U n sp e c i…e d J o u rn a l
Vo lu m e 0 0 , N u m b e r 0 , P a g e s 0 0 0 –0 0 0
S ? ? ? ? -? ? ? ? (X X )0 0 0 0 -0
HOPF’S RATIO ERGODIC THEOREM BY INDUCING
ROLAND ZWEIMÜLLER
Abstract. We present a very quick and easy proof of the classical StepanovHopf ratio ergodic theorem, deriving it from Birkho¤’s ergodic theorem by a
simple inducing argument.
During the last few years, there has been some interest in short and easy proofs
of (pointwise) ergodic theorems, naturally focussing on the most fundamental one,
i.e. on Birkho¤’s result for probability preserving transformations, see e.g. [KW],
[Ke], [P], and [Sh]. In [KK] a similar proof of an important extension was given,
which came shortly after the discovery of the …rst ergodic theorems ([N] and [B],
see [Z] for historical comments): the Stepanov-Hopf ratio ergodic theorem ([St],
[H]) which is the proper version of the pointwise ergodic theorem for in…nite measure preserving transformations (there is no way to get a.e. convergence for ergodic
sums normalized by a sequence of constants, cf. [A], §2.4). The aim of the present
note is to point out that this result can also be derived as a direct consequence of
Birkho¤’s theorem, via a (very) simple inducing argument (which doesn’t seem to
be available or hinted at in the literature I know).
We are going to prove
Theorem 1 (Hopf ’s Ratio Ergodic Theorem). Let T be a measure preserving
transformation
on the
…nite measure space (X; A; ). Let f; g 2 L1 ( ) with
R
g 0 and X g d > 0. Then there exists a measurable function Q(f; g) : X ! R
such that
Pn 1
f Tk
Sn (f )
= Pk=0
! Q(f; g) a.e. on fsupn Sn (g) > 0g as n ! 1.
n
1
k
Sn (g)
k=0 g T
On the conservative part the limit function
w.r.t. the
algebra
R Q(f; g) is measurable
R
I A of T -invariant sets and satis…es I Q(f; g) g d = I f d for all I 2 I.
In particular, if g > 0 a.e., then
f
kI , where d g := g d ,
g
R
R
and if T is ergodic, then Q(f; g) = X f d = X g d a.e.
Q(f; g) = E
g
2000 Mathematics Subject Classi…cation. Primary 28D05, 37A30.
Key words and phrases. pointwise ratio ergodic theorem, induced transformation.
May 19, 2004; revised October 13, 2004.
c 2 0 0 4 R .Z .
1
2
ROLAND ZW EIM ÜLLER
Pn 1
Proof. For the dissipative part of T , where k=0 f T k < 1 a.e. for any f 2 L1 ( ),
the assertion is trivial. We can therefore assume w.l.o.g. that T is conservative. By
linearity it is enough to consider nonnegative f .
a) To emphasize the simplicity of the argument, we …rst consider the special
case of an ergodic map T . The main step will be to prove that for any Y 2 A with
0 < (Y ) < 1,
R
fd
Sn (f )
(1)
! X
a.e. on Y .
Sn (1Y )
(Y )
R
As the set fSn (f )=Sn (1Y ) ! X f d = (Y )g is T -invariant, we then see that this
convergence in fact holds a.e. on X. Applying the same to g, the assertion of the
theorem follows.
To verify (1) we consider the …rst return (or induced ) map TY : Y ! Y given
by TY x := T '(x) x, where '(x) := minfn 1 : T n x 2 Y g is the …rst return time
of Y , cf. [Ka]. According to basic classical results, TY is a measure preserving
transformation on the …nite measure space (Y; A \ Y; jA\Y ), ergodic since T is.
Moreover, it is well known that can be reconstructed from jA\Y via
X
(E) =
(Y \ f' > jg \ T j E) for E 2 A.
j 0
R
R P' 1
In other words, X 1E d = Y ( j=0 1E T j ) d . An obvious argument using linearity and monotone convergence shows that this extends from indicator functions
1E to arbitrary measurable f : X ! [0; 1), i.e. that
Z
Z
fd =
fY d ,
X
Y
where fY : Y ! [0; 1) is de…ned by fY :=
P'
1
j=0
f
T j.
