Continuous flow

2020 Mathematics Subject Classification: Primary: 37A10 [MSN][ZBL]

A continuous flow in ergodic theory is a family $\{T^t\}$ (where $t$ ranges over the real axis $\R$) of automorphisms modulo 0 of a measure space $(M,\mu)$ such that:

a) $T^t T^s(x) = T^{t+s}(x)$ for any $t,s \in \R$ and all $x \in M$, except possibly for a set of $x$ belonging to an exceptional set of measure 0 (which may depend on $t$ and $s$); in other words, $T^t T^s = T^{t+s} \bmod 0$;

b) for each measurable set $A \subset M$ the measure of the symmetric difference $\mu(A \Delta T^t A)$ depends continuously on $t$.

Let $\mathfrak{A}$ be the set of all automorphisms modulo 0 of the space $(M, \mu)$ with the usual identification: if $T$ and $S$ coincide almost-everywhere, then they determine the same element of $\mathfrak{A}$. If $\mathfrak{A}$ is endowed with the weak topology (see ), then b) means that the mapping $\R\to\mathfrak{A}$ that takes $t$ to $T^t$ is continuous.

If $(M,\mu)$ is a Lebesgue space, then the concept of a continuous flow is practically the same as that of a measurable flow: The latter is always a continuous flow (see ), and for any continuous flow $\{T^t\}$ there is a measurable flow $\{S^t\}$ such that $T^t = S^t \bmod 0$ for all $t$ (see ; a related result is proved in , but see also the correction in ). The converse to any of these results depends on the character of the problem in question and the methods used.

In another sense the term "continuous flow" can be used to emphasize that the flow is considered in the context of topological dynamics. In this meaning a continuous flow is a collection of homeomorphisms $\{T^t\}$ of a topological space $M$ such that $T^t(T^s(x)) = T^{t+s}(x)$ for all $t,s \in \R$ and $x \in M$; the mapping $M \times \R\to M$ taking $(x,t)$ to $T^t x$ is continuous.

To avoid confusion with 1) it is better to talk in this case of a topological flow and in the case of 1) of a metric continuity.

References

[Ha]	P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
[Ho]	E. Hopf, "Ergodentheorie" , Springer (1970) MR0024581 Zbl 0185.29001
[V]	A.M. Vershik, "Measurable realization of continuous automorphism groups of a unitary ring" Izv. Akad. Nauk. SSSR Ser. Mat. , 29 : 1 (1965) pp. 127–136 Zbl 0194.16302
[M]	G.W. Mackey, "Point realizations of transformation groups" Illinois J. Math. , 6 : 2 (1962) pp. 327–335 MR0143874 Zbl 0178.17203
[R]	A. Ramsay, "Virtual groups and group actions" Advances in Math. , 6 : 3 (1971) pp. 253–322 MR0281876 Zbl 0216.14902 Zbl 1085.54027