special flows and Rudolph's theorem

Question

The Rudolph's theorem confirm the existence of a special representation of an ergodic flow on the Lebesgue space. (In the book of I.P.Cornfeld entitled Ergodic theory). My question is: what is the benefit and the use of the special flows? And of the Rudolph's theorem?

Answer 1

A classical technique in differential equations theory is to take a section, to identify a set where trajectories hit the set in a discrete set of times. A theorem of this type in ergodic theory is the Ambrose-Kakutani theorem: for every ergodic measure-preserving flow, there exists a measurable section: a subset $A$ of $X$ such that for $\mu$-a.e. $x$, $\{t\colon T_t(x)\in A\}$ is a discrete subset of $\mathbb R$, unbounded above and below. A section allows you to study the continuous time system by looking at returns to the section. The map from the section to itself mapping a point to the next place on the section where it returns is the so-called induced map, or first return map. In this way, a continuous time dynamical system gives rise to a discrete time dynamical system.

A converse construction in ergodic theory is that of a suspension flow: starting from a discrete time dynamical system, $T\colon X\to X$, one can build a flow by taking an arbitrary integrable positive "roof function", $h(x)$ and building a measurable flow on $\bar X=\{(x,t)\colon x\in X,\ 0\le t<h(x)\}$. The picture is that the second coordinate, the "fibre", is arranged vertically over each point of $X$. In the flow, you move up the fibre until you hit the roof, $(x,h(x))$, at which point you move to $(T(x),0)$. Broadly, these two constructions show that you can move backwards and forwards between discrete time and continuous time dynamical systems (the section in this case is the set $X\times\{0\}$).

The purpose of the Rudolph theorem is to show that for an arbitrary ergodic measure-preserving flow, it is isomorphic to a suspension flow over a discrete time dynamical system where the roof function takes exactly two values, $\alpha$ and $\beta$ (where these have to be chosen to be rationally independent).

The significance is twofold: (1) it might give a convenient picture to work in, rather than working with a general measure-preserving flow; (2) in the Ambrose-Kakutani scenario or the differential equation section picture, one starts by finding the section, and the return-time/height function is then forced. In particular, one might wonder if the structure of the height function gives important information about the dynamical system. Rudolph's theorem tells us that this is not the case, because you can find in any system a height function of a very prescribed type.

Answer 2

Note that the flow must not have fixed points. Global sections allow to reduce the study of a flow to the study of a transformation. This is often the reason invoked in differential dynamics in order to explain that behaviors observed for transformations in dimension n appear for flows in dimension n+1.

Tere is no continuous global section for a continuous flow in general. The fact that it holds in the measurable setting is remarkable in my opinion.

An example of the property made evident by a special representation of a flow is the $L^2$-continuity of the flow: $f\circ \phi_t \rightarrow f$ in $L^2$ norm when $t$ goes to 0. No continuity assumption is needed on the flow $\phi_t$ here, so that's not completely obvious.