Metric transitivity

of a dynamical system $\{T_t\}$ with a (quasi-) invariant measure $\mu$

2020 Mathematics Subject Classification: Primary: 37Axx [MSN][ZBL]

The property of $\{T_t\}$ that any measurable subset $A$ of the phase space $W$ which is invariant relative to $T_t$ (in the sense that it coincides with its complete inverse images $(T_t)^{-1}A$) either has measure zero or coincides with $W$ up to a set of measure zero. A formally stronger version of this property is obtained if in the definition instead of invariant sets one considers sets which are invariant modulo 0 (a set $A$ differing for each $t$ from $(T_t)^{-1}A$ by a set of measure zero). These two versions are equivalent if $\mu$ is a $\sigma$-finite measure and if the "time" $t$ runs through a locally compact group with a countable base (see the proof for flows in [H]), but for arbitrary transformation groups this is not the case (see the example, given for another reason, in [Fo]).

Instead of metric transitivity one may also speak of metric indecomposability, or ergodicity. If several (quasi-) invariant measures are considered in a given system and $\mu$ is one of them, then instead of speaking of metric transitivity (ergodicity) relative to $\mu$ one speaks of metric transitivity (ergodicity) of $\mu$. Of course, decomposability of a normalized invariant measure $\mu$ means something different, i.e. the possibility to represent $\mu$ as $a_1\mu_1+a_2\mu_2$, where $\mu_i$ are normalized invariant measures different from $\mu$ and $a_i>0$. Indecomposability of $\mu$ is equivalent to the second (stronger) version of metric transitivity (see [Fo], [B]).

If there is a topology on $W$, then under natural assumptions (see the discussion for flows in [H]) metric transitivity implies that for almost-all $w\in W$ the trajectory of $w$ is everywhere dense (in this sense, metric transitivity implies topological transitivity). The converse is false.

Starting with J. von Neumann , a number of results were obtained concerning the decomposition of non-ergodic systems into ergodic components. For the traditional, purely metric version ($W$ a Lebesgue space and only one (quasi-) invariant measure involved) see [R], [R2], . N.M. Krylov and N.N. Bogolybov [BK] obtained stronger results for topological flows and cascades on a metric compactum $W$, of a twofold kind (see also [NS], [O]): 1) a decomposition of normalized invariant measures, compatible with the topology, into ergodic measures; and 2) a "geometric" realization of all these decompositions at once by means of ergodic sets (cf. Ergodic set). Regarding the generalization of these results to other transformation groups, it is known that 1) is valid under broader assumptions than 2) (see [Fo], [B]). There are also similar results for dynamical systems (at present only cascades and flows) in appropriate measurable spaces (cf. Measurable space) and (or) for quasi-invariant measures (see [O]–[KP]).

Metric transitivity of a partition $\xi$ of a measure space $(W,\mu)$ is the property that any measurable subset $A\subset W$ consisting entirely of elements of $\xi$ either has measure zero or coincides with $W$ up to a set of measure zero. Instead of metric transitivity one also speaks of absolute non-measurability of $\xi$. Metric transitivity of a dynamical system generated by a group of transformations, whose trajectories form a partition of the phase space, coincides with metric transitivity of this partition. If the transformations forming the dynamical system are not invertible, then their metric transitivity also reduces to that of some partition; however, its description is more complicated: the element containing some point consists of all images of the point and all inverse images of these images.

