\magnification = 1200
\hfuzz=10pt
\hsize=4.8in
\vsize=7.3in
\baselineskip=18pt
\hoffset=0.35in
\voffset=0.1in
\parindent=3pt
\def\v{\par\noindent}
\def\di{\displaystyle}
\def\seq#1#2{(#1_0, #1_1, \ldots , #1_{#2})}
\def\rest#1#2{\sigma|_{\di [ #1, #2]}}
\def\R{I\!\!R}
\def\C{I\!\!\!\!C}
\def\N{I\!\!N}
\def\Q{I\!\!\!\!Q}
\def\Z{I\!\!\!\!Z}
\def\ui{[0,1]}
\def\O{\Omega_{\geq}}
\def\o{\omega}
\def\t{\theta}
\def\z{\zeta}
\def\limsup{\mathop{\overline{\rm lim}}}
\def\liminf{\mathop{\underline{\rm lim}}}
\def\ut{{\tilde u}}
\def\dz{\zeta^{\prime}}
\def\S{\Sigma_{\geq}}
\def\SE{\Sigma}
\def\s{\sigma}
\def\a{\alpha}
\def\g{\gamma}
\def\k{\kappa}
\def\b#1#2{\exp ( #1 ( \log #2) ^\a )}
\def\eps{\epsilon}
\def\sp{\sigma^{\prime}}
\def\A{{\cal A}}
\def\L{{\cal L}}
\def\P{{\cal P}}
\def\I{{\cal I}}
\def\df{f^{\prime}}
\def\ddf{f^{\prime \prime}}
\def\dphi{\phi^{\prime}}
\def\dpsi{\psi^{\prime}}
\def\dg{g^{\prime}}
\def\St{\tilde S}
\def\Dt{\tilde D}
\def\Xt{\tilde X}
\def\F{{\cal F}_{\g}}
\def\ub{ u'}
\def\ubt{{\tilde {\ub}}}
\centerline{\bf ON THE INVARIANCE PRINCIPLE FOR}
\centerline{\bf NON-UNIFORMLY EXPANDING TRANSFORMATIONS OF $\ui$}
\vglue 0.2cm
\vglue 1.0cm
\centerline{ Massimo Campanino and Stefano Isola}
\centerline{\sl Universit\'a di Bologna}
\vskip 1cm
{\bf Abstract.} We consider a class of maps of $\ui$
with an indifferent fixed point at $0$ and
expanding everywhere else. Using a suitable uniformly expanding induced map we
prove a functional central limit theorem (invariance principle)
with anomalous scaling $n/\log n$ for the random stationary process generated
by this dynamical system.
\vskip 1cm
{\it Keywords}: Donsker theorem, infinite invariant measure, Markov partition,
induced transformation.
\vskip 1cm
{\it AMS 1991 Subject classification}: Primary 60F05, Secondary: 28D05, 58F11
\footnote{}{
Postal address: Dipartimento di Matematica, piazza di
Porta S.Donato 5, I-40127 Bologna, Italy.
e-mail address: campanin@dm.unibo.it, isola@dm.unibo.it.
The first author's research was partially supported by EU
contract CHRX-CT93-0411 and by Italian G.~N.~A.~F.~A. }
\vfill \eject
We first introduce the basic setting.
Let $f$ be the map of the unit interval $\ui$ defined as follows:
\item{(i)} $f(0)=0, \; f(1)=1$;
\item{(ii)} $f$ is monotone and non decreasing on $I_0 = [0,{1\over
2}[$ and $I_1=]{1\over 2},1]$;
\item {(iii)} For each $i=0,1$, $f_{|I_i}$ extends to a $C^2$ function
$f_i$ on its closure which is onto $\ui$.
\item{(iv)} There are two numbers $\alpha > 1$ and $L > 0$ such that:
$$
\df_{|I_1} \geq \alpha, \;\; \df(0)=1,\;\; \df_{|]0,1/2[}
\geq 1, \;\; \sup_{x\in \ui}|\ddf(x)/\df(x)|\leq L.
$$
\item{(v)} $\ddf(0)\not= 0$ which implies $\ddf (0) >0$.
