On Generators for Ergodic Flows

293

Then

Н { а ) й - I m(ßJlogm(ßJ+ ^ m(^г)^og4.

This number can be made arbitrarily small if we choose Ê^ (i^l) small.

V . Proof of Theorem 7

1 . First we show all statements under the hypothesis that all T, (f + O) are ergodic. ^ is then called totally ergodic or weakly mixing.

a ) Let ^ be represented as the special flow under a function / satisfying the general conditions (13.1), (13.2). Let ß = {B^,B2,...) = {BXei be a generator for the base transformation T^ without immediate return (see Theorem 5). We neglect the T^-invariant null set Nj^aE and thus may assume that ß separates all points of E under T^. Observe that the set

n - , 'Nj - = {{x,y)eQ\xeNj-}

is .^-invariant and has measure 0. We put aL = {AXei with

A=n~ , ' B, = {{x,y)\xEB^,Quy<f{x)}.

Clearly the passage time for a is at least 1. Let 0<s^l, z = (x,y)eß,

KW = ^:^ and {Ь)^,^ = Ф1-х.

Evidently а^ = Ь^. When it leaves A^^, the positive .T-orbit of z enters into A^^ and stays there at least one unit of time. At least one of the points of the positive r,-orbit of z must fall into this interval since s^l; the same can be said about the negative orbit. Repeating this consideration we see that the sequence (a^)jez decomposes into

. . . b_i . . . b_ibobo - - - ^obi ...bibi-'-^i-'- with ao = bo. where each block bj..,b^ has at least length 1. Therefore, Ф^! determines ФJ^7ГlZ and x = n^z, i.e. if 0<s^l, z = {x,y\ z' = (x',/) and Ф12 = Ф^/, then x = x'. This even holds if T, is not ergodic.

For an ergodic T^ also y=y' a.s. To show this, let y'-y>^. Then for any ;gZ, n,Tj^z = n^Tj^z' and n2Tj,z'-n2Tj^z = y-y. By the ergodicity of T^, / - у g ess inf / = inf/. The set

U = {{u,v)EQ\ueEJ{u)-{y-y)uv<f{u)}

has measure m(l7) = m£(£)(/-y)>0. Since T, is ergodic, it is almost sure that

Г z enters into U. In this case we have a contradiction because

J *

Hence a separates (under T^) the points outside of the Tj-invariant null set

N : = [ J r]{zEQ\0un2Tjs^<f{n,T^,z)-l/n}.