Modern Ergodic Theory: From a PhysicsHypothesis to a Mathematical Theory withTransformative Interdisciplinary Impact

Dogan Çömez

Contents

Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Consequence of the Ergodic Theorem and Other Significant Results . . . . . . . . . . . . . . . . . . . . 7Interdisciplinary Aspects of Ergodic Theory in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Functional Analysis and Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Interdisciplinary Aspects of Ergodic Theory with Other Disciplines . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Abstract

Ergodic theory emerged as a statistical mechanics hypothesis and has quicklyreached into a mature and influential mathematical theory. Beginning with abrief historical account on the origins of the theory, the first two sections of thischapter aim to provide a broad exposé of some major results in ergodic theoryand dynamical systems. The remaining sections are devoted to the discussionof the interdisciplinary nature of ergodic theory from a broad perspective.Select applications of the theory are outlined, and its interactions with othermathematical fields and with some nonmathematical disciplines are brieflyillustrated.

D. Çömez (�)Department of Mathematics, North Dakota State University, Fargo, ND, USAe-mail: dogan.comez@ndsu.edu

© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_31-1

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2 D. Çömez

KeywordsErgodic theory · Dynamical system · Ergodic transformation ·Measure-preserving transformation · Invariant measure · Interdisciplinaryaspects of ergodic theory

Prelude

The history of the immense body of knowledge and technology that humans createdclearly points out that mathematics emerged as the first body of knowledge asan independent discipline. In the beginning, particularly in ancient Egypt andMesopotamia, mathematics consisted merely of a bunch of “technical methods”that were useful in solving practical problems in surveying, construction, and astro-nomical prediction. Greek thinkers managed making the quantum leap to transformthese practical methods into an abstract and organized collection of concepts, asexemplified in Elements of Euclid or in the Conics of Apollonius. The conditionsthat led to this transformation are peculiar to the social and cultural environmentof Greek society, which valued engaging in intellectual pursuits. This intellectualpursuit also took the form of making connections; hence, we see attempts beingmade to describe musical harmony using properties of numbers, applying geometricconcepts to geography and astronomy. During the Roman and Islamic periods thatfollowed, due to various political and cultural changes, development of mathematicscontinued, albeit with a slower pace and with a shift to ideas more aligned topractical purposes rather than a pursuit of knowledge for its own sake. The emphasiswas more on applications, especially, for Romans who favored the engineeringaspect. Islamic intellectuals continued the Greek tradition, as exemplified in al-jabrof Al Khwarizmi, but they were as much interested in applications that would makesocial life orderly by calculating praying times and predicting phases of the moonaccurately; hence, their other contributions were focused more on trigonometry.Even during the early developmental ages, one encounters significant interactionsof mathematical ideas and other disciplines like astronomy, music, geography, andengineering. Needless to say, these interactions have continued with increasingintensity, resulting in numerous interdisciplinary subjects.

In broadest terms, a discipline is considered as interdisciplinary if it is acombination of or involves two or more (rather distinct) fields of study. In today’sscientific and intellectual environment, this description is rather restrictive. Somedisciplines have evolved to a stage so as to include several sub-disciplines whichhave their own methodologies and perspectives. In such a case, one can alsospeak about “interdisciplinary” subfields within a vast field. The evolution of manySTEM disciplines suggest that emergence of many intellectual and technologicaldisciplines, as an independent discipline or as a subfield, is rarely an isolatedoccurrence; rather, it is a result of interaction among several disciplines. Therefore,it is not an exaggeration if we posit that many disciplines first come into being asan interdisciplinary field and grow into an independent one over time. In one casean idea, method, or practice in a certain discipline finds a fruitful application to a

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 3

situation/problem in another discipline, often in a transformative way. Over timethis leads to new/different methodologies and practices within the latter. Indeed,sometimes the evolution of the newly formed discipline results in an independentdiscipline in itself. There are also cases where a problem or a phenomena in adiscipline begs ideas from one or more others, which leads to a new sub-disciplinethat utilizes methodologies and ideas from all involved disciplines while generatingits own in the meantime. Occasionally, a new idea or development within a fieldgives rise to new practices and eventually grows apart from the mother disciplineinto an independent one. As an example we can consider the emergence anddevelopment of computer science out of a subfield of mathematics.

Whichever the manner disciplines or sub-disciplines emerge and evolve, theynaturally keep having strong interaction with their “mother” disciplines in one formor another. Hence, not surprisingly, one might consider them as an interdisciplinaryfield, whereas others may object to this view. Depending on the discipline or thesubfield, the arguments pro or con may have merit. In this chapter, we will take thebroad view of the disciplines under discussion as commonly accepted today. Wewill also measure the level of the interdisciplinary aspect of a field by the impactand contributions of its interaction with other sub-disciplines in the same broaddiscipline or with the other disciplines.

Mathematics at large, and many of its sub-disciplines in particular, is a primeexample of a body of knowledge that has always been at the heart of many interdis-ciplinary activities. This is partly due to the fact that it provides the most definitiveand effective language for other disciplines. Mostly it is because its methodologiesand concepts are devoid of any ambiguities due to its deductive nature. During itslong history, it has experienced a significant growth, particularly from the Age ofEnlightenment to present. Now it is a vast discipline containing numerous diversefields. If one speaks about the impact or application of mathematics, often it is notdescriptive, unless the particular subfield and particular idea/concept or methodis mentioned. Our central field will be ergodic theory, which initially emergedindependently of classical dynamical systems, and over time it has reached a levelto encompass measurable dynamical systems. We will exhibit the interactions ofergodic theory, together with its dynamical system component, within mathematicsas well as with other nonmathematical disciplines.

Origins

Physics has always been one of the most prolific sources of ideas and conceptsfor mathematicians. This is particularly the case if one looks at the developmentsof many mathematical theories in the last two centuries. For instance, Fourieranalysis, the theory of distributions, and string theory all have their origins tiedto some physical concepts. Likewise, having originated from a physical idea, withthe beginnings and growth as a mathematical field and its interactions with otherfields within mathematics as well as other (nonmathematical) disciplines including

4 D. Çömez

physics, ergodic theory is a prime case of such a field of mathematics that hassignificant impact in this regard.

Ergodic theory, which focuses on statistical properties of a system, initiallyemerged independently of classical dynamical systems and over time reached alevel that became synonymous with, or inseparable from, measurable dynamicalsystems. Earliest ideas in ergodic theory have their origins in the attempts madeby the pioneers of statistical mechanics, like Boltzmann, Maxwell, and Gibbs,who investigated connections between the ensembles typically studied in statisticalmechanics and the properties of single systems. More specifically, their work ledthem to postulate the existence and equality of infinite time averages and phaseaverages.

