数理解析研究所講究録
1042 巻 1998 年 1-10

1

REMARKS ON GIBBS MEASURES FOR FIBRED SYSTEMSMANFRED DENKER AND MIKHAIL GORDIN

1. Gibbs measures for fibred systems

In [4] we investigated the existence and uniqueness of a system of conditional
measures for fibred systems which plays a similar role as conformal measures for
non-fibred one-dimensional dynamical systems. In this note we add some generalproperties and make some additional remarks about this concept.
consists of two dynamical systems
Recall that a fibred system
which commutes withand : $Xarrow X$ and a projection ::and if has the form
and . In particular, is a skew product if$\tau((y\mathrm{o}, x))=(T_{x}(y_{0),S())}X$ . We make the further assumption that both spaces areandand that all
Polish spaces and are equipped with the Borel -algebras$T,$concepts
below this is not a
and are continuous, although for somemaps
; the restriction of $T^{n}(n\geq 1)$ topreserves the fibres
prerequisite.
, so. If we need to specify awill be denoted by:
the fibre
$d(y,
y’)$
,
by
.
will
be
denotedonitmetricThe main result in [4] is for fibred systems which have the expanding and exactness property fibrewise. The fibred system is called fibre expanding, if there
for every
is a local homeomorphisms on
exists a constant $a>0$ such$x\in X$ and expands the metric uniformly.
is topologically exact along
for any
and any $x\in X$ where $N$ depends
fibres if
on alone.
Under these assumptions we proved (cf. [1] for a similar result when is invertible)
andTheorem 1.1: Let be compact, be
there:
for $x\in X$ . Then for every H\"older continuous function$\mu_{x}(x\inX)$
and a uniqueexists a unique family of conditional probability measures
measurable function $A:Xarrow \mathbb{R}$ satisfying$\mathcal{Y}=(\mathrm{Y}, T,X, S, \pi)$$T$$\mathrm{Y}arrow \mathrm{Y}$

$T$

$S$

$S$

$\pi$

$\mathrm{Y}arrow X$$B_{X}$$B_{Y}$$\sigma$$S$$T$$\mathrm{Y}=\mathrm{Y}_{0}\cross X$$\mathcal{Y}$

$\pi$

$T$

$\mathrm{Y}_{x}=\pi^{-1}(x)$$T_{x}^{n}$$T_{x}^{n}$$\mathrm{Y}_{x}$$\mathrm{Y}_{x}arrow \mathrm{Y}_{S^{n}(x)}$$\mathrm{Y}$$\mathcal{Y}$$B(y,a)\cap \mathrm{Y}_{x}$$T_{x}$$y\in \mathrm{Y}_{x},$$\mathcal{Y}$$\epsilon>0$$T^{N}(B(x, \epsilon)\cap \mathrm{Y}_{x})\supset \mathrm{Y}_{S^{N}(x)}$$\epsilon$$S$$\mathrm{Y}$$T$$\mathrm{b}_{0}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}_{- \mathrm{t}\mathrm{o}^{-}}\mathrm{o}\mathrm{n}\mathrm{e},$$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{Y}_{x}\geq 2$$\phi$$\mathrm{Y}_{S(x)}$(1)$A(x) \int g(y)\mu x(dy)=\int V_{x}^{(1)}g(y)\mu S(x)(dy)$for every bounded measurable function(2)$T(\mathrm{Y}_{x})=$$\mathrm{Y}arrow \mathbb{R}$$g:\mathrm{Y}arrow \mathbb{R}$, where$V_{x}^{(k)}g(y)= \sum_{(y’\in\tau^{-k}(y)\mathrm{n}\pi^{-}1x\rangle}g(y)\mathrm{e}\mathrm{x}\mathrm{p}’[\phi(y’)+\ldots+\emptyset(T^{k1}-(\phi(y’)\mathrm{I}]$$(k\geq 1)$.

This research was supported by the Deutsche Forschungsgemeinschaft, JSPS, and by INTAS,
grant 93-0570
Typeset by$A_{\mathcal{M}}S- \mathrm{u}\mathrm{E}^{\mathrm{X}}$

2
if it
is called Gibbs for the potential
A family of conditional measuresatisfies (1) for some measurable function $A:Xarrow \mathbb{R}$ . For short we also say thatis $(\phi, A)$ -Gibbs.
Under some additional asumptions one can prove continuity of the function. These conditions are openess of and , and the property
and the mapthat the mapping $y\vdash+(\pi(y), T(y))$ is a local homeomorphism onto its image.
$\phi$$\mu_{x}$$\mu_{x}$

$A$

$S$

$x\vdash+\mu_{x}$Example 1.2: Let-regularthat for every$(d_{1},d_{2})$$T:\mathbb{C}^{n}arrow \mathbb{C}^{n}$$(d_{1}\in \mathbb{Q}_{+},d_{2}\in \mathbb{N})$$\pi$

be an entire mapping. Following [6], it is called$k_{2}>0$ and $r\geq 0$ such
if there are constants$k_{1},$$||z||\geq r$$z\in \mathbb{C}^{n},$$k_{1}||z||^{d_{1}}\leq||T(Z)||\leq k_{2}||z||^{d_{2}}$.

