24 results
8 - Equivalent systems
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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This chapter is devoted to the isomorphism problem in ergodic theory: under what conditions should two systems (f, μ) and (g,ν) be considered “the same” and how does one decide, for given systems, whether they are in those conditions?
The fundamental notion is called ergodic equivalence: two systems are said to be ergodically equivalent if, restricted to subsets with full measure, the corresponding transformations are conjugated by some invertible map that preserves the invariant measures. Through such a map, properties of either system may be translated to corresponding properties of the other system.
Although this is a natural notion of isomorphism in the context of ergodic theory, it is not an easy one to handle. In general, the only way to prove that two given systems are equivalent is by exhibiting the equivalence map more or less explicitly. On the other hand, the most usual way to show that two systems are not equivalent is by finding some property that holds for one but not the other.
Thus, it is useful to consider a weaker notion, called spectral equivalence: two systems are spectrally equivalent if their Koopman operators are conjugated by some unitary operator. Two ergodically equivalent systems are always spectrally equivalent, but the converse is not true.
The idea of spectral equivalence leads to a rich family of invariants, related to the spectrum of the Koopman operator, that must have the same value for any two systems that are equivalent and, thus, may be used to exclude that possibility. Other invariants, of non-spectral nature, have an equally crucial role. The most important of all, the entropy, will be treated in Chapter 9.
The notions of ergodic equivalence and spectral equivalence, and the relations between them, are studied in Sections 8.1 and 8.2, respectively. In Sections 8.3 and 8.4 we study two classes of systems with opposite dynamical features: transformations with discrete spectrum, that include the ergodic translations on compact abelian groups, and transformations with a Lebesgue spectrum, which have the Bernoulli shifts as the main example.
These two classes of systems, as well as others that we introduced previously (ergodicity, strong mixing, weak mixing) are invariants of spectral equivalence and, hence, also of ergodic equivalence. Finally, in Section 8.5 we discuss a third notion of equivalence, that we call ergodic isomorphism, especially in the context of Lebesgue spaces.
3 - Ergodic theorems
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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5 - Ergodic decomposition
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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For convex subsets of vector spaces with finite dimension, it is clear that every element of the convex set may be written as a convex combination of the extremal elements. For example, every point in a triangle may be written as a convex combination of the vertices of the triangle. In view of the results in Section 4.3, it is natural to ask whether a similar property holds in the space of invariant probability measures, that is, whether every invariant measure is a convex combination of ergodic measures.
The ergodic decomposition theorem, which we prove in this chapter (Theorem 5.1.3), asserts that the answer is positive, except that the number of “terms” in this combination is not necessarily finite, not even countable. This theorem has several important applications; in particular, it permits the reduction of the proof of many results to the case when the system is ergodic.
We are going to deduce the ergodic decomposition theorem from another important result from measure theory, the Rokhlin disintegration theorem. The simplest instance of this theorem holds when we have a partition of a probability space (M,μ) into finitely many measurable subsets P1,…,PN with positive measure. Then, obviously, we may write μ as a linear combination
μ = μ(P1)μ1+· · ·+μ(PN)μN
of its normalized restrictions μi(E) = μ(E ⋂Pi)/μ(Pi) to each of the partition elements. The Rokhlin disintegration theorem (Theorem 5.1.11) states that this type of disintegration of the probability measure is possible for any partition P (possibly uncountable!) that can be obtained as the limit of an increasing sequence of finite partitions.
Ergodic decomposition theorem
Before stating the ergodic decomposition theorem, let us analyze a couple of examples that help motivate and clarify its content:
Example 5.1.1. Let f : [0,1] → [0,1] be given by f(x) = x2. The Dirac measures δ0 and δ1 are invariant and ergodic for f. It is also clear that x = 0 and x = 1 are the unique recurrent points for f and so every invariant probability measure μ must satisfy μ({0,1}) = 1. Then, μ = μ({0})δ0 + μ({1})δ1 is a (finite) convex combination of the ergodic measures.
Frontmatter
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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1 - Recurrence
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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Ergodic theory studies the behavior of dynamical systems with respect to measures that remain invariant under time evolution. Indeed, it aims to describe those properties that are valid for the trajectories of almost all initial states of the system, that is, all but a subset that has zero weight for the invariant measure. Our first task, in Section 1.1, will be to explain what we mean by ‘dynamical system’ and ‘invariant measure’.