Consider a physical system consisting of a (large) number of particles, say N,

confined in a compact phase space X; this is usually an energy surface. The state ofa single particle moving in this space can be described by the trajectory of a pointx = (p, q), where p, q ∈ R

N represent the position and momenta of all N particlesin the system. We can also assume that X ⊂ R

2N is a compact manifold inheritingits topological structure from R

2N. By the same token, we can also consider X ashaving a measurable structure inherited from R

2N ; in other words, X is a probabilityspace (X,B, μ), where B is the Borel σ -algebra of subsets of X and μ is thenormalized Lebesgue measure. Then, the quadruple (X,B, μ, τ) is a measurableas well as topological dynamical system.

Now, given an initial state x, such a system always has a unique solution, whichdetermines the state Tt (p, q) = (p(t), q(t)) at any time t ≥ 0. In particular, thisgives us a one-parameter continuous flow of transformations τ = {Tt }t∈R on thephase space X ⊂ R

2N that describes the evolution of the system. Consequently, theorbit of a particle x is the set Ox = {Tt (x)}t∈R ⊂ R

2N. By Liouville’s theorem,τ preserves the normalized Lebesgue measure on X, i.e., each Tt is a (Lebesgue)measure-preserving transformation.

If f : X → R denotes a function of a physical quantity, measured during anexperiment, for any t ≥ 0, f (Ttx) is the value it takes at the instant t provided thatthe system is at x at time t = 0. It is argued that the measurements of the precisevalues of f (Ttx) is not possible since it requires knowing the detailed positions andmomenta of all N particles. Hence, it is assumed that the result of a measurement isactually the time average of f, i.e.,

1

t

∫ t

0f (Ttx)dt (time average of f ).

Indeed, since macroscopic interval of time for the measurements is extremely largefrom the microscopic point of view, one may actually consider the limit of the timeaverages:

limt→∞

1

t

∫ t

0f (Ttx)dt.

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 5

Originally, Boltzmann argued that such a system left to itself will pass through allthe points of the phase space; hence, the phase space is completely filled by the orbitof a single particle. Then, he deduced that the time average should coincide with theaverage value of f over X, which is

∫X

f (x)dμ (space average of f ). Thus, hehypothesized that

limt→∞

1

t

∫ t

0f (Ttx)dt =

∫X

f (x)dμ.

This is the ergodic hypothesis of Boltzmann.Although it was instrumental in laying the foundation of statistical mechanics,

this was a controversial hypothesis since its inception. For example, some influentialfigures of that time, such as Landau, raised doubts about verifiability of thishypothesis for many systems. There is a very good account of these historicalarguments in the sources (Patrascioiu, 1987; Sklar, 1993; van Leth, 2001).

The original formulation of the hypothesis has some mathematical issues inregard to rigor. First of all, f must be integrable on X, which is usually the case.However, there’s a deeper problem. Mathematically, a curve is the image of aninterval in R, and any curve in R

n is one dimensional. Therefore, such a curvecannot fill a space with dimension greater than one. This is an apparent contradictionto Boltzmann’s assumption that “the orbit of a single point in the phase spacevisits every point in the space.” Having recognized this contradiction, around 1911P. Ehrenfest (a student of Boltzmann) modified this assumption into what is alsoknown today as the quasi-ergodic hypothesis:

“The orbit of a single point comes arbitrarily close to any point in the phasespace.”

Mathematically speaking, this means that the orbit of a point is dense in the phasespace. This is a reasonable assumption, which is accepted as the actual and workablehypothesis by adherents of the theory and many mathematicians.

While working on some problems in celestial mechanics, around 1893, well-known mathematician H. Poincare observed an important property of measure-preserving transformations on probability spaces:

Theorem 1 (Poincare recurrence theorem). If E ∈ B with μ(E) > 0, then foralmost every x ∈ E, there exists k ≥ 1 such that T kx ∈ E.

That is, almost every point of E returns to E (and does so infinitely often). Soonafter its appearance, this innocent but powerful result stirred quite a controversyamong many scholars in mathematics, physics, and philosophy. For instance, itsuggests that in a box containing large number of gas molecules, left on its ownfor a long period of time, there will be an instant at which all molecules will occupythe right half of the box while no molecule residing in the other half! Having suchan interesting result, which has many interesting consequences within mathematics,

6 D. Çömez

physics, and astronomy, investigations on the qualitative behavior of orbits arefurther continued by Lyapunov (1892), Perron (1929), Kolmogorov (1954), Sinai(1959), and many others. Observations that small changes in the initial conditionof some differential equations resulted in large deviations led to concepts likebifurcation and chaotic behavior as we know it today. Their pioneering work hasbeen extended to the concept of entropy, which, roughly speaking, is a measure ofcomplexity of a system, by Kolmogorov and his students.

Recall that while Poincare recurrence theorem guaranties that an initial state ina set of positive measure returns infinitely often to this set. Naturally, one questionsthe rate at which such a state returns to the set. With a clever use of Birkhoff’sergodic theorem, Kac proved that the average return time to the set can be calculatedas inversely proportional to the size of the set (Kac, 1947). Furthermore, if the pointoriginally was outside of the given set A of positive measure, there is no guaranteeabout its return to A. The condition that guarantees this for almost every point inthe system is what is known as the ergodicity of the transformation, which statesthat the only sets invariant under the map (up to sets of measure zero) are the wholespace and the empty set.

Having defined ergodicity, now it’s time to check if we can expect the timeaverage be equal to space average in such systems as claimed. Well, the result statingthis fact, known as the ergodic theorem or Birkhoff’s ergodic theorem, is the jewelof ergodic theory.

Theorem 2 (Birkhoff’s ergodic theorem – discrete parameter version). Let(X,μ) be a probability space and T : X → X be a measure-preservingtransformation and f ∈ L1(X). Then

(a) limn→∞ 1n

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

(b) f ∗(T x) = f ∗(x) for almost every x ∈ X, (i.e., the limit is T -invariant),(c) if T is ergodic, then f ∗(x) = ∫

Xf (x)dμ.