Let $n=2$ . For $d_{1}=1/2$ and $d_{2}=2$ these transformations include all H\’enon maps
(these are automorphisms, but there are also endomorphisms in this class) and for$d:=d_{1}=d_{2}=2$ the polynomial map$T(x, y)=(p(x,y),q(x, y))$$(x, y\in \mathbb{C})$is called strict (where and are polynomials). A special case are skew products
where :
denotes thewhen does not depend on . Thenprojection map onto the first factor.
is called weakly normal if there exists an open
According to [6], a pointof at least one-dimensional complex
neighborhood of and a family
analytic setssuch thatand the family
is normal in .
The complement of the set of weakly normal points plays the same role as the Julia
set in one dimension (and in fact reduces to this set for $n=1$ ) and is denoted by. It is shown in [6] that for strict polynomials
is compact, fully invariant
and contained in (but not equal in general)
where
$p$

$p$

$q$

$\pi \mathrm{o}T=p\mathrm{o}\pi$$y$$\mathbb{C}^{2}arrow \mathbb{C}$$\pi$$z\in \mathbb{C}^{2}$$V$$\mathcal{K}_{x}$$\{\mathcal{K}_{x} :$z$x\in V\}$$x\in \mathcal{K}_{x}$$x$$\{T_{1\cap}^{n}\kappa_{x}V : n\geq 0\}$$\mathcal{H}(T)$$\mathcal{H}(T)$$\partial K(T)$$K(T)= \{z\in \mathbb{C}^{2} :\sup_{k}||Tk(Z)||<\infty\}$.

It follows that a strict skew product restricted to
is a fibred system, but not$\mathcal{H}_{x}=\pi^{-1}(x)\cap
H(T)$ is the fibrea skew product in general. It is worth noting thatover , and is the Julia set of
in case is periodic with period .
This example is the basic motivation for us. The notion of normality introduced
by Heinemann seems to be the appropriate notion to study repellers for higher
dimensional polynomial mappings and their dynamical properties (despite of otherattempts by considering these maps on projective spaces; however, this causes unnecessary difficulties). In some cases, it is known (see [6], [7], [3]) thatisequal to each of the following sets: (a) the closure of the set of repelling periodic
points, (b) the Shilov boundary of $K(T),$the support of the measure of maximal entropy (this measure can be obtained using the Green current). These cases
include Cantor skew products, noodle type maps (which can be studies throughgeneralized Mandelbrot sets) and Torus like maps. Two of these classes belong
to the class of fibred systems as considered here, and we expect that our result
has further application to the ergodic theory of these dynamical systems as thosementioned in [4].
$\mathcal{H}(T)$$x$$T_{x}^{n}$

$n$

$x$

$\mathcal{H}(T)$$(\mathrm{c})$3
2. Equivalent conditions for the Gibbs propertyis
In this and the following sections we make the general assumption thatbounded-.to-one on fibres. This means there exists a constant $M>0$ such that for.
all
In what follows measurability is always meant with respect to the Borel a-fieldsof the Polish spaces under consideration. There is no necessity to consider any
extension of Borel a-fields. General results on countable-to-one Borel maps contained in [8] (and cited there) can be applied to verify the Borel measurability of allsets and functions considered here (’countable’ means ’countably infinite or finite’).
The function in under the integral in the right-hand side of (1) may serve as an
example:denote the Borel a-field of and the spaceandFor a Polish space , let, respectively.
of bounded measurable functions :The first result gives equivalent conditions for the Gibbs property. Its proof isstandard and omitted.
$T$$y\in \mathrm{Y}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}T-1(y)\mathrm{n}\pi^{-1}(\pi(y))\leq M$

$y$

$Z$

$Z$

$B_{Z}$$B_{Z}$$Zarrow \mathbb{R}$$f$Proposition 2.1: Let $\{\mu_{x} : x\in X\}$ be a system of conditional probabilities for ,be measurable functions. Then the following:
andand let :conditions are equivalent and each of the conditions expresses the Gibbs property:
is invertible and for all
on which(a) For every $x\in X$ and every
vanishing outside of $T(E)$ ,$\mathcal{Y}$$\mathrm{Y}arrow \mathbb{R}$$A$$Xarrow \mathbb{R}$$\varphi$$T$$E\in B_{\mathrm{Y}}$$f\in B_{\mathrm{Y}}$(3)$\int f(y)\mu s(x)(dy)--A(X)\int f(T(y))\exp[-\varphi(y)]\mu_{x}(dy)$.

(b) For all $h\in B_{Y}$ vanishing outside a set $E$ as above, for all $x\in X$ ,(4)

$A(x) \int h(y)\mu x(dy)=\int h(T^{*}(y))\exp[\varphi(T*(y))]\mu s(x)(dy)$.
denotes the inverse of
where(c) For all $x\in X$ the Jacobian with respect to the map$\tau*$,$T_{|E}$(5)$\frac{d\mu_{S(x})\mathrm{o}Tx}{d\mu_{x}}=A(x)\exp[-\varphi]$

is given by$T$$\mu_{x}\mathrm{a}.\mathrm{e}$.

