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[[Category:Ergodic theory]]
 
[[Category:Ergodic theory]]
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The branch of the theory of dynamical systems that studies systems with an
 +
[[Invariant measure|invariant measure]] and related problems.
  
The branch of the theory of dynamical systems that studies systems with an [[Invariant measure|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|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 $G$)
  
1) In the "abstract" or "general" part of ergodic theory one examines measurable dynamical systems. In the most general sense this is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361501.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361502.png" /> is a [[Measurable space|measurable space]] (the "phase space" ), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361503.png" /> is a locally compact Hausdorff group (or semi-group) with a countable base and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361504.png" /> is a measurable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361505.png" /> defining a (left) action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361506.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361507.png" />: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e0361509.png" /> is the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615011.png" />, then (in multiplicative notation for the operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615012.png" />)
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$$F(e,w) = w \quad \textrm{and}\quad F(gh,w) = F(g,F(h,w)).\tag{$*$}$$
 +
(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|Borel set]]) are taken to be measurable. Under the assumptions above on $G$ several versions of the latter concept (see
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[[Borel measure|Borel measure]];
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[[Baire set|Baire set]]) turn out to be equivalent.)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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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|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|Lebesgue space]]. As regards $G$, the basic cases are $G=\Z$ or $G=\N$ (a
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[[Cascade|cascade]]) or $G=\R$ (a
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[[Flow (continuous-time dynamical system)|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|measurable flow]] or a cascade in a Lebesgue space $(W,\mu)$, preserving $\mu$.
  
(It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615014.png" /> is endowed with the structure of a measurable space as the direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615016.png" /> and that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615017.png" /> the Borel sets (cf. [[Borel set|Borel set]]) are taken to be measurable. Under the assumptions above on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615018.png" /> several versions of the latter concept (see [[Borel measure|Borel measure]]; [[Baire set|Baire set]]) turn out to be equivalent.)
+
In "abstract" ergodic theory one studies various statistical properties of dynamical systems reflecting their behaviour over long periods of time (for example,
 +
[[Ergodicity|ergodicity]] or
 +
[[Mixing|mixing]]) as well as problems connected with the metric classification of systems (with respect to a
 +
[[Metric isomorphism|metric isomorphism]]), and the two groups of problems turn out to be closely connected.
  
Denoting the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615019.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615020.png" />, one can write (*) in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615021.png" />. (One may also consider a right action, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615022.png" />.) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615023.png" /> is a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615024.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615025.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615026.png" />-finite or finite measure that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615027.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615028.png" /> is a measurable set, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615029.png" />. A finite measure is usually normalized; most often <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615030.png" /> is a [[Lebesgue space|Lebesgue space]]. As regards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615031.png" />, the basic cases are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615033.png" /> (a [[Cascade|cascade]]) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615034.png" /> (a [[Flow (continuous-time dynamical system)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615036.png" />; in many respects (though not in all) there is then an analogy with the case of classical time. For non-amenable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615037.png" /> the situation is different: it has been less thoroughly studied.) Below the basic case is considered: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615038.png" /> is a [[Measurable flow|measurable flow]] or a cascade in a Lebesgue space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615039.png" />, preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615040.png" />.
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Since a non-ergodic system splits into ergodic components (cf.
 +
[[Metric transitivity|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.
  
In "abstract" ergodic theory one studies various statistical properties of dynamical systems reflecting their behaviour over long periods of time (for example, [[Ergodicity|ergodicity]] or [[Mixing|mixing]]) as well as problems connected with the metric classification of systems (with respect to a [[Metric isomorphism|metric isomorphism]]), and the two groups of problems turn out to be closely connected.
+
a) The appearance of ergodic theory as an independent branch is connected with the
 +
[[Von Neumann ergodic theorem|von Neumann ergodic theorem]] and the
 +
[[Birkhoff ergodic theorem|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|Maximal ergodic theorem]];
 +
[[Operator ergodic theorem|Operator ergodic theorem]];
 +
[[Ornstein–Chacon ergodic theorem|Ornstein–Chacon ergodic theorem]]). But for ergodic theory itself their elaboration was of lesser significance.
  