Hence, Birkhoff’s ergodic theorem, proved in 1931 (Birkhoff, 1931), confirms thatfor ergodic dynamical systems the time average must be equal to the space average.However, having said so, one should notice that the assertions may not hold on a setof measure zero. This may be a concern for the philosophically minded, since thereare sets that are topologically dense in X while having measure zero.

If one shifts focus to the mathematical side of the issue, it is a mild statementto assert that the ergodic hypothesis created tremendous interest among mathemati-cians of the early twentieth century. For mathematicians, main problems of interestwere:

1. Which (measurable) dynamical systems satisfy the ergodic hypothesis?2. In a dynamical system, can we always expect the time average be equal to space

average? If not, what is the limit of the time averages?3. What is the structure of dynamical systems satisfying ergodicity?

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 7

Each of these questions (and subsequent inquiries) led to a prolific output ofresults that not only provided answers to these questions but also paved the wayto new ideas and techniques that impacted many fields within mathematics andnonmathematical disciplines. To name a few, Birkhoff’s ergodic theorem has beenextended and generalized to many different settings; numerous systems (both purelymathematical and physical) proved to satisfy ergodicity. Deeper investigations of thestructure of measure-preserving systems lead to many other finer properties beyondergodicity such as mixing, weak mixing, and exact transformations, which providefiner statistical properties of measure-preserving transformations. More specifically,a measure-preserving transformation T is

• mixing if limn→∞ μ(T −nA ∩ B) = μ(A)μ(B),

• weak-mixing if limn→∞ 1n

∑n−1k=0 |μ(T −kA ∩ B) − μ(A)μ(B)| = 0,

for all measurable sets A and B. Furthermore, the hierarchy between these types oftransformations is determined as

T is exact ⇒ T is mixing ⇒ T is weak-mixing ⇒ T is ergodic.

Indeed, there are also several types of transformations between each of thesecategories, and the inclusions these implications provide are not reversible. Also,it turns out that by a remarkable theorem of Rohlin (1967) any measure-preservingtransformation can be decomposed into ergodic components, although this decom-position can be very involved.

Consequence of the Ergodic Theorem and Other SignificantResults

Being, undoubtedly, the most fundamental result in dynamical systems and ergodictheory, over the years Birkhoff’s ergodic theorem has been generalized and extendedto numerous settings. For easy reference in discussions in the next sections, wewill outline some of these. The first two are operator-theoretic generalizations. Acontraction T on L1(X), where (X,μ) is a σ -finite measure space, is a linearoperator that satisfies the condition ‖T ‖1 ≤ 1; if it also satisfies ‖T ‖∞ ≤ 1, itis called an L1 − L∞-contraction.

Theorem 3 (Dunford and Schwartz 1956). Let (X,μ) be a σ -finite measurespace and T : L1(X) → L1(X) be an L1 − L∞-contraction. Then, for allf ∈ Lp(X), 1 ≤ p < ∞, limn→∞ 1

n

∑n−1k=0 T kf (x) exists for a.e. x ∈ X.

If T is an L1-contraction only, then the assertion is not valid anymore; hence, thecondition of T being an L1 −L∞-contraction in Dunford-Schwartz theorem cannotbe dropped.

8 D. Çömez

There are other operator theoretical generalizations of Birkhoff’s ergodic the-orem; notable one is the similar result for T is an Lp-contraction (i.e., ‖T ‖p ≤1), 1 < p < ∞. By Riesz interpolation theorem, an L1 − L∞-contraction is alsoan Lp-contraction for all 1 < p < ∞ (Koopman operators induced by measure-preserving transformations are prime examples of such operators). On the otherhand, this is not the case for Lp-contractions when 1 < p < ∞; namely, an Lp-contraction need not be an Lq -contraction if p �= q and 1 < p, q < ∞.

Theorem 4 (Akcoglu 1975). Let (X,μ) be a σ -finite measure space, and let T :Lp(X) → Lp(X) be a positive contraction for some 1 < p < ∞. Then, for allf ∈ Lp(X), limn→∞ 1

n

∑n−1k=0 T kf (x) exists for a.e. x ∈ X.

It is known that the condition of positivity in Akcoglu’s theorem cannot bedropped when p = 2, and it is still an open problem if positivity can be droppedwhen p �= 2.

Many physical and mathematical situations involve multiple operators; hence,obtaining multiparameter versions of (one-parameter) ergodic theorems is a naturalendeavor. In some fields of mathematics, extending a one-parameter statementto higher parameters is merely a simple exercise; however, this is not the casefor multiparameter extensions of ergodic theorems, especially those involving a.e.convergence of ergodic averages. The following are two extensions of the ergodictheorem to multiparameter setting. Assume T : X → X and S : X → X betwo measure-preserving transformations or T , S : Lp(X) → Lp(X) be linearcontractions, 1 ≤ p ≤ ∞. Then, it makes sense to consider multiparameter averagesof the form 1

mn

∑m,ni,j=1 T iSjf (x).

Theorem 5 (Fava 1972; Zygmund 1951). Let (X,μ) be a σ -finite measure spaceand T , S : L1(X) → L1(X) be positive L1 − L∞-contractions. Then for all f ∈L log L(X), the limit of multiparameter averages exists a.e. and

limm,n→∞

1

mn

m,n∑i,j=1

T iSjf (x) = ( limn→∞

1

n

n−1∑k=0

T kf (x))( limn→∞

1

n

n−1∑k=0

Skf (x)).

This statement is not true if f ∈ L1. On the other hand, if the operators commuteand the averaging is limited to “squares,” then multiparameter averages converge forf ∈ L1 as well:

Theorem 6 (Brunel 1973; Dunford and Schwartz 1956). If (X,μ) is a σ -finitemeasure space and T , S : L1(X) → L1(X) are commuting L1 − L∞-contractions,then, for all f ∈ L1(X), limn→∞ 1

n2

∑ni,j=1 T iSjf (x) exists a.e.

Dunford and Schwartz proved this theorem in the setting of commuting measure-preserving flows; later, Brunel provided the discrete version stated above.

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 9

One can view Lebesgue differentiation theorem as a result that recovers afunction from its averages. Following this approach one can ask if it is possibleto recover f from the averages 1

t

∫ t

0 f (Tsx)ds? Well, the answer is, surprisingly,“Yes” for many cases, which are known as local ergodic theorems.

Theorem 7 (Local ergodic theorem, Wiener 1939). Let {Tt } be a continuous one-parameter flow of measure-preserving transformations on X such that T0 = I. Then,for all f ∈ L1(X),

limt→0+

1

t

∫ t

0f (Tsx)ds = f (x) exists a.e.