Remark 2.2:a) The Jacobian in the left-hand side of (5) is, by definition, any Borel measatisfiyng) $(\tau(E))=$on the support ofsurable nonnegative function
is injective (seewith the property that$\int_{E}J(y)\mu x(dy)$ for every Borel set
be a probability measure on a standard Borel space $X_{i}(\dot{i}=1,2)$ .
(3) . Letis a Borel map which is nonsingular with respect to:Assume that-sets toand which maps
one on the support ofandand isexists, takes values in. Then the Jacobian-sets. It can be characterized by the property thatuniquely defined up to
$J$

$)$

$\mu_{S(x}$$\mu_{x}$$T|_{E}$$E$$\mu_{i}$$R$$X_{1}arrow X_{2}$$\mu_{1^{-}}\mathrm{n}\mathrm{u}\mathrm{l}1$$\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}-\mathrm{t}_{0}$$\mu_{1}$$\mu_{1}$$\mu_{2},$$J_{R}=J_{R}^{\mu_{1},\mu_{2}}$$\mu_{2^{-\mathrm{n}\mathrm{u}\mathrm{l}1}}- \mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s}$$\mu_{1^{-}}\mathrm{n}\mathrm{u}\mathrm{l}1$$\int_{X_{1}}f(x)\mu_{1}(dx)=\int_{X_{2}}\sum_{R(x’)=x}f(x’)J_{R}(X’)-1\mu_{2}(dx)$$(0, \infty)$

4for every $f\in L_{1}(\mu_{1})$ . Also the following holds for the Jacobian and the Radon$(x\in X_{2})$ .
Nikodym derivative:$\frac{d\mu_{1}\mathrm{o}R^{-1}}{d\mu_{2}}(x)=\sum_{x\in R^{-1}(x},)J_{R}(X^{l})^{-1}$b) Again, consider two standard Borel spaces
andwith -finite measuresand , respectively, and a countable-to-one Borel map $R:X_{1}arrow X_{2}$ . If is areal-valued Borel function on
and A is a constant then is called Gibbs withrespect toif for every function $f\in B_{X_{1}}$ we have (cf. (1))
$X_{1}$$\mu_{1}$$X_{2}$$\sigma$$\mu_{2}$$\varphi$$R$$X_{1}$$(\varphi, A)$$A \int f(y)\mu 1(dy)=\int\sum_{y\in R(y)}f(y)\prime \mathrm{p}\mathrm{e}\mathrm{x}[\varphi(y’)]\mu 2(d’-1y)$.

In case of a self-map of . probability space one obtains the definition of aconformal measure in [2].
Moreover, this definition applied to any triple
for $x\in X$ reduces to (1). From a general measure-theoretical
viewpoint the Gibbs property in the above general form (for some$A)$ ) is the
combination of two properties of :(1) is nonsingular ( $\mu_{2}(A)=0$ implies $\mu_{1}(T^{-1}(A)=0)$ ;(2) if $\mu_{1}(A)=0$ , then $\mu_{2}(T(A))=0$ .
More precisely, in this case is Gibbs
respect to $(-\log J_{R}, 1)$ .
$\phi-$$\mathrm{a}$$((\mathrm{Y}, \mu_{x}),$$(\mathrm{Y},\mu_{S(x})),$ $\tau_{x})$$(\varphi,$

$R$

$R$

$R$

$\mathrm{w}\mathrm{i}\mathrm{t}\dot{\mathrm{h}}$c) The definition of a conditional Gibbs measure does not imply that preimages ofsets of positive measure have positive measure. But this is the case if the image of
the whole space has full measure.
d) Consider the situation in b) again. It follows from Rokhlin’s theory [9] of conditional probabilities for measurable partitions that a measure preserving countableto-one map has the property that images of null-sets are null-sets and, in view ofb), that the measure is Gibbs. Conversely, a map $R$ which is Gibbs with respect to, is measure preserving
if and only if $A^{-1} \sum_{R()=x}x’\exp[\varphi(X’)]=$$(\varphi, 1)$$1\mu_{2}-\mathrm{a}.\mathrm{e}$$(\mu_{1}\mathrm{o}R=\mu_{2})$.