We can therefore apply Birkho¤’
theorem to TY and fY , thus considPs mergodic
1
ering the ergodic sums SYm (fY ) := k=0 fY TYk , m 1, to see that
R
R
f d
fd
SYm (fY )
Y Y
(2)
!
= X
a.e. on Y .
m
(Y )
(Y )
Pm 1
Let 'm := SYm (') = k=0 ' TYk , m 1, denote the m-th return time to Y . Then,
on Y , Sn (1Y ) = m for n 2 f'm 1 + 1; : : : ; 'm g and S'm (f ) = SYm (fY ), so that
S'm (f )
SYm (fY )
=
,
m
S'm (1Y )
for m
1, a.e. on Y ,
showing that (2) is equivalent to (1) along the subsequence of indices n = 'm ,
m 1. To prove convergence of the full sequence, we need only observe that Sn (f )
is nondecreasing in n since f 0. Hence,
m 1 SYm 1 (fY )
m
m 1
and (1) follows from (2).
Sn (f )
Sn (1Y )
SYm (fY )
m
for n 2 f'm 1 + 1; : : : ; 'm g,
m 1, a.e. on Y ,
HOPF’S RATIO ERGODIC THEOREM BY INDUCING
3
b) If T is not necessarily ergodic, we …rst observe that as fsupn Sn (g) > 0g is
invariant, we may assume w.l.o.g. that it equals X. Also, the set M on which
Sn (f )=Sn (g) does not converge to a function Q with the advertized properties belongs to I. Due to -…niteness, every set of positive measure has a subset Y with
0 < (Y ) < 1, and we prove that (M ) > 0 is impossible by showing that on any
such Y the desired convergence holds a.e.
Restricting our attention
S to the (smallest) invariant set generated by Y , we
suppose w.l.o.g. that X = n 0 T n Y . By the general form of Birkho¤’s theorem,
SYm (fY )=m ! E jA\Y [fY k IY ]= (Y ) a.e. on Y , where IY is the -algebra of
TY -invariant sets in A \ Y . It is a standard fact about …rst return maps that
IY = I \ Y = fI \ Y : I 2 Ig. By exactly the same argument as before we obtain
the following parallel to (1),
Sn (f )
Sn (1Y )
(3)
!
E
jA\Y
[fY k IY ]
(Y )
a.e. on Y ,
and analogously for g. Observe now that
fE
jA\Y
[gY k IY ] > 0g = fsupm SYm (gY ) > 0g = Y \ fsupn Sn (g) > 0g.
Exploiting T -invariance of lim inf and lim sup of the ratios Sn (f )=Sn (g), we therefore conclude that their sequence converges a.e. on X to the (uniquely de…ned)
I-measurable extension Q = Q(f; g) to X of E jA\Y [fY k IY ]=E jA\Y [gY k IY ].
R
R
It remains to verify the property I Q g d = I f d for all I 2 I, which uniquely
characterizes the T -invariant limit Q. To do so, we notice that by T -invariance, we
have (Q g)Y = Q gY , and hence, for any I 2 I,
Z
Z
Z
Z
Q gd
=
(Q g)Y d =
Q gY d =
Q E jA\Y [gY k IY ] d
I
I\Y
I\Y
I\Y
Z
Z
Z
=
E jA\Y [fY k IY ] d =
fY d = f d ,
I\Y
I\Y
I
as required.
Acknowledgements. This work was supported by an APART [Austrian programme for advanced research and technology] fellowship of the Austrian Academy
of Sciences.
References
[A]
[B]
[H]
[Ka]
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4
ROLAND ZW EIM ÜLLER
[P]
K. Petersen: Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal
Ergodic Theorem. preprint 2000.
[Sh] P. Shields: The Ergodic Theory of Discrete Sample Paths. AMS 1996.
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Mathematics Department, Imperial College London,, 180 Queen’s Gate, London SW7
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