Extension of this result to the multiparameter setting was proved by Terrell (1971)and to positive linear L1-contractions was proved by Ornstein (1970) and Akcogluand del Junco (1981).

Another extension of the ergodic theorem is along subsequences. Imagine thatwe’re able to make precise measurements at each point T kx along the orbit of a pointin our dynamical system. Suppose that you asked one of your graduate studentsto make the measurements. But the poor student is sleep deprived, and, ratherthan making measurements at each T nx, n = 0, 1, 2, . . . , precisely, he/she mademeasurements at times n1, n2, n3, . . . Having realized this mistake, the student wasafraid to tell the truth; hence, you ended up with the averages 1

N

∑N−1k=0 f (T nkx).

Does limN1N

∑N−1k=0 f (T nkx) converge a.e.? If so, does it converge to the right

value? If not, what are conditions on the sequence (nk) that guarantees an affirmativeresult?

The answer turns out to be affirmative in some cases and negative in some others.Below is a sample list (out of a vast literature) of sequences {nk} along which a.e.convergence holds in the space of functions (i.e., limN

1N

∑N−1k=0 f (T nkx) exists

a.e).

• The sequence of square-free integers in L1 (Boshernitzan and Wierdl, 1996),• The sequence [n3/2] and [n log n], in Lp, 1 < p < ∞ (Boshernitzan and Wierdl,

1996),• Return time sequences, in L2 (Bourgain, 1989),• Randomly generated sequences of positive density in L1, randomly generated

sequences of zero density, in L2 (Boshernitzan and Wierdl, 1996; Bourgain,1988a),

• Sequences of squares, in Lp, 1 < p < ∞ (Bourgain, 1988a),• Sequences of primes, in Lp, 1 < p < ∞ (Bourgain, 1988b; Wierdl, 1988).

Furthermore, the limit along the first three sequences is the right one, namely, tothe space average! A more comprehensive list of sequences along which ergodicaverages exist a.e. (as well as those along which averages fail to converge in anysystem) is available in the sources (Eisner et al., 2015; Petersen and Salama, 1995).

10 D. Çömez

We all know well that no measuring device is capable of making perfectmeasurements. Hence, many measurements are somewhat “tainted” or modulated.That is, instead of obtaining the values f (T kx) along the orbit of the point x, wewould be getting values like akf (T kx) for some sequence {ak}. Then, we’re lookingfor modulated averages

1

n

n−1∑k=0

akf (T kx).

For which sequences {ak} does limn1n

∑n−1k=0 akf (T kx) converge a.e.? As in the

previous cases, the answer is affirmative for some sequences. The following is a listof sequences {ak} along which a.e. convergence holds, i.e., limn

1n

∑n−1k=0 akf (T kx)

exists a.e. for f ∈ L1(X) :

• ak = λk, where |λ| = 1 (Wiener, 1939; Wiener and Wintner, 1941),• {ak} is a bounded Besicovitch sequence (Bellow and Losert, 1985)• {ak} is a sequence having a mean (Bellow and Losert, 1985)• {ak} is a dynamically generated sequence (Çömez et al., 1998).

The first of these two results has been extended to various settings, to L1 − L∞-contraction case (Çömez et al., 1998; Lin et al., 1999), to the case of function valuedsequences (Çömez and Litvinov, 2013), and to the setting of noncommutative vonNeumann algebras (Litvinov, 2012). Since the setup and the statements of theseresults require introduction of many technical concepts, we will refrain from doingso.

Convergence of the time averages also exists in norm, which was proved aroundthe same time as Birkhoff’s ergodic theorem, in 1932, by J. von Neumann, which isknown as the mean ergodic theorem (von Neumann, 1932). In the case of probabilityspaces, convergence in norm follows from Birkhoff’s ergodic theorem. However, amore general convergence result is also the case:

Theorem 8 (von Neumann’s mean ergodic theorem). If U is a unitary operatoron a Hilbert space H, then for any f ∈ H, limn→∞ 1

n

∑n−1k=0 Ukf = f ∗ exists in

norm and f ∗ ∈ F where F ⊂ H is the subspace of U -invariant elements.

If H = L2(X) of a probability space, this is exactly the norm convergenceversion of Birkhoff’s ergodic theorem. As expected various extensions and gen-eralizations of mean ergodic theorem are also proved, such as the multiparameterversion by Dunford and Schwartz (1988), the analogue of Brunel’s theorem fornorm convergence by Çömez and Lin (1991), and, recently, a generalization of itto non-conventional averages by Tao (2008).

In his quest to provide an ergodic theoretical proof of Szemeredi’s theorem, in1977 H. Fürstenberg proved a generalization of Poincare recurrence theorem whichproved to have far-reaching consequences.

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 11

Theorem 9 (Multiple recurrence theorem (Fürstenberg, 1977)). Let (X,μ) bea probability space and : X → X be a measure-preserving transformation. If k ≥ 1and μ(A) > 0, then

lim infn→∞

1

n

n∑i=1

μ(A ∩ T −iA ∩ T −2iA ∩ . . . ∩ T −(k−1)iA) > 0.

In particular, for some n ≥ 1, μ(A ∩ T −nA ∩ T −2nA ∩ . . . ∩ T −(k−1)nA) > 0.

This is a significant strengthening of Poincare recurrence theorem.Actually, Fürstenberg proved a more general version, namely, if T is weakly

mixing, then for every integer k ≥ 1 and for all {fj }kj=1 ⊂ L∞(X),

1

N

N∑n=1

f1(Tnx)f2(T

2nx) . . . fk(Tknx) →

∫f1

∫f2 . . .

∫fk in L2,

which can be considered a special multiparameter von Neumann mean ergodictheorem ( 1

nk is replaced by 1n

). Over the years various generalizations of this resulthave been obtained. The first type of generalization is proving the convergenceof multiple ergodic averages along integer sequences. Belgelson, Fürstenberg andWeiss, and Host and Kra obtained results of this type. The most general one sofar was proved by Leibman (2005), who proved that if T is an invertible measure-preserving transformation, then for every integer k ≥ 1, for any integer polynomialsp1, p2, . . . , pk with pi(0) = 0, and for all {fj }kj=1 ⊂ L∞(X),

1

N

N∑n=1

f1(Tp1(n)x)f2(T

p2(n)x) . . . fk(Tpk(n)x) converges in L2 as N → ∞.