3. Uniqueness of Gibbs measures for fibred systems
In this section we state conditions for the uniqueness of Gibbs measures which,in fact, are weaker than those of theorem 1.1. (An interesting peoblem is to weaken
the conditions in theorem 1.1 for the existence of Gibbs measures). A system$\{\mu_{x} : x\in X\}$ of probability measureson
is called (weakly) continuous, if
for every continuous bounded function on
the function $x arrow\int f(y)\mu_{x}(dy)$ is
continuous. Also, two systems $\{\mu_{x} : x\in X\}$ and $\{\nu_{x} : x\in X\}$ are called equivalent
if for every $x\in X$ the measuresandare equivalent (have the same null-sets).
Definition 3.1: The fibred system
is called expanding along
fibres (with respect to the metric ) if there exist constants $\Lambda>1,$, and an$N_{0}>0$
integersuch that$\mathrm{Y}$$\mu_{x}$$f$$\mu_{x}$$\mathrm{Y}$$\nu_{x}$$\mathcal{Y}=(\mathrm{Y}, T, X, S, \pi)$$d$$d(T^{N\mathrm{o}}(y), \tau N\mathrm{O}(y’))\geq\Lambda^{N_{0}}d(y,y’)$$\epsilon_{0}>0$5.
such that $\pi(y)=\pi(y’),$for every $y,$
The next property of fibred systems (see (6) below) needs some terminology
related to covering theory with applications to differentiation of set functions. Except for the definition of the universal Vitali relation we follow section 2.8 of [5]. A
covering relation on a metric space is a subset of $\{(z, S) : z\in S\subset Z\}$ and it. For $A\subset Z$ let $C(A)=\{S$ : $(z, S)\in$
is fine at if
for some $z\in A$ }. A Vitali relation for a Borel measure $m$ (finite on bounded sets)
is a family of Borel sets, is fine at eachis a covering relation such thatis a covering relation, $A\subset Z$
point of and the following condition holds: if
and if is fine at each point of , then $C(A)$ has a countable disjoint subfamily
which covers a subset of of full measure in . A covering relation is said to be auniversal Vitali relation if it is a Vitali relation for any measure $m$ as above.
We consider the following property for expanding fibred systems
where is a compact metric space:
There exist $r_{0}>0$ and a covering relation $C\subset\{(y,B):y\in B\in B\}$ on such thatfor any $x\in X$ the following two conditions are satisfied:(6a) $C_{x}=\{(y, B\cap \mathrm{Y}_{x}) : y\in \mathrm{Y}_{x}, (y, B)\in C\}$ is a universal Vitali relation on .
(6b) For every $(y, B)\in C_{x}$ there exists $n\geq 0$ such that(where is the same as: $Barrow T^{n}(B)$ is invertible, diam
(i)
and
3.1)
in definition(ii) $T^{n}(B)\supset B_{2}(T^{n}y, r_{0})$ , where $B_{2}(z, r)$ denotes the ball of radius incentered at .
There are two basic examples satisfying (6) and which are expanding along fibres.
be continuous with a:
be a compact space andExample 3.2: Let
Markov partition for , respecting fibres and for which the fibre maps are uniformly$TR$ is a
expanding (if $R$ is a Markov partition then for every $x\in X,$
can be assumeded; also note that the sets fromunion of sets into have arbitrary small diameters). Let us suppose also that each element of 71 is
the closure of its interior. Then property (6) can be verified (see theorem 2.8.19 of[5] for the relation$d(\tau^{N_{\mathrm{O}}}(y), T^{N}0(y’))<\epsilon_{0}$$y’\in \mathrm{Y}$

$Z$

$C$

$\inf\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(S) : (z, S)\in C\}=0$

$z$

$C$

$\mathcal{V}$$\mathcal{V}(Z)$$\mathcal{V}$$Z$$C\subset \mathcal{V}$

$A$

$C$

$A$

$A$

$\mathcal{Y}=(\mathrm{Y}, T, x, S, \pi)$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}_{x}$$(\tau^{n}(B))\leq\epsilon_{0}$$T^{n}|_{B}$$\epsilon_{0}$$r$$\mathrm{Y}_{\pi(z)}$

$z$

$T$

$\mathrm{Y}$$\mathrm{Y}arrow \mathrm{Y}$$T$$R\in \mathcal{R}\cap \mathrm{Y}_{x},$$\mathcal{R}\cap \mathrm{Y}_{x}$$\mathcal{R}\cap \mathrm{Y}s_{(x)}$$)$$C= \{(y,B) : y\in B=\bigcap_{k=0}^{mk}\tau-R_{i_{k}}, R_{i_{k}}\in R;0\leq k\leq m;m\geq 0\}$.

and
Example 3.3: The second example is from conformal dynamics. Let
that
,
and
assume,denoted bybe contained in the complex plane
let each. If is expanding:
map
extends to a holomorphic
each mapalong fibres (we conjecture that it is true for strict polynomials if the forward orbitof the set of critical points of does not have any accumulation point in Y), thencan be shown to satisfy (6) for the relation defined by all pairs $(y, B)$ where$B$ is any ball (with respect to the maximum metric of) of sufficiently small
consist of pairs of the(note that in this casediameter and centered at$(y,B_{2}(y,r)),$(6a) is implied by
Property. This can be seen as follows.
form
theorem 2.8.18 and section 2.8.9 of [5]. Furthermore, there exist constants $K>0$of
all inverse branches
and $r_{1}>0$ such that for each $n\geq 1$ and every
are well defined on $\hat{B}_{2}(y, r_{1})=\{y’\in\hat{\mathrm{Y}}_{\pi(y)} : d_{\pi(y)}(y,y’)<r_{1}\}$ and, by Koebe’s
denotes the partial, wheretheorem,. This estimate bounds the distortion of fibres underderivative in the fibre$\mathrm{Y}\subset \mathbb{C}^{2}$$\hat{\mathrm{Y}}_{x}$$\mathbb{C}$$\mathrm{Y}_{x}$$\hat{T}_{x}$$T_{x}$$\hat{\mathrm{Y}}_{x}arrow\hat{\mathrm{Y}}_{S(x)}$

$T$

$T$

$C$

$T$

$\mathbb{C}^{2}$$C_{x}$$y\in \mathrm{Y}$$y\in \mathrm{Y}_{x})$$\hat{T}_{\nu}^{-n}$$y\in \mathrm{Y}$$\hat{T}^{n}$$|D_{2}\hat{T}_{\nu}^{-n}(y)|\leq K|D_{2\nu}\hat{\tau}^{-n}(y’)|y’\in\hat{B}_{2}(y, r_{1})$$\hat{\mathrm{Y}}_{\pi(y)}$$D_{2}$