\vskip 0.2cm
In [CI2] we proved that under the above assumptions the following
ergodic theorem holds: there exist an increasing sequence $c_n =
\kappa \, n / \log n$, where $\kappa$ is a positive constant
depending on $f$, such that for
any real function $u$ compactly supported on $]0,1]$
$$
{1\over c_n} \, \sum_{k=0}^{n-1}\, u(f^kx) \rightarrow \nu(u) \qquad
\hbox{in probability}
$$
where $\nu$ is a $\s$-finite absolutely continuous $f$-invariant measure
whose density $e$ satisfies $C_1/x \leq e(x) \leq C_2/x$ for any $x\in \ui$
and $C_1,C_2$ suitable positive constants
(for related results see [T1],[T2],[CF]).
In this paper we want to study the fluctuations of the finite sums
${1\over \di c_n} \, \sum_{k=0}^{n-1}\, u(f^kx)$.
In particular, we shall prove a functional central limit theorem
for the random variables
${1\over \di \sqrt{c_n}} \, \sum_{k=0}^{n-1}\, u(f^kx)$ (see below, Theorem 1).
A central limit theorem for the case of a finite measure has been
proved in [LSV] and, in a slightly different context, in [ADU] (see
also [HK] for a more general situation).
\vskip 0.5cm
Let us consider the sequence of points $c_k$, $k\geq 0$,
given by $$
c_0=1, \;\; c_k=f_0^{-1}(c_{k-1}),\;\;\; k\geq 1.
$$
This sequence generates a countable partition of
$\ui$ into the intervals $A_k=[c_{k},c_{k-1}]$, $k\geq 1$,
which is a Markov partition. In particular,
$f(A_k)=A_{k-1}$, $k\geq 1$.
\vskip 0.2cm
Let $\O$ be the set of one-sided
sequences $\o = (\o_0,\o_1,\dots )$, $\o_i\in\{1,2,\dots\}$
satisfying the compatibility condition: given $\o_i$ then either
$\o_{i-1}=\o_i +1$ or $\o_{i-1}=1$. Then, the map
$$
\phi :\o \rightarrow \phi(\o) =x\quad\hbox{according to}\quad
f^i(x)\in A_{\o_i}, i\geq 1
$$
is a
bijection between $\O$ and the points of $\ui$ which are not
preimages of the origin. Moreover, $\phi$ conjugates the map
$f$ with the shift $\tau$ on $\O$.
For every integer $i\geq 1$ we denote by $x_i$ the projection on
the $i^{th}$ symbol, i.e. $x_i(\o)=\o_i$,
and define the "free" probability measure $\mu$ by
$$
\mu(\o_i) = |A_{\o_i}|,\;\; i\geq 1 \eqno(1)
$$
With slight abuse of language we shall again denote by $\mu$ the measure
$\mu\circ\phi^{-1}$, i.e. the Lebesgue measure on $\ui$.
\vskip 0.5cm
We now introduce the infinite sequence
$\tau_j$, $j\geq 1$, of successive entrance times in the state $1$:
$\tau_1(\o)=\inf\{i\geq 0\;:\; x_i(\o)=1\}$ and, for $j\geq 2$,
$\tau_j(\o)=\inf\{i>\tau_{i-1}\;:\; x_i(\o)=1\}$.
Furthermore, we define a sequence of integer valued random
variables by
$$
\s_j(\o)=\tau_{j+1}-\tau_j,\;\; j\geq 0\eqno(2)
$$
with the convention that $\tau_0=-1$.