Since a non-ergodic system splits into ergodic components (cf. [[Metric transitivity|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.
+
b) The spectral theory of dynamical systems, that is, the investigation of problems connected with the
 +
[[Spectrum of a dynamical system|spectrum of a dynamical system]].
  
a) The appearance of ergodic theory as an independent branch is connected with the [[Von Neumann ergodic theorem|von Neumann ergodic theorem]] and the [[Birkhoff ergodic theorem|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|Maximal ergodic theorem]]; [[Operator ergodic theorem|Operator ergodic theorem]]; [[Ornstein–Chacon ergodic theorem|Ornstein–Chacon ergodic theorem]]). But for ergodic theory itself their elaboration was of lesser significance.
+
c) The
 +
[[Entropy theory of a dynamical system|entropy theory of a dynamical system]].
  
b) The spectral theory of dynamical systems, that is, the investigation of problems connected with the [[Spectrum of a dynamical system|spectrum of a dynamical system]].
+
d) The
 
+
[[Approximation by periodic transformations|approximation by periodic transformations]].
c) The [[Entropy theory of a dynamical system|entropy theory of a dynamical system]].
 
 
 
d) The [[Approximation by periodic transformations|approximation by periodic transformations]].
 
  
 
e) Change of time and monotone equivalence (Kakutani equivalence).
 
e) Change of time and monotone equivalence (Kakutani equivalence).
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615041.png" /> consists in a transition to a new flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615042.png" />; for the new flow the time for which a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615043.png" /> falls into the position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615044.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615047.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615048.png" /> (the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615049.png" /> has invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615050.png" />). One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615052.png" /> are monotonically equivalent. An equivalent definition is: Two flows are monotonically equivalent if they are metrically isomorphic to special flows (cf. [[Special flow|Special flow]]) constructed from one and the same automorphism of some measure space (but, generally speaking, from distinct positive functions). Two automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615054.png" /> (as well as the cascades <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036150/e03615056.png" />) are called monotonically equivalent if they are metrically isomorphic to special automorphisms (cf. [[Special automorphism|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).
+
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|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|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?
 
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?
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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.
 
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|Dynamical system]]; [[Statistical physics, mathematical problems in|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.
+
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|Dynamical system]];
 +
[[Statistical physics, mathematical problems in|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|Strange attractor]]).
+
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|Strange attractor]]).
  
====References====
+
====Comments====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Hopf, "Ergodentheorie" , '''4''' , Springer (1937) {{MR|0024581}} {{ZBL|0017.28301}} {{ZBL|63.0786.07}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" ''Transl. Amer. Math. Soc. Ser. 2'' , '''49''' (1966) pp. 171–240 ''Uspekhi Mat. Nauk'' , '''4''' : 2 (1949) pp. 57–128 {{MR|}} {{ZBL|0185.21802}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> "Summer school on ergodic theory" ''Russian Math. Surveys'' , '''22''' : 5 (1967) pp. 1–167 ''Uspekhi Mat. Nauk.'' , '''22''' : 5 (1967) pp. 3–127</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Billingsley, "Ergodic theory and information" , Wiley (1965) {{MR|0192027}} {{ZBL|0141.16702}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with invariant measure" ''Progress in Math.'' , '''8''' (1970) pp. 151–215 ''Itogi Nauk. Mat. Anal. 1967'' (1969) pp. 133–187 {{MR|0286981}} {{ZBL|0252.28006}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Ya.G. Sinai, "Introduction to ergodic theory" , Princeton Univ. Press (1976) (Translated from Russian) {{MR|0584788}} {{ZBL|0375.28011}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" ''J. Soviet Math.'' , '''7''' (1977) pp. 974–1065 ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''13''' (1975) pp. 129–262 {{MR|0584389}} {{ZBL|0399.28011}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> D. Ornstein, "Ergodic theory, randomness, and dynamical systems" , Yale Univ. Press (1974) {{MR|0447525}} {{ZBL|0296.28016}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) {{MR|832433}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press (1981) {{MR|0603625}} {{ZBL|0459.28023}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> R.J. Zimmer, "Ergodic theory and semisimple groups" , Birkhäuser (1984) {{MR|0776417}} {{ZBL|0571.58015}} </TD></TR></table>
+
For the relationship of ergodic theory with other branches of the theory of dynamical systems (homeomorphisms on compact spaces, smooth flows, etc.) see
 +
{{Cite|Bo}},
 +
{{Cite|DeGrSi}},
 +
{{Cite|Ma2}}, and
 +
{{Cite|Ve}}. For (almost) all known ergodic theorems consult
 +
{{Cite|Kr}}. Applications of ergodic theory to number theory can be found in
 +
{{Cite|Fu}}, and to the theory of lattices in semi-simple groups (work of Margulis) in
 +
{{Cite|Za}}.
  