6
itself (restricted to the range of an inverseand under
the inverse branches ofbranch) uniformly in and over all fibres. We may (and shall) suppose that). It is not difficult to show thatis sufficiently small (in particular,
holds for the family .
$\hat{T}^{n}$$\hat{T}^{n}$$n$$r_{1}$$r_{1}\leq\epsilon_{0}/2$$(6\mathrm{b}\rangle$$C$

be a fibred system with compact metric
Proposition 3.4: Let: $x\in$space Y. Assume that is expanding along fibres and satisfies (6). Let {$X\}$ be a conformal system of conditional probabilities for some continuous function
which is also uniformly H\"older continuous in each fibre for some exponent $s>0$ .
Assume that$\mathcal{Y}=(\mathrm{Y}, T, X, S, \pi)$$T$$\mu_{x}$$\varphi$(7)$\inf${$\mu_{x}(B)$: $x\in X;B$ is a ball of radius$r_{0}$

with center in$\mathrm{Y}_{x}$} $>0$ .

Then $\{\mu_{x} : x\in X\}$ is uniquely determined b.y the above properties up to equivalence.

.
-Gibbs. Note that by (6)Proof. We may assume that the system isthe inverse of
Fix $x\in X$ and $(y, B)\in C_{x}$ . Set $D=T^{n}(B)$ and denote byon
sending $T^{n}(y)$ to . Then by the expanding property in definition 3.1 for$y’,y”\in D$ we have that $d(y’,y”)=d(\tau^{n-k}(\tau k(\tau^{n*}(y’))),Tn-k(\tau^{k}(\tau n*(y’’))))$ andhence$r_{0}\leq\epsilon 0$$(\varphi, 1)$$T^{n*}$$T^{n}$$B$$y$$d(Tk(T^{n*}(y)), \tau k’(\tau n*(y\prime\prime)))\leq\Lambda^{-n+k}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(D)$.

Then, using H\"older continuity, we obtain$|_{k=0} \sum^{n}\varphi(T^{k}(\tau n*(y’)))-\varphi(T^{k}(\tau^{n}*(y’)’))|\leq||\varphi||_{s}\sum_{k=0}^{n}\Lambda^{-s}k(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(D))^{S}=\log K_{1}$

,

where
denotes the upper bound of all H\"older seminorms of exponent takenover the fibres. Therefore, $\mu_{x}(B)\exp[-\varphi(y) -... -\varphi(T^{n}(y))]=K_{1}\mu_{S^{n}(}x)$ $(T^{n}(B))\leq$. Similarly, $\mu_{x}(B)\exp[-\varphi(y) -... -\varphi(Tn(y))]\geq K_{1}^{-1}\inf_{B}’\{\mu_{S^{n}()}x(B’)\}>0$ ,of radius(as in (6)).
where $B’$ denotes a ball in
Let $\{\nu_{x} : x\in X\}$ denote another Gibbs system of conditional measures for thefunction satisfying (7). We derive for some constant $K_{2}>0$$s$$||\varphi||_{s}$$K_{1}$$\mathrm{Y}_{S^{n}(\pi())}y$$r_{0}$$\varphi$(8)$K_{2}^{-1}\nu_{x}(B)\leq\mu_{x}(B)\leq K_{2x}\nu(B)$for all sets $B$ such that $(y,B)\in C$ for some $y\in$ Y. In order to extend (8) toY. There exists a
arbitrary Borel sets, consider a relatively open
which is fine at any $y\in G$ and such that $B\subset G$ for
covering relation
$(y,
B)\in
C_{x}^{G}$. Thencontains a countable disjoint subfamily $\{B_{i} : i\in I\}$every
, hencewith$G\subset \mathrm{Y}_{x}\subset$$C_{x}^{G}\subset C_{x}$$C_{x}^{G}$$\mu_{x}(\bigcup_{i\in I}B_{i})=\mu_{x}(G)$$\mu_{x}(G)=\mu_{x}(\bigcup_{i\in I}Bi)=\sum\mu x(B_{i}i\in I)\leq K_{2}\sum_{i\in I}\nu_{x}(Bi)=\nu_{x}(i\bigcup_{\in I}Bi)\leq\nu_{x}(G)$,by suchand, by symmetry arguments, $\nu_{x}(G)\leq K_{2}\mu_{x}(G)$ . Approximating
we obtain (8) for any measurable set
andsimultaneously with respect to$B$ . This proves the proposition.
$B\in \mathcal{B}_{\mathrm{Y}_{x}}$$G$$\mu_{x}$$\nu_{x}$7Corollary 3.5: Let $\{\mu_{x} : x\in X\}$ be in addition continuous and assume that theset of periodic points of : $Xarrow X$ is dense in $X$ . Then $\{\mu_{x} : x\in X\}$ is unique
as a continuous Gibbs measure for .
$S$$\phi$:
Proof. Let $x\in X$ be periodic for with period . Thenand the function $\Phi_{x}(y)=\varphi(y)+\ldots+\varphi(T^{n-1}(y))$ .
is Gibbs forand henceis H\"older continuous. Thisis expanding andis unique because
Suchfollows from the previous proposition together with the ergodic decomposition of
Gibbs measures (this can be shown as in [10]). Now let $\{\nu_{x} : x\in X\}$ be anotherfor every periodic point of , andcontinuous Gibbs system for . Thensince the systems are continuous, they must coincide because periodic points aredense.
$S$$n$$\mathrm{Y}_{x}arrow \mathrm{Y}_{x}$$T_{0}:=T_{|\mathrm{Y}_{x}}^{n}$$T_{0}$$\mu_{x}$$\Phi_{x}$$T_{0}$$\mu_{x}$

