Ergodic theory

metric theory of dynamical systems

2020 Mathematics Subject Classification: Primary: 37Axx [MSN][ZBL] The branch of the theory of dynamical systems that studies systems with an invariant measure and related problems.

1) In the "abstract" or "general" part of ergodic theory one examines measurable dynamical systems. In the most general sense this is a triple $(W,G,F)$, where $W$ is a measurable space (the "phase space" ), $G$ is a locally compact Hausdorff group (or semi-group) with a countable base and $F$ is a measurable mapping $F:G\times W \to W$ defining a (left) action of $G$ on $W$: if $w\in W$, $e$ is the unit element of $G$ and $g,h\in G$, then (in multiplicative notation for the operation in $G$)

$$F(e,w) = w \quad \textrm{and}\quad F(gh,w) = F(g,F(h,w)).\tag{$*$}$$ (It is assumed that $G\times W$ is endowed with the structure of a measurable space as the direct product of $G$ and $W$ and that in $G$ the Borel sets (cf. Borel set) are taken to be measurable. Under the assumptions above on $G$ several versions of the latter concept (see Borel measure; Baire set) turn out to be equivalent.)

Denoting the transformation $w\mapsto F(g,w)$ by $T_g$, one can write (*) in the form $T_{gh} = T_g T_h$. (One may also consider a right action, for which $T_{gh} = T_h T_g$.) Except in those cases where the existence of an invariant measure (possibly with certain specific properties) has to be specially discussed, in ergodic theory one usually assumes that $W$ is a measure space $(W,\mu)$. Here $\mu$ is a $\sigma$-finite or finite measure that is invariant under $T_g$: If $A\subset W$ is a measurable set, then $\mu(T_g^{-1} A) = \mu(A)$. A finite measure is usually normalized; most often $(W,\mu)$ is a Lebesgue space. As regards $G$, the basic cases are $G=\Z$ or $G=\N$ (a cascade) or $G=\R$ (a flow (continuous-time dynamical system)). Of these one can speak as of cases with "classical time" (discrete or continuous) in accordance with the meaning which $g$ really has in specific examples. (By analogy, in other cases one sometimes speaks also of "time" (but "non-classical" ); not being the time in the ordinary sense of the word it can have another physical meaning denoting, for example, spatial shifts of a translation-invariant physical system. An ergodic theory has been developed especially for amenable (and, a fortiori, commutative) groups $G$; in many respects (though not in all) there is then an analogy with the case of classical time. For non-amenable $G$ the situation is different: it has been less thoroughly studied.) Below the basic case is considered: $\{T_t\}$ is a measurable flow or a cascade in a Lebesgue space $(W,\mu)$, preserving $\mu$.

In "abstract" ergodic theory one studies various statistical properties of dynamical systems reflecting their behaviour over long periods of time (for example, ergodicity or mixing) as well as problems connected with the metric classification of systems (with respect to a metric isomorphism), and the two groups of problems turn out to be closely connected.

Since a non-ergodic system splits into ergodic components (cf. Metric transitivity), both groups of problems need to be investigated only for ergodic systems. The basic part of "abstract" ergodic theory comprises the following six directions.

a) The appearance of ergodic theory as an independent branch is connected with the von Neumann ergodic theorem and the Birkhoff ergodic theorem and the recognition of their metric nature. Subsequently, various modifications and generalizations of these theorems emerged, frequently without a connection with dynamical systems (in this sense they go beyond the framework of ergodic theory), nevertheless they are called ergodic theorems (see Maximal ergodic theorem; Operator ergodic theorem; Ornstein–Chacon ergodic theorem). But for ergodic theory itself their elaboration was of lesser significance.

b) The spectral theory of dynamical systems, that is, the investigation of problems connected with the spectrum of a dynamical system.

c) The entropy theory of a dynamical system.

d) The approximation by periodic transformations.

e) Change of time and monotone equivalence (Kakutani equivalence).

f) The trajectory theory and related problems.

Most important for applications are b) and c). (With respect to flows, the idea of e) and f) is, roughly speaking, to separate the properties of a flow that depend on the location of the trajectories in the phase space from those depending on the parametrization of the trajectories by time. The difference between e) and f) is that in e) a trajectory is regarded as a continuous curve with a distinguished positive direction and, accordingly, the class of admissible parametrizations is restricted, whereas in f) a trajectory is regarded simply as a point set and, accordingly, the parametrizations can be discontinuous and need not be monotone relative to each other. Precise definitions are given below.

A change of time in a flow $\{T_t\}$ consists in a transition to a new flow $\{S_s\}$; for the new flow the time for which a point $w$ falls into the position $T_tw$ is $\int_0^ta(T_\tau w)d\tau$, where $a, 1/a \in L_1(W,\mu)$, $a>0 \mod 0$ (the flow $\{S_s\}$ has invariant measure $\lambda(A)=\int_A 1/a d\mu$). One says that $\{T_t\}$ and $\{S_s\}$ are monotonically equivalent. An equivalent definition is: Two flows are monotonically equivalent if they are metrically isomorphic to special flows (cf. Special flow) constructed from one and the same automorphism of some measure space (but, generally speaking, from distinct positive functions). Two automorphism $T$ and $S$ (as well as the cascades $\{T^n\}$ and $\{S^n\}$) are called monotonically equivalent if they are metrically isomorphic to special automorphisms (cf. Special automorphism) constructed from one and the same automorphism. Dynamical systems are trajectory equivalent if there exists a metric isomorphism of their phase spaces taking the trajectories of one system into those of the other (as point sets).

In e) one analyzes the following problems: How much can the properties of a flow change under a change of time? In particular, can one perhaps find a change such that the new flow has some special property? (The problem can be raised in the general case or for a concrete flow; the change of time can be subject to certain special conditions.) And, what can be said about the classification of systems relative to monotone equivalence?

Trajectory equivalence for systems with "classical" time is uninteresting: If the invariant measure is continuous, then any two ergodic flows or cascades are trajectory equivalent. However, for systems with "non-classical" time trajectory equivalence leads to a substantial theory.

2) In the "applied" part of ergodic theory one examines diverse specific dynamical systems (and classes of them) which arise in various branches of mathematics and physics. (Historically, the birth of ergodic theory is linked with statistical physics (see Dynamical system; Statistical physics, mathematical problems in). Recently, new connections with this discipline have come to light; see, for example, about Gibbs measures in the last-named article.) Here one studies for the relevant systems the same questions about statistical properties and classifications as in 1), but now one cannot assume from the very beginning that the system in question is ergodic. On the contrary, the elucidation of the problem of its ergodicity is, as a rule, a necessary (and frequently difficult) stage of the investigation, even when ultimately it is established that stronger statistical properties are present.

There are also cases (in number theory and statistical physics) where one is concerned not with the application of concepts or results of ergodic theory, but with the use of arguments having some affinity with ergodic theory. Finally, ideas of the theory of dynamical systems, in particular, of ergodic theory, lend themselves to the interpretation of results of certain numerical experiments (see Strange attractor).

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For the relationship of ergodic theory with other branches of the theory of dynamical systems (homeomorphisms on compact spaces, smooth flows, etc.) see [Bo], [DeGrSi], [Ma2], and [Ve]. For (almost) all known ergodic theorems consult [Kr]. Applications of ergodic theory to number theory can be found in [Fu], and to the theory of lattices in semi-simple groups (work of Margulis) in [Za].

As to other applications of ergodic theory see [ArAv], [Ma] and [Ru]. The problem of ergodic and mixing properties of physical systems is dealt with in [Zi].

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