References

[H]	E. Hopf, "Ergodentheorie" , Springer (1937) MR0024581 Zbl 0017.28301 Zbl 63.0786.07
[Fo]	S.V. Fomin, "On measures invariant under certain groups of transformations" Izv. Akad. Nauk SSSR Ser. Mat. , 14 : 3 (1950) pp. 261–274 (In Russian) MR0035925
[B]	N.N. Bogolyubov, "On some ergodic properties of continuous transformation groups" , Selected Works , 1 , Kiev (1969) pp. 561–569 (In Russian)
[N]	J. von Neumann, "Zur Operatorenmethode in der klassischen Mechanik" Ann. of Math. , 33 : 3 (1932) pp. 587–642 Zbl 0005.31504 Zbl 0005.12203 Zbl 58.1270.04
[N2]	J. von Neumann, "Zusätze zur Arbeit "Zur Operatorenmethode in der klassischen Mechanik" " Ann. of Math. , 33 : 4 (1933) pp. 789–792 Zbl 0005.31504 Zbl 58.1271.01
[R]	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
[R2]	V.A. Rokhlin, "On the decomposition of a dynamical system into transitive components" Mat. Sb. , 25 : 2 (1949) pp. 235–249 (In Russian)
[BK]	N.N. Bogolyubov, N.M. Krylov, "La theorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non-linéaire" Ann. of Math. (2) , 38 (1937) pp. 65–113 MR1503326
[NS]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[O]	J. Oxtoby, "Ergodic sets" Bull. Amer. Math. Soc. , 58 (1952) pp. 116–136 MR0047262 Zbl 0046.11504
[KP]	Yu.I. Kifer, S.A. Pirogov, "The decomposition of quasi-ergodic measures into ergodic components" Uspekhi Mat. Nauk , 27 : 5 (1972) pp. 239–240 (In Russian)

Comments

The term "ergodic" is mostly used for the first version of metric transitivity; for the second (stranger) version sometimes the term "irreducible" is used; see e.g. [M]. The two notions coincide if for every measurable subset $A$ of $W$ which is invariant $\bmod\,0$ there exists an invariant measurable subset $A_1$ of $W$ such that $\mu(A\Delta A_1)=0$. For conditions such that this holds, see also (the proof of) Lemma 1 in Chapt. 1, § 2 of [CFS], and e.g. Lemma 3.3 in [V]. A simple example where the two notions do not coincide is in [P], p. 84.

What is called above "indecomposability" is usually expressed by saying that $\mu$ is an extreme point of the (convex) set $J(W)$ of all invariant probability measures on $W$ (a probability measure is a normalized measure). Proofs of the fact that an invariant probability measure is indecomposable (i.e., is an extreme point of $J(W)$) if and only if it is irreducible (according to the above definition), under various conditions for $W$ or $\{T_t\}$, can also be found in [BH] and in [V]; also the proof of theorem 6.10(iii) in [W] can easily be adapted to the case of a general semi-group $\{T_t\}$ and arbitrary $W$.

As to the decomposition of the phase space into ergodic components, see also [V] and [Fa]. Such decompositions lead to representations of invariant measures as integrals of ergodic measures; if the phase space $W$ is compact, this is an easy consequence of Choquet theory: see [P], p. 82. For such a representation in the context of random transformations, see Appendix A.1 in [K].

References

[BH]	J.R. Blum, D.L. Hanson, "On invariant probability measures I" Pacific J. Math. , 10 (1960) pp. 1125–1129 MR0133426 Zbl 0104.11006
[Fa]	R.H. Farrell, "Representation of invariant measures" Illinois J. Math. , 6 (1962) pp. 447–467 MR0150264 Zbl 0108.14203
[K]	Y. Kifer, "Ergodic theory of random transformations" , Birkhäuser (1986) MR0884892 Zbl 0604.28014
[M]	G.W. Mackey, "Point realizations of transformation groups" Illinois J. Math. , 6 (1962) pp. 327–335 MR0143874 Zbl 0178.17203
[P]	R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966) MR193470
[V]	V.S. Varadarajan, "Groups of automorphisms of Borel spaces" Trans. Amer. Math. Soc. , 109 (1963) pp. 191–220 MR0159923 Zbl 0192.14203
[W]	P. Walters, "An introduction to ergodic theory" , Springer (1982) MR0648108 Zbl 0475.28009
[CFS]	I.P. Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) MR832433