$S$

$x$

$\mu_{x}=\nu_{x}$$\varphi$

4. Absolutely continuous invariant measures

In this section we construct invariant and other measures of special interest. It is clear that for a
from a given system of conditional probabilities forthe measure $\mu=\nu 0\pi^{-1}$ is S-invariant.
-invariant probability measure on
Conversely, every -invariant probability measure on $X$ can be lifted to a Tinvariant probability measure on Y. However, we are interested in such liftings
when the conditional measures given are prescribed. A system $\{\mu_{x} : x\in X\}$ ofconditional probabilities for is called invariant if) for all $x\in X$ .
Integrating such a system over with respect to any -invariant probability one
obtains a -invariant measure . Moreover, assuming the hypotheses of corollary
3.5 it can be easily seen that a system $\{\mu_{x} : x\in X\}$ is invariant if such an integral
is a -invariant measure for any -invariant probability . For invertible any Tby integration of a (measurable) invariant
invariant probability measure isfamily of conditional probabilities. But this is not the case for non-invertible .
In this case invariant systems only form a subclass within the class of systems ofconditional measures on fibres arising from -invariant probabilities on Y. The
following proposition gives some necessary and sufficient conditions that a Gibbs
measure admits an equivalent invariant system.
..
operators
sfer
:
the
conditional. We
Assume that$(k\geq 0, x\in X)$ , by (2) where.
$(\mathrm{Y}, T)$

$T$

$\nu$

$\mathrm{Y}$

$S$

$\mu$

$\nu$

$\pi$

$\mu_{x}\mathrm{o}T^{-1}=\mu_{S(x}$$\mathcal{Y}$

$S$

$x$

$T$

$\mu$

$\nu$

$T$

$S$

$\mu$

$S$

$\nu$

$\mathrm{o}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\dot{\mathrm{e}}\mathrm{d}$

$S$

$T$

$V_{x}^{(k)}$$\mathrm{t}\mathrm{r}\mathrm{a}\dot{\mathrm{n}}$$\mathrm{d}\dot{\mathrm{e}}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}$$\varphi\in B_{Y}$$B_{x}arrow$$B_{x}:=B_{\mathrm{Y}_{x}}$$B_{S^{k}(x)},$

Proposition 4.1: Let $\{\mu_{x} : x\in X\}$ be(A) Assume that there exists a family
measurable nonnegative functions satisfying$(\varphi, A)$$\{h_{x} :$V_{x}^{(1)}h_{x}=\lambda(x)h_{S(x})$

and-Gibbs.

\mathrm{Y}_{x}arrow \mathbb{R}, x\in X\}\cup\{\lambda :$\int h_{x}(y)\mu_{x}(dy)>0$,$(x\in X)$

Xarrow \mathbb{R}\}$.

Then$A(x)= \lambda(x)\frac{\int h_{S(x)}(y)\mu s_{(}x)(dy)}{\int hx(y)\mu x(dy)}$and$\tilde{\mu}_{x}=\frac{h_{x}}{\int hx(y)\mu x(dy)}\mu_{x}$,$(x\in X)$of8defines an invariant system of conditional probabilities absolutely continuous withrespect to $\{\mu_{x} : x\in X\}$ .
(B) If there exists a -invariant system of conditional probabilities
absolutely continuous with respect to $\{\mu_{x} : x\in X\}$ so thatfor some
family of nonnegative measurable functions, then:$T$$\{\tilde{\mu}_{x} :x\in X\}$$\tilde{\mu}_{x}=h_{x}\mu_{x}$$h_{x}$$\mathrm{Y}_{x}arrow \mathbb{R}$,$(x\in X)$$V_{x}^{(1)}h_{x}=A(x)h_{S(x)}\mu_{S(x)}- \mathrm{a}.\mathrm{e}.$.

Proof. Our assumptions and (1) imply that
. Replacby $\varphi(y)-\log A(\pi(y))$ andby $h_{x}/ \int h_{x}(y)\mu_{x}(dy)$ we may assume thating$\{\mu_{x} : x\in X\}$ is a
-Gibbs measure and thatsatisfies $V_{x}^{(1)}h_{x}=h_{S(x)}$ and$\int h(xy)\mu x(dy)=1$ , $(x\in X)$ . Let. Then, by (1) and the definitionof the conditional transfer operator, we have for any:$A(x)= \lambda(x)\frac{\int h_{S(x)}(y)\mu s(x)(dy)}{\int h_{x}(y)\mu x(dy)}$$\varphi(y)$$h_{x}$$(\varphi, 1)$$h_{x}$$d\tilde{\mu}_{x}=h_{x}d\mu_{x}$$f\in B_{\mathrm{Y}}$$\int f(T(y))\overline{\mu}x(dy)=I^{f}(T(y))hx(y)\mu_{x}(dy)=\int V_{x}^{(1)}[h_{x}f\circ T](y)\mu_{g}(x)(dy)$$= \int f(y)V^{()}1h(y)\mu_{S(}x)(dy)=xx\int f(y)hs(x)(y)\mu_{S(}x)(dy)=\int f(y)\overline{\mu}_{S}(x)(dy)$This proves the invariance ofand (A).
In order to show (B), we obtain from the above$\tilde{\mu}_{x}$$\int f(y)V_{x}^{(1})hx(y)\mu_{S}(x)(dy)=\int f_{0}Td\tilde{\mu}x=\int fd\tilde{\mu}_{S}(x)=\int fh_{S(x})d\mu s(x)$for every integrable $f$ . Thus

$V_{x}^{(1)}h_{x}=h_{S(x)}\mu_{S(x)}- \mathrm{a}.\mathrm{e}$.