The case when pm(n) = mn, 1 ≤ m ≤ k, was obtained by Host and Kra(2005). Also, they proved that if T n is ergodic for all n ≥ 1 and pi’s are rationallyindependent, then the L2-limit is

∫f1 . . .

∫fk.

Another way to generalize Fürstenberg multiple recurrence theorem is byconsidering simultaneous intersection of iterates of the same set under differentmeasure-preserving transformations. A generalization of this kind has been studiedby Fürstenberg and Katznelson, Conze and Lesigne, Frantzikinakis and Kra.One of the latest results of this type is due to Tao (2008); he proved that ifT1, T2, . . . , Tk are commuting invertible measure-preserving transformations, thenfor all {fj }kj=1 ⊂ L∞(X),

1

N

N∑n=1

f1(Tn1 x)f2(T

n2 x) . . . fk(T

nk x) converges in L2 as N → ∞.

12 D. Çömez

Although Birkhoff’s ergodic theorem guarantees a.e. convergence of the averagesAn := 1

n

∑n−1k=0 f (T kx) for f ∈ L1, it does not provide any information on the rate

of convergence. By Abel’s summation by parts formula,

n∑k=1

f (T kx)

k= f (T nx)

n− f +

n−1∑k=1

Ak

k(k + 1);

which suggest that one can obtain the rate of convergence once the left side of thisequality is known to converge a.e. However, this is far from the case; also it has beenshown that one cannot have any knowledge on the rate of convergence in ergodictheorem unless additional conditions are assumed. On the other hand, the left-handside of this equality has some connotations with the well-known Hilbert transform.Inspired by this, M. Cotlar proved in 1955 that:

Theorem 10 (Existence of the ergodic Hilbert transform (Cotlar, 1955)). Let(X,μ) be a probability space and T be an invertible measure-preserving transfor-mation. Then, for all f ∈ L1(X),

limn→∞

n∑k=−n,k �=0

f (T kx)

kexists a.e. on X.

As in the case of others, this theorem has also been extended to various settings; forinstance, Campbell and Petersen (1989), Sato (1987), Jones et al. (1998), and othersproved that it is also valid in the operator setting when T is a unitary operator onL2 and when T is an L1 and L∞-contraction. Modulated version of this result hasrecently been proved by Akhmedov and Çömez (2015).

The statements mentioned in this section, which are some of the important resultsproved in ergodic theory, constitute a small sample from a vast collection of resultsin ergodic theory. For example, some other important types of statements in ergodictheory are omitted: ergodic theorems for semigroup of operators, ergodic theoremsof group actions, noncommutative ergodic theorems, superadditive ergodic theo-rems, etc. For a more comprehensive study, the reader is referred to the sources(Eisner et al., 2015; Goldstein and Litvinov, 2000; Petersen and Salama, 1995).The article Moore (2015) provides a nice historical perspective of the theorems ofBirkhoff and von Neumann.

Interdisciplinary Aspects of Ergodic Theory in Mathematics

In this section we will explore the interdisciplinary aspect of ergodic theory withinmathematics. Ergodic theory is intimately intertwined with topological dynamicsand probability theory. Many measurable dynamical concepts are amenable to betransferred to the topological setting, sometimes verbatim or in some other caseswith some modifications. In the same way, sequences of independent random

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 13

variables can be obtained using measurable dynamical systems. For example, withT (x) = 2x(mod1), and with the function

f (x) =

⎧⎪⎪⎨⎪⎪⎩

1 if x ∈ [0,1

2)

0 if x ∈ [1

2, 1],

f (T n(x))s are independent random variables for most x ∈ [0, 1]. This is nothingbut the Bernoulli shift on two letters. Indeed, both weak law of large numbersand strong law of large numbers can be deduced from the mean ergodic theoremand Birkhoff’s ergodic theorem, respectively. Naturally, there are many straight-forward applications of ergodic theory statements in topological dynamics andprobability.

As it also happens naturally in many fields of mathematics, in the developmentof ergodic theory, many theorems or methods from other fields of mathematicswere instrumental. In return, many statements of ergodic theory, particularly theresults stated in the previous section, have contributed significantly to other fieldsof mathematics, in some cases by leading to the creation of a new line of researchwithin that field. In this regard, we will focus on the interaction with number theory,combinatorics, functional analysis and harmonic analysis, and fractal geometry. Onecan continue to add many other fields with significant ergodic theory impact to thislist, but we will limit ourselves to those mentioned above.

Number Theory

A sequence of numbers {an} is called equidistributed (modulo 1) if for any interval[a, b] ⊂ [0, 1]

limn

|{k : 1 ≤ k ≤ n,< ak >∈ [a, b]}| = b − a,

where |A| denotes the cardinality of the set A and < x > denotes the fractional partof x ∈ R. In 1916 Weyl proved that the sequence {< nα >} is equidistributed if andonly if α is irrational. Later, he also showed that {< n2α >} is equidistributed if andonly if α is irrational. Equidistribution of {< pnα >}, where pn is the n-th prime,was proved by Vinogradov. Let T (x) = x + α(mod1), and with the function

f (x) ={

χ[a,b](x) if b < 1

χ{0}∪[a,b](x) if b = 1,

14 D. Çömez

then

1

n

n−1∑k=0

f (T k(0)) = 1

n

n−1∑k=0

f (< kα >) = 1

n|{k : 1 ≤ k ≤ n,< kα >∈ [a, b]}|.

By the mean ergodic theorem (uniform version), the limit of these averages existsand is equal to

∫ 10 f (x)dx = b − a, implying Weyl’s equidistribution theorem.

Similarly, the equidistribution of the sequences {< n2α >} and {< pnα >}and many other such sequences are also obtained by reducing the problem intothe convergence of ergodic averages and applying an appropriate version of meanergodic theorem mentioned in the previous section.

A number 0 < x < 1 is called normal to the base k if every finite combination oflength m of the digits {0, 1, . . . , k − 1} appears in the expansion of x (to the base k)with asymptotic frequency of 1

km . The famous theorem of Borel (1909) states that inalmost every real number (with respect to Lebesgue measure) in its binary expansionthe frequency of appearances of 0 (or 1) is 1

2 . This is equivalent to the statement forall m ≥ 1, l = 0, 1, . . . , 2m − 1,

limn→∞

1

n

n−1∑i=0

χCm,l(T ix) = 1

2m,

where Cm,l = [ l2m , l+1

2m ] and T (x) = 2x(mod1). Then Borel’s result is a simpleconsequence of a.e. convergence of ergodic averages. The more general statementthat almost every number in [0,1] is normal to any base k follows from the ergodictheorem similarly by using the transformation T (x) = kx(mod1).