\vskip 0.5cm
{\bf Definition 1.} {\it The `first passage' map (on the interval $A_1$),
is the map $g\; :\;\ui\to \ui$
induced by $f$ in the following way}:
$$
x\rightarrow g(x) = f^{n(x)}(x)
\quad\hbox{where}\quad n(x)=1+\min \{n\geq 0 \;:\; f^n(x)\in
A_1\;\}\eqno(3)
$$
\vskip 0.2cm
{\bf Definition 2.} {\it Let $u:\; ]0,1] \to \R$ be any real function.
Its} induced {\it version $\ut$ is defined by }
$$
\ut (x) = \sum_{s=0}^{n(x) -1}u(f^sx)\eqno(4)
$$
\vskip 0.2cm
{\bf Remark 1.}
The map $g$ is uniformly
expanding and surjective on each $A_k$, and enjoys the following property:
let $x=\phi(\o)$, where $\o\in \O$, then
$g^j(x)\in A_{\s_j}$, $j\geq 1$, where
the integers $\s_j=\s_j(\o)$ are defined in (2).
It has been proved in [CI1] that the dynamical system $(\ui, g)$ leaves invariant an
ergodic absolutely continuous probability measure $d\rho = h \, d\mu$,
such that $h$ is H\"older continuous and satisfies $d^{-1}\leq h \leq d$
for some $d>0$. Furthermore, $\rho$ satisfies the exponential uniform
mixing property [R].
The $g$-invariant probability measure $\rho$ is related to the $f$-invariant
infinite measure $\nu$ by the identity $$\nu (u) = \rho(\ut),$$ valid for
any $u$ of compact support in $]0,1]$ (see [CI2]).
\vskip 0.2cm
We now introduce a space of locally H\"older continuous functions.
Let $x, x' \in I \subseteq A_k$, for some $k\geq 1$, and write
$$
{\rm var}_{I} \, u = \sup \, \{ \, |u(x)-u(x')|\, : \, x,x'\in I\, \}
$$
Let $\F$ be the space of bounded continuous functions $u : \ui \to \R$ with compact support
in $]0,1]$ such that
$$
\sup_{k} \, \sup_{I\subseteq A_k} \,
\left(\, {{\rm var}_I \, u \over |I|^{\g}} \, \right)
\leq M <\infty
$$
for some $0<\gamma \leq 1$ and $M>0$. Notice that, since $u$ is of compact support
over $]0,1]$ the first sup above is actually taken over a finite set of $k$'s.
\vskip 0.2cm
{\bf Lemma 0.} {\it
\item{1)} If $u\in \F$ then $\ut \in \F$.
\item{2)} $u\, : \, \ui \to \R$ is a cocycle with respect
to the map $f$, i.e. $u(x)=v(f(x))-v(x)$ for some $v$,
if and only if $\ut$ is a cocycle with respect to the map $g$. }
\vskip 0.2cm
{\it Proof.} Let $u\in \F$. If $x, x' \in I \subseteq A_k$
we have $n(x)=n(x')=k$ and
$$
|\ut (x) - \ut (x') | \leq \sum_{0\leq j0$ is a constant independent of $j$ and $k$.
Hence
$$
{{\rm var}_{I} \, \ut \over |I|^{\g} } \leq \, M \,
\sum_{0\leq j < k} \,\left({|f^j(I)|\over |I|}\right)^{\g}\,
\leq \, M \, R\, \sum_{0\leq j < k} \,\left({|f^j(A_k)|\over |A_k|}\right)^{\g}
$$
so that, taking the sup over $k$ and recalling that $u$ is of
compact support over $]0,1]$, it follows that $\ut\in \F$.
To show the last assertion notice first that Definition 2 implies
at once that if $u=v(f(x))-v(x)$ then $\ut = v(g(x))-v(x)$.