 +
As to other applications of ergodic theory see
 +
{{Cite|ArAv}},
 +
{{Cite|Ma}} and
 +
{{Cite|Ru}}. The problem of ergodic and mixing properties of physical systems is dealt with in
 +
{{Cite|Zi}}.
  
  
====Comments====
 
For the relationship of ergodic theory with other branches of the theory of dynamical systems (homeomorphisms on compact spaces, smooth flows, etc.) see [[#References|[a2]]], [[#References|[a3]]], [[#References|[a7]]], and [[#References|[a9]]]. For (almost) all known ergodic theorems consult [[#References|[a5]]]. Applications of ergodic theory to number theory can be found in [[#References|[11]]], and to the theory of lattices in semi-simple groups (work of Margulis) in [[#References|[a4]]].
 
 
As to other applications of ergodic theory see [[#References|[a1]]], [[#References|[a6]]] and [[#References|[a8]]]. The problem of ergodic and mixing properties of physical systems is dealt with in [[#References|[12]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, V. Avez, "Ergodic problems of classical mechanics" , Benjamin (1968) (Translated from Russian) {{MR|232910}} {{ZBL|0167.22901}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" , ''Lect. notes in math.'' , '''470''' , Springer (1975) {{MR|0442989}} {{ZBL|0308.28010}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Denken, C. Grillenberg, K. Sigmund, "Ergodic theory on compact spaces" , Springer (1976) {{MR|457675}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> G.M. Zaslowsky, "Chaos in dynamic systems" , Harwood (1985) (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> U. Krengel, "Ergodic theorems" , de Gruyter (1985) pp. 261 {{MR|0797411}} {{ZBL|0575.28009}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G.W. Mackey, "Ergodic theory and its significance for statistical mechanics and probability theory" ''Adv. in Math.'' , '''12''' (1974) pp. 178–268 {{MR|0346131}} {{ZBL|0326.60001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Mañé, "Ergodic theory and differentiable dynamics" , Springer (1987) ((Translated from the Portuguese)) {{MR|0889254}} {{ZBL|0616.28007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Ruelle, "Thermodynamic formalism" , Addison-Wesley (1978) {{MR|0511655}} {{ZBL|0401.28016}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> W.A. Veech, "Topological dynamics" ''Bull. Amer. Math. Soc.'' , '''83''' (1977) pp. 775–830 {{MR|0467705}} {{ZBL|0384.28018}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> V.I. Oseledec, "Multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems" ''Trudy Moskov Mat. Obshch.'' , '''19''' (1968) pp. 179–210 (In Russian) {{MR|0240280}} {{ZBL|}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|ArAv}}||valign="top"| V.I. Arnol'd, V. Avez, "Ergodic problems of classical mechanics", Benjamin (1968) (Translated from Russian) {{MR|232910}} {{ZBL|0167.22901}}
 +
|-
 +
|valign="top"|{{Ref|Bi}}||valign="top"| P. Billingsley, "Ergodic theory and information", Wiley (1965) {{MR|0192027}} {{ZBL|0141.16702}}
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms", ''Lect. notes in math.'', '''470''', Springer (1975) {{MR|0442989}} {{ZBL|0308.28010}}
 +
|-
 +