Two efficient ways of approximating a real number in [0, 1), in particular anirrational number, are via decimal expansion and continued fraction expansion. Bothof these expansions can be studied as dynamical systems ([0, 1), T ), where T :[0, 1) → [0, 1) is the ten-fold map T (x) = 10x(mod1) in the case of decimalexpansion and in the case of continued fraction expansion T : [0, 1) → [0, 1) is theGauss map

T (x) =

⎧⎪⎨⎪⎩

1

x− � 1

x� if x ∈ (0, 1)

0 if x = 0,

where �x� is the floor function. Clearly, these maps have significantly differentdynamical behavior; hence, one would be inclined to deduce that this differencein behavior would reflect on the corresponding expansions for a given irrationalnumber. In 1964 G. Lochs proved that contrary to this expectation, for almost allirrationals for large enough n, the n digits of decimal expansion of an irrationalnumber x determine close to first n digits (partial quotients) of its continued fraction

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 15

expansion! Let x ∈ (0, 1) be an irrational number with decimal expansion (which isunique)

x = 0.d1d2 . . . dn . . . .

For any n ≥ 1, let xn = 0.d1d2 . . . dn, the rational number determined by the first n

decimal digits of x. Let

c(xn) = [0; c1, c2, . . . , ck]

be the continued fraction expansion of xn, which is unique if ck = 1 is not allowed.Define k(x, n) as the largest integer for which digits of c(xn) coincide with the digitsof actual continued fraction expansion of x.

Theorem 11 (Lochs 1964). For almost all irrational numbers (with respect toLebesgue measure)

limn→∞

k(x, n)

n= 6 log 2 log 10

π2 ≈ 0.97027014.

Since the limiting value is very close to 1, it follows that for a large enough n,

knowing n decimal digits of x completely determines its k(x, n) (almost closeto n) partial quotients. Indeed, Lochs himself showed that first 1000 decimals of π

determine its first 968 partial quotients (Lochs, 1963). The proof of Lochs’ theoremfollows from Shannon-McMillan-Breiman which is, in turn, proved via Birkhoff’sergodic theorem. It turns out that one can replace continued fraction by other typeof expansions, such as Lüroth expansion and β-continued fraction expansion, withsimilar conclusions (Bosma et al., 2006; Dajani and Fieldsteel, 2001).

Combinatorics

As mentioned above, Fürstenberg’s multiple ergodic theorem was obtained inconnection with Szemeredi’s theorem. In 1927, van der Waerden proved that in anyfinite partition of N there is an element of the partition that contains arbitrarily longarithmetic progressions (namely, a sequence of the form {a + kd}nk=0, a, d ∈ N).Knowing this result, Erdös and Turan conjectured that any subset E of N thatfails to contain an arithmetic progression of length k must be of 0 density (i.e.,limn→∞ |E∩{1,2,...,n}|

n= 0). This is the same as stating that any subset of N with

positive upper density contains arbitrary long arithmetic progressions. Roth (1952)proved the conjecture affirmatively if k = 3, and so did Szemeredi when k = 4first (1969) and then for all k ≥ 1 in 1975. Since the proof is very involved andis not easily accessible, naturally mathematicians sought for simpler argumentsthat yielded the same conclusion. Fürstenberg realized a deep connection betweena type of Poincare recurrence and arithmetic progressions, which led him to the

16 D. Çömez

multiple recurrence theorem from which Szemeredi’s theorem follows (Fürstenberg,1977). The key ingredient in this connection is also known as the Fürstenbergcorrespondence principle:

Let E ⊂ Z with d(E) = lim supn→∞|E∩{1,2,...,n}|

n> 0. Then there exists

an invertible measure-preserving system (X,μ, T ) and a measurable set A withμ(A) = d(E) such that for all finite subset C of Z

d( ∩n∈C (E − n)

)≥ μ

(∩n∈C T −nA

).

This result, as well as being much more accessible, opened up a new avenue ofresearch in ergodic theory as well as combinatorics. Simply put, by the correspon-dence principle, in order to claim the property about arithmetic sequences, all oneneeds is to establish the right side of the inequality. Since all generalizations andextensions of Fürstenberg multiple recurrence establish various extensions of theinequality in the correspondence principle, it begs to see the associated configurationabout arithmetic sequences in Z. First of all, Fürstenberg multiple recurrencetheorem (and correspondence principle) paved way to a subfield of ergodic theory,also termed as ergodic Ramsey theory, containing many results of similar nature.Some notable ones are:

• IP-multiple recurrence theorem. A set I = {ni1 +ni2 +· · ·+nik : k ∈ N, i1 <

i2 < · · · < ik} is called an IP-set. Fürstenberg and Katznelson proved an IP-multiple recurrence theorem which states that the recurrence always takes placealong an IP-sets (Fürstenberg and Katznelson, 1985).

• Polynomial Szemeredi theorem. This is the consequence of polynomial exten-sion of multiple recurrence theorem, from which the polynomial van derWaerden theorem is also obtained by Bergelson and Leibman (2003).

There are other such results, some of which have no counterpart within combinato-rial number theory, that opened up a new challenge in finding associated structureswithin Z. Furthermore, proving existence of these new structural theorems solelywith combinatorial techniques has been another new task for mathematicians.Another notable outcome is a recent result of Tao and Ziegler (2008), whoproved, via appropriate ergodic theorem and correspondence principle, a remarkabletheorem which states that the set of primes contains arbitrarily long arithmeticprogressions!

Functional Analysis and Harmonic Analysis

Ergodic theorems are in general difficult to prove, or their proofs are very“involved,” with a few exceptions, like the Poincare recurrence theorem. Naturally,these proofs employ various theorems and/or techniques from other fields ofmathematics, notably from functional analysis and harmonic analysis. One such

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 17

is the Banach principle, which states that, if a convergence result takes place on adense subset of a function space and if the (averaging) operators involved in theprocess are bounded, then the convergence holds for all functions. This functionalanalytic result has been utilized, with some necessary modifications, in all a.e.ergodic theorems stated in the previous section. However, it requires boundednessof the (averaging) operators, which is known as maximal inequality. It turns outthat, in the majority of theorems, establishing the necessary maximal inequality isone of the most difficult tasks. Indeed, proof of this inequality, despite following thesame broad pattern, is significantly difficult, as well as challenging, in each case.For instance, all the subsequential and modulated ergodic theorems stated abovefall into this category. These various types of maximal inequalities have, in turn,become an indispensable tool for many convergence results in harmonic analysis(Eisner et al., 2015; Krengel, 1985; Rosenblatt and Wierdl, 1992). Establishing asuitable dense subset of the function space involved has also been fruitful in provingnumerous decomposition results. Starting with mean ergodic theorem, one suchdecomposition is expressing the function space as the sum of invariant functionsand the “coboundary," which consists of all functions of the form f − f ◦ T .