The roots of the theory date back to the first half of the 19th century. By 1838, the French mathematician Joseph Liouville observed that every energy-preserving system in classical (Newtonian) mechanics admits a natural invariant volume measure in the space of configurations. Just a bit later, in 1845, the great German mathematician Carl Friedrich Gauss pointed out that the transformation
(0,1]→ ℝ, x ↦ fractional part of 1/x,
which has an important role in number theory, admits an invariant measure equivalent to the Lebesgue measure (in the sense that the two have the same zero measure sets). These are two of the examples of applications of ergodic theory that we discuss in Section 1.3. Many others are introduced throughout this book.
The first important result was found by the great French mathematician Henri Poincaré by the end of the 19th century. Poincaré was particularly interested in the motion of celestial bodies, such as planets and comets, which is described by certain differential equations originating from Newton's law of universal gravitation. Starting from Liouville's observation, Poincaré realized that for almost every initial state of the system, that is, almost every value of the initial position and velocity, the solution of the differential equation comes back arbitrarily close to that initial state, unless it goes to infinity. Even more, this recurrence property is not specific to (celestial) mechanics: it is shared by any dynamical system that admits a finite invariant measure. That is the theme of Section 1.2.
The same theme reappears in Section 1.5, in a more elaborate context: there, we deal with any finite number of dynamical systems commuting with each other, and we seek simultaneous returns of the orbits of all those systems to the neighborhood of the initial state. This kind of result has important applications in combinatorics and number theory, as we will see.
Preface
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- By Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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In short terms, ergodic theory is the mathematical discipline that deals with dynamical systems endowed with invariant measures. Let us begin by explaining what we mean by this and why these mathematical objects are so worth studying. Next, we highlight some of the major achievements in this field, whose roots go back to the physics of the late 19th century. Near the end of the preface, we outline the content of this book, its structure and its prerequisites.
What is a dynamical system?
There are several definitions of what a dynamical system is some more general than others. We restrict ourselves to two main models.
The first one, to which we refer most of the time, is a transformation f : M → M in some space M. Heuristically, we think of M as the space of all possible states of a given system. Then f is the evolution law, associating with each state x ∈ M the one state f(x) ∈ M the system will be in a unit of time later. Thus, time is a discrete parameter in this model.
We also consider models of dynamical systems with continuous time, namely flows. Recall that a flow in a space M is a family f t : M→M, t ∈ ℝ of transformations satisfying
f0 = identity and f t ∘ f s = f t+s for all t, s ∈ ℝ.
Flows appear, most notably, in connection with differential equations: take f t to be the transformation associating with each x ∈ M the value at time t of the solution of the equation that passes through x at time zero.
We always assume that the dynamical system is measurable, that is, that the space M carries a σ-algebra of measurable subsets that is preserved by the dynamics, in the sense that the pre-image of any measurable subset is still a measurable subset. Often, we take M to be a topological space, or even a metric space, endowed with the Borel σ-algebra, that is, the smallest σ-algebra that contains all open sets. Even more, in many of the situations we consider in this book, M is a smooth manifold and the dynamical system is taken to be differentiable.
11 - Expanding maps
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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The distinctive feature of the transformations f : M → M that we study in the last two chapters of this book is that they expand the distance between nearby points: there exists a constant σ >1 such that
d(f(x), f(y)) ≥ σd(x, y)
whenever the distance between x and y is small (a precise definition will be given shortly). There is more than one reason why this class of transformations has an important role in ergodic theory.
On the one hand, as we are going to see, expanding maps exhibit very rich dynamical behavior, from the metric and topological point of view as well as from the ergodic point of view. Thus, they provide a natural and interesting context for utilizing many of the ideas and methods that have been introduced so far.
On the other hand, expanding maps lead to paradigms that are useful for understanding many other systems, technically more complex. A good illustration of this is the ergodic theory of uniformly hyperbolic systems, for which an excellent presentation can be found in Bowen [Bow75a].