To see the converse, observe that,
again from Definition 2, one has
$$
u(x) = \cases{ \ut(x), &if $x\in A_1$ \cr
\ut(x)-\ut(f(x)), &if $x\notin A_1$. \cr }
$$
Suppose now that $u(x)\equiv 0$
only for $x\notin A_1$ and assume that $\ut(x) = V(g(x))-V(x)$. Then,
since $f(x)=g(x)$ for $x\in A_1$, one also has
$u(x)=V(f(x))-V(x)$. On the other hand, if $x\notin A_1$ then
$\ut(x)=\ut(f(x))$ and $g(f(x))=g(x)$. Hence
$$
\ut(x)=\ut(f(x)) = V(g(x))-V(x) = V(g(f(x)))-V(x)=V(g(f(x)))-V(f(x))
$$
so that
$$
V(f(x))=V(x)
$$
and this implies that $u(x) = V(f(x))-V(x)$ for any $x\in \ui$.
Now, for any $u$ compactly supported in $]0,1]$, one can reduce
to the previous case by observing that using an induction procedure
$u$ can be always decomposed as
$\ub + \phi$ where $\ub(x)\equiv 0$ for $x\notin A_1$
and $\phi$ is a cocycle with respect to the map $f$.
Moreover, if $\ut$ is a cocycle then
$\ubt$ is a cocycle as well and the argument above can be applied.
Q.E.D.
\vskip 0.2cm
Let $\S$ be the set of {\it all} one-sided
sequences $\s$ of the form $\s =(\s_0,\s_1,\dots )$,
$\s_j\in\{1,2,\dots\}$. Then, the map
$$
\pi :\s \rightarrow \pi(\s) =x\quad\hbox{according to}\quad
g^j(x)\in A_{\s_j},\;\; j\geq 1\eqno(6)
$$
is a
bijection between $\S$ and the points of $\ui$ which are not
preimages of zero. Moreover, $\pi$ conjugates the map
$g$ with the shift $\tau$ on $\S$.
Notice also (cf. (4)) that
$n(g^k(x))=\s_k$ where $\s=(\s_0,\s_1,\dots)=\pi^{-1}(x)$.
\vskip 0.2cm
Let us now consider an orbit $\{f^kx\}_{k=0}^{n-1}$, for some $x\in ]0,1]$
and
denote by $N(n,x)$ the number of its passages in $A_1$, or, in other terms,
the number of symbols in its (truncated) $\s$-coding $(\s_0,\s_1,\dots \s_{N(n,x)-1})$.
We have
$$
\sum_{k=0}^{n-1}u(f^kx) = \sum_{s=0}^{N(n,x)-1}\ut (g^sx) + R_n(x,u)
\eqno(7)
$$
where the remainder is given by
$$
R_n(x,u) = \sum_{s=m(n,x)}^{n-1}u(f^sx)\quad\hbox{with}\quad
m(n,x)=\sum_{k=0}^{N(n,x)-1}n(g^kx)\eqno(8)
$$
Consider now a continuous function $u$ compactly supported on $]0,1]$.
The remainder $R_n(x,u)$ is then uniformly bounded in $n$ and $x$.
For such an $u$, we define
$$
S_n(x)=\sum_{k=0}^{n-1}u(f^kx),\qquad \St_n(x)=\sum_{s=0}^{n-1}\ut (g^sx)
\eqno(9)
$$
In order to deal with a continuous process we define for $t\geq 0$
$$
S(x,t) = \cases{ S_{n-1}(x) + (t-n+1)(S_n(x)-S_{n-1}(x)), &if
$n-1 \leq t < n$. \cr
S_n(x), &if $t=n$ \cr } \eqno(10)
$$
with the convention $S_0(x)=0$. An identical definition with
$\St_n$ in place of $S_n$ yields $\St(x,t)$.
We are now in the position to state the main result.
\vskip 0.2cm
{\bf Theorem 1.} {\it Let $u\in \F$ be such that $\nu (u)=0$ and not a cocycle.
Let moreover $a_n=n/\log n$.
