction to Ergodic Theory,Springer-Verlag, 1982.


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