An important special case of expanding maps are the differentiable transformations on manifolds such that
∥Df(x)v∥ ≥ σ∥v∥
for every x ∈ M and every vector v tangent to M at the point x. We focus on this case in Section 11.1. The main result (Theorem 11.1.2) is that, under the hypothesis that the Jacobian detDf is Hölder, the transformation f admits a unique invariant probability measure absolutely continuous with respect to the Lebesgue measure. Moreover, that probability measure is ergodic and positive on the open subsets of M.
In Section 11.2 we extend the notion of an expanding map to metric spaces and we give a global description of the topological dynamics of such maps, starting from the study of their periodic points. The main objective is to show that the global dynamics may always be reduced to the topologically exact case (Theorem 11.2.15). In Section 11.3 we complement this analysis by showing that for these transformations the topological entropy coincides with the growth rate of the number of periodic points.
The study of expanding maps will proceed in Chapter 12, where we will develop the so-called thermodynamic formalism for such systems.
2 - Existence of invariant measures
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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In this chapter we prove the following result, which guarantees the existence of invariant measures for a broad class of transformations:
Theorem 2.1 (Existence of invariant measures). Let f : M → M be a continuous transformation on a compact metric space. Then there exists some probability measure on M invariant under f.
The main point in the proof is to introduce a certain topology in the set M1(M) of probability measures on M, that we call weak* topology. The idea is that two measures are close, with respect to this topology, if the integrals they assign to (many) bounded continuous functions are close. The precise definition and some of the properties of the weak* topology are presented in Section 2.1. The crucial property, that makes this topology so useful for proving the existence theorem, is that it turns M1(M) into a compact space (Theorem 2.1.5).
The proof of Theorem 2.1 is given in Section 2.2. We will also see, through examples, that the hypotheses of continuity and compactness cannot be omitted.
In Section 2.3 we insert the construction of the weak* topology into a broader framework from functional analysis and we also take the opportunity to introduce the notion of the Koopman operator of a transformation, which will be very useful in the sequel. In particular, as we are going to see, it allows us to give an alternative proof of Theorem 2.1, based on tools from functional analysis.
In Section 2.4 we describe certain explicit constructions of invariant measures for two important classes of systems: skew-products and natural extensions (or inverse limits) of non-invertible transformations.
Finally, in Section 2.5 we discuss some important applications of the idea of multiple recurrence (Section 1.5) in the context of combinatorial arithmetics. Theorem 2.1.5 has an important role in the arguments, which is the reason why this discussion was postponed to the present chapter.
Weak*topology
In this section M will always be a metric space. Our goal is to define the so-called weak* topology in the set M1(M) of Borel probability measures on M and to discuss its main properties.
10 - Variational principle
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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In 1965, the IBM researchers R. Adler, A. Konheim and M. McAndrew proposed [AKM65] a notion of topological entropy, inspired by the Kolmogorov–Sinai entropy that we studied in the previous chapter, but whose definition does not involve any invariant measure. This notion applies to any continuous transformation in a compact topological space.
Subsequently, Efim Dinaburg [Din70] and Rufus Bowen [Bow71, Bow75a] gave a different, yet equivalent, definition for continuous transformations in compact metric spaces. Despite being a bit more restrictive, the Bowen– Dinaburg definition has the advantage of making more transparent the meaning of this concept: the topological entropy is the rate of exponential growth of the number of orbits that can be distinguished within a certain precision, arbitrarily small. Moreover, Bowen extended the definition to non-compact spaces, which is also very useful for applications.
These definitions of topological entropy and their properties are studied in Section 10.1 where, in particular, we observe that the topological entropy is an invariant of topological equivalence (topological conjugacy). In Section 10.2 we analyze several concrete examples.
The main result is the following remarkable relation between the topological entropy and the entropies of the transformation with respect to its invariant measures:
Theorem 10.1 (Variational principle). If f : M → M is a continuous transformation in a compact metric space then its topological entropy h(f) coincides with the supremum of the entropies hμ(f) of f with respect to all the invariant probability measures.
This theorem was proved by Dinaburg [Din70, Din71], Goodman [Goo71a] and Goodwin [Goo71b]. Here, it arises as a special case of a more general statement, the variational principle for the pressure, which is due to Walters [Wal75].