Then, there exist a positive constant $D$ such that the random
element $X_n$ of $C(\ui)$ defined on the probability space $(\ui, d\rho)$ by
$$
X_n(x,t) = {1\over \sqrt{D\, a_n}} S(x,nt),\qquad 0\leq t \leq 1
$$
converges in law to the Brownian motion $B(t)$.}
\vskip 0.2cm
We shall prove Theorem 1 through a sequence intermediate results.
\vskip 0.2cm
{\bf Theorem 1'.} {\it Let $\ut \in \F$ be such that $\rho (\ut)=0$
and not a cocycle.
Then, there is a positive constant $\Dt$ such that the random
element $\Xt_n$ of $C(\ui)$ defined on the probability space $(\ui, d\rho)$ by
$$
\Xt_n(x,t) = {1\over \sqrt{\Dt \,n}} \St(x,nt),\qquad 0\leq t \leq 1
$$
converges in law to the Brownian motion $B(t)$.}
\vskip 0.2cm
{\it Proof.} Let $\{ \xi_i \}$ be the sequence of random variables defined by
$\xi_i = \ut(\pi (\tau^i\s))$ where $\pi(\s)=x$ and $\tau$ is the shift on
$\S$ (see (6)). Set
$$
\Xt_n(\s ,t) = {1\over \sqrt{\Dt \,n}} \St(\s,nt),\qquad 0\leq t \leq 1
$$
with the identification $\St(\s,nt) = \St(\pi(\s),nt)$ and
$$
\Dt = \rho\{\xi_0^2\} + 2\sum_{j=0}^{\infty}\,\rho\{\xi_0\xi_j\}
$$
where $\rho (\xi_i)=0$ by assumption. The exponential
uniform mixing property for the random variables $\{ \s_i \}$,
proved in [CI1], and
the smoothness hypothesis on $\ut$ entail that the above series
converges absolutely. Moreover the assumption that $\ut$ is not a cocycle
implies that $\Dt >0$ (see, e.g., [Bo]).
Now the result follows from the functional central limit
theorem (Donsker's theorem) for dependent variables proved,
e.g., in [Bi] page 174. Q.E.D.
\vskip 0.2cm
This result enables us to prove limit results for various functions of the
partial sums $\St_n$. In particular, we have the following result:
\vskip 0.2cm
{\bf Lemma 1.} {\it Set $b_n = \left[ \kappa \, a_n\right ]$
where again $a_n=n/\log n$. Then, there are two positve constants $C,c$ such that
for any $\epsilon >0$ and $n$ large enough }
$$
\rho \, \{\, x\in \ui \, : \, \max_{|m-b_n|\leq \epsilon \, a_n}\,
\left| \, \St_m(x) - \St_{b_n}(x) \, \right|\, < \,{\epsilon}^{1/4}
\sqrt{a_n} \, \}\, \geq \, 1 - C\, \exp{(-c\, {\epsilon}^{-1/2})}
$$
{\it Proof.} The proof is a trivial adaptation to the present situation
of the argument given in [Bi], Section 10. Q.E.D.
\vskip 0.2cm
{\bf Lemma 2.} {\it There is a constant $\kappa>0$ such that for any $\epsilon >0$,
$$
\lim_{n\to \infty}\rho \biggl(\left\{ x:
1-\epsilon < {N(n,x)\over \kappa\, a_n}< 1+ \epsilon
\right\} \biggr) = 1
$$
where $a_n = n /\log n$.}
\vskip 0.2cm
{\it Proof.} See [CI2], Lemma 3.3. Q.E.D.
\vskip 0.2cm
{\bf Lemma 3.}
$$
{1\over \sqrt{a_n}}\left(\, S_n(x) - \St_{b_n}(x)\, \right) \rightarrow 0\quad
\hbox{in $\rho$-probability}
$$
{\it and therefore, for any $0\leq t \leq 1$,}
$$
{1\over \sqrt{a_n}}\left(\, S(x,nt) - \St(x,b_nt)\, \right) \rightarrow 0\quad
\hbox{in $\rho$-probability}
$$
{\it Proof.}
Let us write (7) in the form
$$
S_n(x) = \St_{N(n,x)}(x) + R_n(x) \eqno(12)
$$
with the obvious identifications. Recall that under our assumptions
the remainder $R_n(x)$ is uniformly bounded in $n$ and $x$.