|valign="top"|{{Ref|CoFoSi}}||valign="top"| I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory", Springer (1982) (Translated from Russian) {{MR|832433}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|DeGrSi}}||valign="top"| M. Denken, C. Grillenberg, K. Sigmund, "Ergodic theory on compact spaces", Springer (1976) {{MR|457675}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Fu}}||valign="top"| H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory", Princeton Univ. Press (1981) {{MR|0603625}} {{ZBL|0459.28023}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}}
 +
|-
 +
|valign="top"|{{Ref|Ho}}||valign="top"| E. Hopf, "Ergodentheorie", '''4''', Springer (1937) {{MR|0024581}} {{ZBL|0017.28301}} {{ZBL|63.0786.07}}
 +
|-
 +
|valign="top"|{{Ref|KaSiSt}}||valign="top"| A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" ''J. Soviet Math.'', '''7''' (1977) pp. 974–1065 ''Itogi Nauk. i Tekhn. Mat. Anal.'', '''13''' (1975) pp. 129–262 {{MR|0584389}} {{ZBL|0399.28011}}
 +
|-
 +
|valign="top"|{{Ref|Kr}}||valign="top"| U. Krengel, "Ergodic theorems", de Gruyter (1985) pp. 261 {{MR|0797411}} {{ZBL|0575.28009}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"| G.W. Mackey, "Ergodic theory and its significance for statistical mechanics and probability theory" ''Adv. in Math.'', '''12''' (1974) pp. 178–268 {{MR|0346131}} {{ZBL|0326.60001}}
 +
|-
 +
|valign="top"|{{Ref|Ma2}}||valign="top"| R. Mañé, "Ergodic theory and differentiable dynamics", Springer (1987) ((Translated from the Portuguese)) {{MR|0889254}} {{ZBL|0616.28007}}
 +
|-
 +
|valign="top"|{{Ref|MaSu}}||valign="top"| "Summer school on ergodic theory" ''Russian Math. Surveys'', '''22''' : 5 (1967) pp. 1–167 ''Uspekhi Mat. Nauk.'', '''22''' : 5 (1967) pp. 3–127
 +
|-
 +
|valign="top"|{{Ref|Or}}||valign="top"| D. Ornstein, "Ergodic theory, randomness, and dynamical systems", Yale Univ. Press (1974) {{MR|0447525}} {{ZBL|0296.28016}}
 +
|-
 +
|valign="top"|{{Ref|Os}}||valign="top"| V.I. Oseledec, "Multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems" ''Trudy Moskov Mat. Obshch.'', '''19''' (1968) pp. 179–210 (In Russian) {{MR|0240280}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Ro}}||valign="top"| V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" ''Transl. Amer. Math. Soc. Ser. 2'', '''49''' (1966) pp. 171–240 ''Uspekhi Mat. Nauk'', '''4''' : 2 (1949) pp. 57–128 {{MR|}} {{ZBL|0185.21802}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}||valign="top"| D. Ruelle, "Thermodynamic formalism", Addison-Wesley (1978) {{MR|0511655}} {{ZBL|0401.28016}}
 +
|-
 +
|valign="top"|{{Ref|Si}}||valign="top"| Ya.G. Sinai, "Introduction to ergodic theory", Princeton Univ. Press (1976) (Translated from Russian) {{MR|0584788}} {{ZBL|0375.28011}}
 +
|-
 +
|valign="top"|{{Ref|Ve}}||valign="top"| W.A. Veech, "Topological dynamics" ''Bull. Amer. Math. Soc.'', '''83''' (1977) pp. 775–830 {{MR|0467705}} {{ZBL|0384.28018}}
 +
|-
 +
|valign="top"|{{Ref|VeYu}}||valign="top"| A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with invariant measure" ''Progress in Math.'', '''8''' (1970) pp. 151–215 ''Itogi Nauk. Mat. Anal. 1967'' (1969) pp. 133–187 {{MR|0286981}} {{ZBL|0252.28006}}
 +
|-
 +
|valign="top"|{{Ref|Za}}||valign="top"| G.M. Zaslowsky, "Chaos in dynamic systems", Harwood (1985) (Translated from Russian)
 +
|-
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|valign="top"|{{Ref|Zi}}||valign="top"| R.J. Zimmer, "Ergodic theory and semisimple groups", Birkhäuser (1984) {{MR|0776417}} {{ZBL|0571.58015}}
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|-
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|}

Latest revision as of 22:03, 6 April 2012

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).