The pressure P(f,ϕ) is a weighted version of the topological entropy h(f), where the “weights” are determined by a continuous function ϕ : M → ℝ, which we call a potential. We study these notions and their properties in Section 10.3. The topological entropy corresponds to the special case when the potential is identically zero. The notion of pressure was brought from statistical mechanics to ergodic theory by the Belgium mathematician and theoretical physicist David Ruelle, one of the founders of differentiable ergodic theory, and was then extended by the British mathematician Peter Walters.
12 - Thermodynamic formalism
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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In this chapter we develop the ergodic theory of expanding maps on compact metric spaces. This theory evolved from the kind of ideas in statistical mechanics that we discussed in Section 10.3.4 and, for that reason, is often called thermodynamic formalism. We point out, however, that this last expression is much broader, encompassing not only the original setting of mathematical physics but also applications to other mathematical systems, such as the so-called uniformly hyperbolic diffeomorphisms and flows (in this latter regard, see the excellent monograph of Rufus Bowen [Bow75a]).
The main result in this chapter is the following theorem of David Ruelle, which we prove in Section 12.1 (the notion of Gibbs state is also introduced in Section 12.1):
Theorem 12.1 (Ruelle). Let f : M →M be a topologically exact expanding map on a compact metric space and φ : M → ℝ be a Hölder function. Then there exists a unique equilibrium state μ for φ. Moreover, the measure μ is exact, it is supported on the whole of M and is a Gibbs state.
Recall that an expanding map is topologically exact if (and only if) it is topologically mixing (Exercise 11.2.2). Moreover, a topologically exact map is necessarily surjective.
In the particular case when M is a Riemannian manifold and f is differentiable, the equilibrium state of the potential φ = −log | detDf | coincides with the absolutely continuous invariant measure given by Theorem 11.1.2. In particular, it is the unique physical measure of f. These facts are proved in Section 12.1.8.
The theorem of Livšic that we present in Section 12.2 complements the theorem of Ruelle in a very elegant way. It asserts that two potentials φ and ψ have the same equilibrium state if and only if the difference between them is cohomologous to a constant. In other words, this happens if and only if φ−ψ =c+uof −u for some c∈ℝ and some continuous function u.Moreover, and remarkably, it suffices to check this condition on the periodic orbits of f.
In Section 12.3 we show that the system (f,μ) exhibits exponential decay of correlations in the space of Hölder functions, for every equilibrium state μ of any Hölder potential.
4 - Ergodicity
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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Index
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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9 - Entropy
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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The word entropy was invented in 1865 by the German physicist and mathematician Rudolf Clausius, one of the founding pioneers of thermodynamics. In the theory of systems in thermodynamical equilibrium, the entropy quantifies the degree of “disorder” in the system. The second law of thermodynamics states that, when an isolated system passes from an equilibrium state to another, the entropy of the final state is necessarily bigger than the entropy of the initial state. For example, when we join two containers with different gases (oxygen and nitrogen, say), the two gases mix with one another until reaching a new macroscopic equilibrium, where they are both uniformly distributed in the two containers. The entropy of the new state is larger than the entropy of the initial equilibrium, where the two gases were separate.
The notion of entropy plays a crucial role in different fields of science. An important example, which we explore in our presentation, is the field of information theory, initiated by the work of the American electrical engineer Claude Shannon in the mid 20th century. At roughly the same time, the Russian mathematicians Andrey Kolmogorov and Yakov Sinai were proposing a definition of the entropy of a system in ergodic theory. The main purpose was to provide an invariant of ergodic equivalence that, in particular, could distinguish two Bernoulli shifts. This Kolmogorov–Sinai entropy is the subject of this chapter.
In Section 9.1 we define the entropy of a transformation with respect to an invariant probability measure, by analogy with a similar notion in information theory. The theorem of Kolmogorov–Sinai, which we discuss in Section 9.2, is a fundamental tool for the actual calculation of the entropy in specific systems. In Section 9.3 we analyze the concept of entropy from a more local viewpoint, which is more closely related to Shannon's formulation of this concept. Next, in Section 9.4, we illustrate a few methods for calculating the entropy, by means of concrete examples.
