\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. 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