Comments

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].


References

[ArAv] V.I. Arnol'd, V. Avez, "Ergodic problems of classical mechanics", Benjamin (1968) (Translated from Russian) MR232910 Zbl 0167.22901
[Bi] P. Billingsley, "Ergodic theory and information", Wiley (1965) MR0192027 Zbl 0141.16702
[Bo] R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms", Lect. notes in math., 470, Springer (1975) MR0442989 Zbl 0308.28010
[CoFoSi] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory", Springer (1982) (Translated from Russian) MR832433
[DeGrSi] M. Denken, C. Grillenberg, K. Sigmund, "Ergodic theory on compact spaces", Springer (1976) MR457675
[Fu] H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory", Princeton Univ. Press (1981) MR0603625 Zbl 0459.28023
[Ha] P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
[Ho] E. Hopf, "Ergodentheorie", 4, Springer (1937) MR0024581 Zbl 0017.28301 Zbl 63.0786.07
[KaSiSt] A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math., 7 (1977) pp. 974–1065 Itogi Nauk. i Tekhn. Mat. Anal., 13 (1975) pp. 129–262 MR0584389 Zbl 0399.28011
[Kr] U. Krengel, "Ergodic theorems", de Gruyter (1985) pp. 261 MR0797411 Zbl 0575.28009
[Ma] G.W. Mackey, "Ergodic theory and its significance for statistical mechanics and probability theory" Adv. in Math., 12 (1974) pp. 178–268 MR0346131 Zbl 0326.60001
[Ma2] R. Mañé, "Ergodic theory and differentiable dynamics", Springer (1987) ((Translated from the Portuguese)) MR0889254 Zbl 0616.28007
[MaSu] "Summer school on ergodic theory" Russian Math. Surveys, 22 : 5 (1967) pp. 1–167 Uspekhi Mat. Nauk., 22 : 5 (1967) pp. 3–127
[Or] D. Ornstein, "Ergodic theory, randomness, and dynamical systems", Yale Univ. Press (1974) MR0447525 Zbl 0296.28016
[Os] V.I. Oseledec, "Multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems" Trudy Moskov Mat. Obshch., 19 (1968) pp. 179–210 (In Russian) MR0240280
[Ro] V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" Transl. Amer. Math. Soc. Ser. 2, 49 (1966) pp. 171–240 Uspekhi Mat. Nauk, 4 : 2 (1949) pp. 57–128 Zbl 0185.21802
[Ru] D. Ruelle, "Thermodynamic formalism", Addison-Wesley (1978) MR0511655 Zbl 0401.28016
[Si] Ya.G. Sinai, "Introduction to ergodic theory", Princeton Univ. Press (1976) (Translated from Russian) MR0584788 Zbl 0375.28011
[Ve] W.A. Veech, "Topological dynamics" Bull. Amer. Math. Soc., 83 (1977) pp. 775–830 MR0467705 Zbl 0384.28018
[VeYu] A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with invariant measure" Progress in Math., 8 (1970) pp. 151–215 Itogi Nauk. Mat. Anal. 1967 (1969) pp. 133–187 MR0286981 Zbl 0252.28006
[Za] G.M. Zaslowsky, "Chaos in dynamic systems", Harwood (1985) (Translated from Russian)
[Zi] R.J. Zimmer, "Ergodic theory and semisimple groups", Birkhäuser (1984) MR0776417 Zbl 0571.58015
How to Cite This Entry:
Ergodic theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ergodic_theory&oldid=23606
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article