In Section 9.5 we discuss the role of the entropy as an invariant of ergodic equivalence. The highlight is the theorem of Ornstein (Theorem 9.5.2), according to which any two-sided Bernoulli shifts are ergodically equivalent if and only if they have the same entropy. In that section we also introduce the class of Kolmogorov systems, which contains the Bernoulli shifts and is contained in the class of systems with Lebesgue spectrum.
Index of notation
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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6 - Unique ergodicity
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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Appendix A - Topics in measure theory, topology and analysis
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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In this series of appendices we recall several basic concepts and facts in measure theory, topology and functional analysis that are useful throughout the book. Our purpose is to provide the reader with a quick, accessible source of references to measure and integration, general and differential topology and spectral theory, to try and make this book as self-contained as possible. We have not attempted to make the material in these appendices completely sequential: it may happen that a notion mentioned in one section is defined or discussed in more depth in a later one (check the index).
As a general rule, we omit the proofs. For Appendices A.1, A.2 and A.5, the reader may find detailed information in the books of Castro [Cas04], Fernandez [Fer02], Halmos [Hal50], Royden [Roy63] and Rudin [Rud87]. The presentation in Appendix A.3 is a bit more complete, including the proofs of most results, but the reader may find additional relevant material in the books of Billingsley [Bil68, Bil71]. We recommend the books of Hirsch [Hir94] and do Carmo [dC79] to all those interested in going further into the topics in Appendix A.4. For more information on the subjects of Appendices A.6 and A.7, including proofs of the results quoted here, check the book of Halmos [Hal51] and the treatise of Dunford and Schwarz [DS57, DS63], especially Section IV.4 of the first volume and the initial sections of the second volume.
Measure spaces
Measure spaces are the natural environment for the definition of the Lebesgue integral, which is the main topic to be presented in Appendix A.2.We begin by introducing the notions of algebra and σ-algebra of subsets of a set, which lead to the concept of measurable space. Next, we present the notion of measure on a σ-algebra and we analyze some of its properties. In particular, we mention a few results on the construction of measures, including Lebesgue measures in Euclidean spaces. The last part is dedicated to measurable maps, which are the maps that preserve the structure of measurable spaces.
Measurable spaces
Given a set X, we often denote by Ac the complement X \ A of each subset A.
References
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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Contents
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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Hints or solutions for selected exercises
- Marcelo Viana, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Krerley Oliveira, Universidade Federal de Alagoas, Brazil
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7 - Correlations
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The models of dynamical systems that interest us the most, transformations and flows, are deterministic: the state of the system at any time determines the whole future trajectory; when the system is invertible, the past trajectory is equally determined. However, these systems may also present stochastic (that is, “partly random”) behavior: at some level coarser than that of individual trajectories, information about the past is gradually lost as the system is iterated. That is the subject of the present chapter.
In probability theory one calls the correlation between two random variables X and Y the number
C(X,Y) = E[(X −E[X])(Y −E[Y])] = E[XY]−E[X]E[Y].
Note that the expression (X − E[X])(Y − E[Y]) is positive if X and Y are on the same side (either larger or smaller) of their respective means, E[X] and E[Y], and it is negative otherwise. Therefore, the sign of C(X,Y) indicates whether the two variables exhibit, predominantly, the same behavior or opposite behaviors, relative to their means. Furthermore, correlation close to zero indicates that the two behaviors are little, if at all, related to each other.
Given an invariant probability measure μ of a dynamical system f : M → M and given measurable functions φ,ψ : M → ℝ, we want to analyze the evolution of the correlations
Cn(φ,ψ) = C(φ o fn,ψ)
when time n goes to infinity. We may think of φ and ψ as quantities that are measured in the system, such as temperature, acidity (pH), kinetic energy, and so forth. Then Cn(φ,ψ) measures how much the value of φ at time n is correlated with the value of ψ at time zero; to what extent one value “influences” the other.
For example, if φ = XA and ψ = XB are characteristic functions, then ψ(x) provides information on the position of the initial point x, whereas φ(fn(x)) informs on the position of its n-th iterate fn(x). If the correlation Cn(φ,ψ) is small, then the first information is of little use to make predictions about the second one. That kind of behavior, where correlations approach zero as time n increases, is quite common in important models, as we are going to see.