The following proposition 4.2 is concerned with general conditions under which ameasurable system of conditional probabilities on fibres, together with a probability
measure on the base, gives rise to a conformal (or invariant conformal) measure on
the total space.

Proposition 4.2: Letbe a fibred system with a bounded-toone map $S:Xarrow X$ and let and be measurable functions on and $X$ .
(A) Let be a-conformal probability on and let $\{\nu_{x} : x\in X\}$ be a version of
the conditional probabilities for with respect to . Then $\mu=\nu 0\pi^{-1}$ is a $(\log B, 1)-$
conformal measure for (X, ) where $B(x)= \int_{\mathrm{Y}_{S(x)}}V_{x}1((1)y)\nu s(x)(dy)$ $(x\in X)$ .
Moreover, for. $x\in X,$ :is a-Gibbs map (cf. remark 2.2, i.e. the Jacobian offoris.
(B) Conversely, if $\{\nu_{x} : x\in X\}$ is,-Gibbs forand if is a
conformal measure for (X, ) then $\nu(dy)=\int_{\mathrm{x}^{\nu_{x}}}(dy)\mu(dx)$ is-conformalforwhere A.
$\mathcal{Y}=(\mathrm{Y},T,X, S,\pi)$$A$$\varphi$$\nu$$\mathrm{Y}$$(\varphi, 1)$$\mathrm{Y}$

$\nu$

$\pi$

$S$

$T_{x}$$\mu- \mathrm{a}.\mathrm{e}$$\mathrm{b}))$$\mathrm{Y}_{x}arrow \mathrm{Y}_{S(x)}$$T_{x}$$(\mathrm{Y}_{x}, \nu_{x})$$(\varphi,B)$$\frac{d\nu_{S(x)}\circ T}{d\nu_{x}}(y)=B(x)\exp[-\varphi(y)]\nu_{x}-\mathrm{a}.\mathrm{e}$$(\varphi,A)$$(\mathrm{Y},T)$

$S$

$\mu$

$(\psi, 1)-$$(\Lambda, 1)$$=\varphi+(\psi_{-\mathrm{l}}\mathrm{o}\mathrm{g}A)0\pi$$(\mathrm{Y}, T)$

Proof. (A) Let be-conformal. Then the conformality of follows fromremark 2.2 b). In fact, we only need to prove thatis a nonsingular map and$\nu$$(\varphi, 1)$

$\mu$

$T$

.

9
sends null-sets to null-sets. If $E\in B_{X}$ then $S^{-1}(E)=\pi(T^{-1}(\pi^{-1}(E)))$ and $S(E)=$is measure preserving,, since is onto. Also, since :it follows that is n.onsingular and sends null-sets to null-sets as well. Henceis(and prove once more thatis Gibbs. Let us calculate the Jacobian of$f\in
L_{1}(X,
\mu)$
and
is
preservingismeasure
Then,
since
.
Gibbs). Take a funcion
conformal, we have$\pi(T(\pi-1(E)))$$(\mathrm{Y}, \nu)arrow(X,\mu)$

$\pi$

$\pi$

$S$

$S$

$S$

$S$

$\nu$

$\pi$

$\int_{X}f(_{X})\mu(dX)=\int_{\mathrm{Y}}f(\pi(y))\nu(dy)=\int_{Y}y’\in\tau-1()\sum_{y}f(\pi(y’))\exp[\varphi(y)’]\nu(dy)$$= \int_{X}(\int_{\mathrm{Y}}\sum_{x’\in S^{-1}(x)}(y\sum_{y’\in\tau^{-1})\mathrm{n}\pi^{-1}(x)}f(\pi(y’)’)\exp[\varphi(y)’]\nu_{x}(dy)\mathrm{I}\mu(dx)$$=I_{X_{x’\in S^{-}}} \sum_{(1x)}f(x’)(\int_{Y_{x}}y’\in\tau^{-1}(y)\sum_{(\cap\pi x’)}\exp[\varphi(y)’]-1\nu_{x}(dy))\mu(dx)$$= \int_{X_{x’}}\sum_{s\in-1(x)}JS(X^{;})^{-1}f(x’)\mu(d_{X)}$,where $J_{S}(X)^{-}1= \int \mathrm{Y}s(x)\sum_{y\in\tau^{-1}},(y)\cap\pi-1(x)\exp[\varphi(y’)]\nu_{S(x)}(dy)$ .
Similarly one shows the second part of (A) using (1) repeatedly and conformality
be bounded measurable functions. Thenand :of . Let :
$\mu$

$f$

$\mathrm{Y}arrow \mathbb{R}$$g$$Xarrow \mathbb{R}$$\int_{X}B(x)g(x)\int_{\mathrm{Y}_{x}}f(y)\nu(dy)x\mu(dX)=\int_{\mathrm{Y}}B(\pi(y))g(\pi(y))f(y)\nu(dy)$$= \int_{X}\sum_{x()}g(x\in\prime S^{-1}X’)B(X’)(\int_{\mathrm{Y}_{x}}y’\in\tau-1\sum_{(y)\mathrm{n}\pi(x);}f(y)’\exp[\varphi-1(y)’]\nu_{x}(dy))\mu(dx)$.