Now, from Lemma 2 we have
that for any $\epsilon >0$
and for $n$ sufficently large
$$
\rho \, \{\, x\in \ui \, : \,
\left| \, N(n,x) - b_n \, \right|\, < \,{\epsilon}\, a_n \, \}\, \geq \,
1 - {\epsilon}\eqno(13)
$$
and the statement follows by putting together (12), (13) and Lemma 1. Q.E.D.
\vskip 0.2cm
An easy consequence of Theorem 1' and Lemma 3 is the following
\vskip 0.2cm
{\bf Lemma 4.} {\it Let $D=\kappa \, \Dt$ and
$$
X_n(x,t) = {1\over \sqrt{D\, a_n}} S(x,nt),\qquad 0\leq t \leq 1
$$
Then, for any finite sequence
$0\leq t_1 \leq \dots \leq t_{\ell} \leq 1$ the random vector
$$
\left( X_n(x,t_1), \ldots , X_n(x,t_{\ell})\right)
$$
converge in law to $\left( B(t_1), \ldots , B(t_{\ell})\right)$
as $n\to \infty$.}
\vskip 0.2cm
{\bf Remark 3.}
We have proved so far that the finite dimensional
distributions of the random element $X_n(x,t)$ converge to those
of the Brownian motion $B(t)$. In particular, this implies the validity
of the central limit theorem for the random variables
${1\over \sqrt{D\, a_n}} S_n$, that is
$$
\lim_{n\to \infty} \rho \, \{ \,
{1\over \sqrt{D\, a_n}} S_n \, \leq \, \alpha \, \} =
{1\over \sqrt{2\pi }} \int_{-\infty}^{\alpha} e^{-y^2/2}dy
$$
To complete the proof of Theorem 1 it remains
to show that the sequence $\{X_n\}$ satisfies a tightness condition [Bi].
\vskip 0.2cm
{\bf Lemma 5.} {\it The sequence of random functions $\{X_n\}$ is tight.}
\vskip 0.2cm
{\it Proof.}
We shall exploit the tightness of the sequence $\{\Xt_n\}$.
Let $\epsilon >0$ and $\eta>0$ be fixed. Then, there exist a $\delta >0$ and an integer $n_0$
such that
$$
\rho \, \{x\,:\, \sup_{0\leq t\leq 1}\sup_{t\leq s\leq t+\delta}
|\Xt_n(x,t)-\Xt_n(x,s)| \,\geq \epsilon \} \leq \eta \eqno(14)
$$
for all $n > n_0$.
Let $M$ be a fixed positive integer such that $1/M < \delta$ and
let $t_1,t_2 \in \ui$ be such that $t_1< t_2$ and
$|t_1-t_2|<1/M$. Define $N(t,x)$ for non integer $t$ as $N(t,x)=N([t],x)$.
Using (12) and the boundedness of the remainders we then find
$$\eqalign{
|X_n(x,t_1)-X_n(x,t_2)| = &{1\over \sqrt{Da_n}}|S(x,nt_1)-S(x,nt_2)|\cr
\leq &{1\over \sqrt{Da_n}}\left( 2C + |\St(x,N(nt_1,x))-S(x,N(nt_2,x)|\right) \cr }
\eqno(15)
$$
for some constant $C>0$.