This main theme has been extended to more general settings in functional analysis(Eisner et al., 2015; Krengel, 1985).

Proving maximal inequalities for ergodic Hilbert transform is as much, if notmore, difficult task as proving those about ergodic averages. This task, both in thelocal and discrete settings, has been an intense subject of research by numerousergodists (Campbell and Petersen, 1989; Campbell et al., 2003; Petersen, 1983).

Since the Banach principle is fundamental in proving a.e. convergence, whenergodic theorems in more general settings are considered, for instance, in spaces likeL1(X)+L∞ or in von Neumann algebras, it would be desirable to have a version ofBanach principle in such spaces. This is a nontrivial task. However, Banach principlewas extended to L1(X) + L∞ by Bellow and Calderón (1999) and to von Neumannalgebras by Goldstein and Litvinov (2000).

Akcoglu’s ergodic theorem (Akcoglu, 1975) was a breakthrough after being anopen problem for more than 40 years. One key ingredient component of the proofis a dilation argument that extended the a.e. convergence from a “simpler” spaceto a general one containing the former. There were several dilation theorems infunctional analysis, particularly in the Hilbert space setting, but they all had alimited scope. Akcoglu’s dilation argument was devoid of many of these restrictions.Not surprisingly, this result has been generalized to many settings by numerousmathematicians (Akcoglu and Sucheston, 1977; Nagel and Palm, 1982).

Fractal Geometry

Before closing, an interesting feature of some dynamical systems in connection withfractal geometry is worth mentioning. For that purpose, consider the tent map, witha modified form as described below. Let X = [0, 1] with Lebesgue measure anddefine

18 D. Çömez

T (x) =

⎧⎪⎪⎨⎪⎪⎩

3x if x ∈ [0,1

2)

3(1 − x) if x ∈ [1

2, 1].

We immediately notice that T does not map [0, 1] into itself. When x ∈( 1

2 , 23 ), T x > 1; hence, it lies outside the interval [0, 1].

The successive images of [0, 1] under T will reveal an interesting outcome.If we define J1 = {x ∈ [0, 1] : T x > 1}, then it is centered around the pointx = 1

2 . Then [0, 1] \ J1 consists of two closed intervals, both of which are mappedone-to-one and onto [0, 1]. Similarly, define J2 = {

x ∈ [0, 1] : T 2x > 1}. Then

[0, 1] \ (J1 ∪ J1) consists of four closed intervals that are mapped one-to-oneand onto [0, 1]. Continuing on we can define In = {x ∈ [0, 1] : T nx ∈ [0, 1]} andJn = {x ∈ In−1 : T nx > 1}. Notice that In = [0, 1] \ ⋃n

k=1 Jk has 2n disjointintervals that T maps one-to-one and onto [0, 1]. Taking the set of all such intervals,define C = {x ∈ [0, 1] : T nx ∈ [0, 1], ∀n ≥ 1}. Since all of these intervals are one-to-one and onto [0, 1], C is mapped to itself. Hence, the pair (C, T ) is a dynamicalsystem. Furthermore from the construction, we can see that C = ⋂

n≥1 In is theCantor set, a fractal!

This is just an example of how one can construct fractals via dynamical systems.The most important consequence of such constructions is that one can bring thepowerful tools of ergodic theory to study the properties of resulting fractals. Indeed,some fundamental features of fractals (such as box dimension, Hausdorff dimension(Pesin, 1997), subfractal structure (Sattler, 2017), etc.) have been studied in depthonly after the introduction of ergodic theory tools into fractal geometry.

Interdisciplinary Aspects of Ergodic Theory with OtherDisciplines

The central mathematical object in ergodic theory is a self-map T , whether it isa measure-preserving transformation on a measure space or a linear operator ona suitable function space. The dynamics that this self-map creates leads up to theconsequences discussed in the previous sections. Basic properties involved in allthese successful results are ergodicity, mixing, and recurrence, and the fundamentalidea is the equality, in some sense, of space average and time average, which requiresexistence of an invariant measure. It is natural to expect that in any disciplinewhich deals with phenomena that changes in time or involves a sequential process,the notions and the central idea of ergodic theory can be utilized in one form oranother. Such disciplines are widespread, many experimental sciences (physics andastronomy, chemistry, biological sciences, climatology), social sciences (populationstudies, sociology), engineering (electrical engineering, aerospace engineering,control theory, transportation studies), economics, and medicine (neural networks).

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 19

The mathematical universe is an ideal one, in the sense that the axioms, terms,and relations on them are well defined and the statements proved are valid withoutany doubt. However, this is not the case in the real world. Consequently, there isalways some kind of discrepancy, or margin of error, or modeling assumptions,etc., in applications to real-world situations. Therefore, it is not unusual thatapplication of a mathematical theory into a real-world problem is contested orcan even be controversial. Indeed, as mentioned earlier, heated debates that tookplace at the earlier stages of ergodic theory is an excellent testimony to this fact.Similarly, today, in some disciplines where ergodic theory concepts or results areimplemented, we witness debates taking place on the necessity or validity of thisimplementation. Paradoxical as it may seem, these conflicts frequently bring newideas or methodologies within the discipline or establish the existing one firmlyin place. The goal of this chapter is to exhibit existing and working connectionsbetween ergodic theory and other disciplines; engaging in any debate about theissues raised on these connections is out of the context. Also, for obvious reasons,rather than providing a comprehensive survey, we will bring up a few selectedcases of interdisciplinary connections. Since the expertise of the author is in ergodictheory only, these connections will be displayed in broad terms.