$= \int_{X}g(x)\int_{\mathrm{Y}_{S(x\rangle}}1)\cap\pi-1()\sum_{xy)}f(y)\prime \mathrm{x}\mathrm{e}\mathrm{p}[\varphi(y’)]\nu s(x)(dy\mu(dx)y\in\tau-(’$. $x \in XB(x)\int_{Y}xf(y)\nu_{x}(dy)=\int_{\mathrm{Y}}s_{1^{x}\rangle}V^{()}xf1(y)\nu_{s(})x(dy)$.
It follows from this that forthe claim follows.
Applying remark 2.2 b) to triples(B) We shall show (1) for a -integrable function and for using conformality
in the form (1).
of and the Gibbs property of$\mu_{- \mathrm{a}}.\mathrm{e}$$((\mathrm{Y}_{x}, \nu_{x}),$$(\mathrm{Y}_{S(x),S}\nu(x)),$ $\tau_{x})$

$f$

$\nu$

$\mu$

$\nu$

$\nu_{x}$$\int f(y)\nu(dy)=\int\sum_{sx’\in(x)}-1\int f(y)\nu_{x’}(dy)\exp[\psi(X’)]\mu(d_{X)}$$= \int\sum_{x’\in S^{-}(x)}A(x)’-1\exp[\psi(x’)]\int\sum_{)y)\cap\pi(-1x\prime}f(y’)\exp[\varphi(y)]\nu s(x’)(d’)1y’\in T^{-}1(y\mu(dx)$$= \int_{y’\in T^{-1}}\sum_{(y)}f(y)\exp[\Lambda’(y)]’(\nu dy)$.

Corollary 4.3:
(A) In the situation of proposition 4.2 (A)$\mu$

is -invariant iff $\int\sum_{y\in\tau^{-1}},(y)\exp[\varphi(y’)]\nu_{x}(dy)$$S$10, and is -invariant iff
v-a.e.
(B) In the situation of proposition 4.2 (B) let be a density with respect to . Let$\overline{\nu}(dy)=h(y)\nu(dy)$ . Thenis -invariant iff satisfies $h(y)= \sum_{y()},\in\tau-1\exp[y\Lambda(y^{;})]h(y);$ ,$=1\mu- \mathrm{a}.\mathrm{e}.$

$\nu$

$T$

$\sum_{y()^{\mathrm{e}\mathrm{x}}},\in\tau-1\mathrm{p}[y\varphi(y’)]=1$$h$$\tilde{\nu}$$\nu- \mathrm{a}.\mathrm{e}$

$T$

$\nu$

$h$

.

References[1] T. Bogensch\"utz, V.M. Gundlach: Ruelle’s transfer operator for random subshiftsof finite type. Ergod. Theory and Dynam. Systems 151995, 413-447[2] M. Denker, M. Urbatski: On the existence of conformal measures. Trans. Amer.
Math. Soc. 3281991, 563-587[3] M. Denker, St. M. Heinemann: Jordan Tori and polynomial endomorphisms in. Preprint.
[4] M. Denker, M. Gordin: Gibbs measures for fibred systems. Preprint.
[5] H. Federer: Geometric measure theory. Grundlehren Band 153, Springer Verlag,
Berlin, Heidelberg, New York 1969.
[6] St. M. Heinemann: Julia sets for endomorphisms of. Ergodic Theory andDyn. Syst. 16 (1996), 1275-1296.
[7] St. M. Heinemann: Julia sets for skew products in. Preprint.
[8] K. Kuratowski: Topology, vol.1, Acad. Press, Warsaw, 1966, Ch.3, No 39: 8[9] V.A. Rokhlin: On the fundamental ideas of measure theory. Mat. Sbornik 25(67) 1949, 107-150 (in Russian). Math. Soc. Translations 711952, (1) 10,$\mathbb{C}^{2}$$\mathbb{C}^{n}$$\mathbb{C}^{2}$$=\mathrm{A}\mathrm{m}\mathrm{e}\mathrm{r}$1-54.
[10] D. Sullivan: Conformal dynamical systems. In: Geometric Dynamics. Lect.
Notes in Math. 10071983, 725-752
DENKER: INSTITUT
MATHEMATISCHE STOCHASTIK, UNIVERSIT\"AT G\"oTTINGEN, LOTZESTR.
13, 37083 G\"OTTINGEN, GERMANY.
$E$-mailaddress: denker@namu01.gwdg.de$\mathrm{F}\ddot{\mathrm{U}}\mathrm{R}$

GoRDlN: ST. PETERSBURG DlvlsloN oF V.A. STEKLOV MATHEMATICAL INSTITUTE, FONTANKA
27, 197011 ST. PERTERSBURG, RUSSIA.
$E$-mailaddress: gordin@pdmi.ras.ru