Let now $ 1\leq k_1,k_2 \in \Z_+$ be such that
$$
0 < {k_1\over M} -t_1 \leq {1\over M}, \qquad 0 < {k_2\over M} -t_2 \leq {1\over M}.
$$
Clearly $|k_1-k_2|\leq 1$ and
$$
N(nt_1,x) \geq N\left({n(k_1-1)\over M},x\right)\quad\hbox{and}\quad N(nt_2,x) \leq
N\left({nk_2\over M},x\right)
$$
so that
$$
N(nt_2,x) - N(nt_1,x) \leq N\left({nk_2\over M},x\right)-N\left({n(k_1-1)\over M},x\right)
$$
On the other hand, if $n$ is large enough, using Lemma 2 we can estimate the r.h.s. as
$$
\eqalign{&N\left({nk_2\over M},x\right)-N\left({n(k_1-1)\over M},x\right) \leq \cr
&(1+{\delta \over 4})\left({ \kappa {nk_2\over M}\over \log{nk_2\over M} }\right) -
(1-{\delta \over 4})
\left({ \kappa {n(k_1-1)\over M}\over \log{n(k_1-1)\over M} }\right) \leq
\delta \, \kappa {n/\log n} \cr }
$$
Finally, using the above inequality and the tightness of $\Xt_{b_n}$ we get
$$
|X_n(x,t_1)-X_n(x,t_2)|
\leq {2C\over \sqrt{Da_n}}+
C\, \sup_{0\leq t\leq 1}\sup_{t\leq s\leq t+\delta}
|\Xt_{b_n}(x,t)-\Xt_{b_n}(x,s)|
$$
which ends the proof. Q.E.D.
\vskip 0.2cm
{\sl Proof of Theorem 1.} The proof now follows putting together
Lemma 0, Lemma 4 and Lemma 5. Q.E.D.
\vfill \eject
{\bf References.}
\vskip 1cm
\item{ [ADU]} Aaronson J., Denker M and Urbanski M., {\sl Ergodic
Theory for Markov fibred systems and parabolic rational maps},
Tans. of Amer. Math. Soc. {\bf 337}, 495-548 (1993)
\vskip 0.2cm
\item{ [Bi]} Billingsley P., {\sl Convergence of Probability Measures},
Wiley, New York, 1968
\vskip 0.2cm
\item{ [Bo]} Bowen R., {\sl Equilibrium states and the ergodic theory
of Anosov diffeomorphisms},
Lect. Notes in Math. {\bf 470}, Springer-Verlag, 1975
\vskip 0.2cm
\item{ [CI1]} Campanino M. and Isola S.,
{\sl Statistical properties of long return times in type I
intermittency}, (1993) Forum Math., to appear.
\vskip 0.2cm
\item{ [CI2]} Campanino M. and Isola S.,
{\sl Infinite invariant measures for non-uniformly expanding transformations of
$\ui$: weak law of large numbers with anomalous scaling}, (1994) Forum Math.,
to appear.
\vskip 0.2cm
\item{ [C.F]} Collet P. and Ferrero P., {\sl Some limit ratio
theorem related to a real endomorphism with a neutral fixed point},
Ann. Inst. H. Poincar\'e {\bf 52}, 283 (1990)
\vskip 0.2cm
\item{ [HK]} Hofbauer F. and and Keller G., {\sl Ergodic properties of invariant
measures for piecewise monotonic transformations}, Math. Z. {\bf 180}, 119-140
(1982)
\vskip 0.2cm
\item{ [LSV]} Lambert A., Siboni S. and Vaienti S., {\sl Statistical properties of
a non-uniformly hyperbolic map of the interval}, Preprint CPT-92/P.2804
\vskip 0.2cm
\item{ [T1]} Thaler M., {\sl Estimates of the invariant densities of
endomorphisms with indifferent fixed points}, Isr. Jour. of Math. {\bf 37},
303-313 (1980)
\vskip 0.2cm
\item{ [T2]} Thaler M., {\sl Transformations in $\ui$ with infinite
invariant measures}, Isr. Jour. of Math. {\bf 46}, 67-96 (1983)
\vskip 0.2cm
\item{ [R]} Ruelle D., {\sl Thermodynamic Formalism},
Addison-Wesley Publ. Co., 1978
\vskip 0.2cm
\bye
\end