Categorically, interactions of ergodic theory with other disciplines happen in twomanners: either the idea of “time average = space average” is utilized in one form oranother, or an ergodic property of the map (ergodicity, mixing, exactness) governingthe dynamical system at hand is assumed. In the first case, one needs a suitableinvariant measure, and in the second case, the property attributed to the map mustrepresent the reality as closely as possible. Both of these can be contentious dueto the extent of accuracy of modeling or adherents of the opposite view. In generalterms, if a real-world phenomenon under investigation gives rise to a dynamicalsystem, often it is a chaotic system; hence, it is not amenable to making precise long-term predictions along trajectories. On the other hand, in many cases, the estimationor measurement of averages is more likely, which leads to derivation of an invariantmeasure. This outcome, naturally, paves way to employing all the machinery ofergodic theory into the study. In practice, deriving an invariant measure meansfinding an algorithm which provides a description of the measure, which, in manycases, takes the form of obtaining the measure as a (finite or countable) discretemeasure. The quest for developing such algorithms, in turn, is a new area of researchin computer science and applied mathematics. Below, in order to provide a contextto interdisciplinary connections of ergodic theory, we will discuss ergodic theoryconnections within climate science, biological science, and economics in somebroad terms.

One can view climate as the expected value of meteorological quantities, such assurface temperature and precipitation, main contributors of which are atmosphere,hydrosphere, biosphere, and geosphere. Each of these is a dynamical system whichevolve under the action of multiple “transformations,” such as solar heating, rotationof the Earth, geological events, etc. Furthermore, these agents of change are highlyinterconnected. Therefore, all the dynamical system models of local, regional, orglobal climatic systems constitute extremely complex systems. The records of

20 D. Çömez

observations of climatic changes over the years and some extensive research onpolar ice and on fossils provide clear evidence on the variability of climatic eventsboth on the short term and the long term. There are weekly or seasonally or yearlycycles; there are also cycles expanding much longer time intervals, some of whichshow measurable differences. Clearly, characterizing this variability is fundamentalin understanding the predictability of climate and weather over both short time andlong time intervals.

Lorenz (1963) was one of the pioneers that used a dynamical systems approachto understand climatic changes and predictability (Lorenz, 1980). His systemwas induced by three nonlinear ordinary differential equations (known as Lorenzequations) whose solutions, few exceptions aside, are inherently chaotic. Hence,while this model would provide quite accurate information in the short term, itwould be unstable beyond a certain range.

Due to the pioneering work of Lorenz and adherents of his ideas, today itis possible to predict atmospheric events within a range of a week to 10 daysaccurately. It is also accepted that beyond a few weeks, such predictions are unlikelyto be accurate. Today, many climatologists agree that Lorenz’s model needs to beimproved; hence, the issue is not whether climate should be modeled as a dynamicalsystem, rather the nature of this dynamical system chosen and to which extent itrepresents the reality (Lucarini, 2009; May, 1987; Ozawa et al., 2003; Pietsch andHasenauer, 2005; Thornton, 1998).

It is a pleasant irony that, besides being a pioneering work in climatology,Lorenz’s work was also inspirational within the ergodic theory and dynamicalsystems. The particular system he developed, in a simplified form given as T x =ax3 + (1 − a)x, is non-periodic one with a very complicated attractor to analyze,a strange attractor and a fractal. It is one of the most studied examples of chaoticdynamical systems; only relatively recently mathematicians have had a firm graspof its features (Tucker, 2002). Furthermore, it turns out that Lorenz equations arenot peculiar to the model he intended to study; they also appear (in some modifiedform, of course) in other models of various phenomena in physics, engineering, andchemistry.

Economics may seem an unlikely discipline that would have ergodic theoryconnections. However, many economical events evolve subject to some agents thatdrive it, such as time, market fluctuations, political events, etc., many of which areinherently unpredictable. On the other hand, over long time scales, there are someregularities that can be measured. So, it can be assumed that, given this statisticalinformation on the past, the future should be predictable. From this perspective,interpreting past fluctuations of prices as time series, Samuelson (1965) modeledmarket future pricing as a stochastic series of events, more specifically, a stationaryprocess with an underlying ergodic probability (Samuelson, 1965). Here, one canutilize the time series analysis tools that ergodic theory provides. This was arevolutionary approach and opened up both avenues of new means of market studiesas well as avenues of criticisms (Blume, 1979). Some of the debate focused onthe dependence of the probability distribution on the initial distribution. This, onthe other hand, is ignorable, by the uniform mean ergodic theorem, if the process

Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical. . . 21

involved has a unique ergodic part. Today, both the objections to the use of ergodictools (by classical economists) and study of various economical processes as anergodic one continue (Jovanovic and Schinckus, 2013; Poitras and Heaney, 2015).

Needless to say, biological science is one of the prime sources of dynamicalsystems as well as a discipline amenable to applications of such. Biological systemshave strong resemblance to those in climate science; often they involve multipleagents of change. Hence, their models can range from a relatively simple systemto a highly complex one. The logistic system, being one of the prototypes of anergodic dynamical system with chaotic behavior, with numerously modified forms,besides providing a model to study some biological systems in depth, also suppliesthe foundation for developing new ones that apply to predator-prey type processesor genetics or many more (May, 1987).

As another example, the plasma membrane of living cells is a porous tissue;hence, diffusion through the membrane, being time dependent, can be studied bymodeling it as a dynamical system. Further, in this model tracking a single moleculecan be interpreted as the trajectory of a point, which naturally gives possibility ofboth time series analysis of individual trajectories and ensemble averages. It turnsout that this is a very complex system that displays seemingly opposing processes,namely, both ergodic and fractal and non-ergodic (Weigel et al., 2011).

Besides statistical mechanics, the originating field, many other systems inphysics are modeled as a dynamical system in which ergodicity naturally emerges.This is not limited to classical dynamics; it also expands to quantum mechanicalsystems (Hofmann, 2015; Klein, 1952). In a quantum system, the equality oftime and ensemble averages is equivalent to having all invariant operators being aconstant multiple of the unit operator, which is essentially von Neumann’s spectralcharacterization of ergodicity. On the other hand, some research also indicate that,due to inherent uncertainty within a quantum system, it would be desirable toinfuse statistical properties into quantum mechanical systems without requiring orassuming the existence of an external probability measure. Such considerations ledto the concept of “breaking ergodicity,” describing systems which are non-ergodic,but their phase space is not necessarily decomposable into regions on which therestricted system is ergodic (Bel and Barkai, 2006; Bouchaud, 1992; Turner et al.,2018). This is not the same as ergodic decomposition of a (non-ergodic measure-preserving) transformation. Ergodic theory emerged from physics and grew into amature and influential field of mathematics with significant applications to otherdisciplines (including physics). Does that mean physics has given birth to anotherergodic-like concept that would become a new theory?

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