1/\la$. Since no point outside of a small neighborhood of the fixed point can hit into this neighborhood, it is enough (as in our previous examples) to study the escape rate from this neighborhood. If $\la=2$ then locally the behaviour of the randomly perturbed system is completely described by the following random walk model on $\IZ$: $$ x \to \function{ 2x + \xi &\mbox{if } x \ge 0 \\ -2x + \xi &\mbox{otherwise} ,} $$ $$ \xi = \function{-1 &\mbox{with probability } q \\ 0 &\mbox{with probability } 1-2q \\ 1 &\mbox{with probability } q .} $$ Clearly, if $|x|>1$ then the trajectory of this point will never return to zero. Consider the part of the transition matrix corresponding to the points $-1$, $0$ and $1$. It can be written as $$ \lrp{ \begin{array}{ccc} q &0 &0 \\ q &1-2q &q \\ q &0 &0 \end{array}} $$ Thus this matrix has an eigenvalue $1-2q$, which proves our statement. \subsection{Instability for a generic multidimensional hyperbolic map} Stability of spectral properties becomes a much more delicate problem in the multidimensional case. Traditionally, to define this spectrum one considers the Perron-Frobenius operator for the expanding map defined on unstable manifolds induced by the original map (see, for example, \cite{Yo,Fr2,BTV}). Another way to calculate the isolated eigenvalues is to study the so called weighted dynamical $\zeta$-function of a map, which counts periodic points of the map weighted by Jacobians in the unstable direction. Zeros and poles of the $\zeta$-function correspond to isolated eigenvalues of the map (see, for example, \cite{Ba} and references therein). Our spectral stability results can be generalized for finite systems of weakly coupled 1D PE maps (see \cite{Bl20} for definitions). On the other hand, even for a more general multidimensional PE map our construction does not work, since there is no good control over coefficients of a Lasota-Yorke type inequality in this case, contrary to the 1D case. \bigskip {\bf Example} 2. Consider a smooth hyperbolic map $f: \IR^2 \to \IR^2$. Let $0$ be a hyperbolic fixed point of the map, and let the horizontal direction be locally unstable with the expanding constant $\la_u>1$, while the vertical direction be contracting with $\la_s \ll 1$. We consider Ulam partitions into equal squares rotated by the angle $\pi/2$ with respect to the coordinate axes. One element of the partition together with its preimage is shown in Figure~\ref{rot-Ulam}. Straightforward calculations show that the probability to remain in the considered Ulam square is of order $$ p {\buildrel {(\la_s\ll1)} \over \approx} 1 - \lrp{1 - \frac1{\la_u}}^2 = \frac1{\la_u} \lrp{2 - \frac1{\la_u}} = \frac34 \big|_{\la_u=2} .$$ One can calculate this probability exactly also for the case of finite values of $\la_s$, for example $p=2/3$ for $\la_s=1/2$. This is not sufficient to prove that $\tilde r_2 > r_2$, but it indicates that the limit behaviour of the approximation might differ from that of the original map. A slightly less striking example of this type was discussed in \cite{Ki4}, where it was claimed that the worst situation is when the angle between the contracting and expanding directions is small. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Picture for the 2D rotated Ulam's partition. \Bfig(150,150) {\bline(0,0)(1,0)(150) \bline(0,0)(0,1)(150) \bline(0,150)(1,0)(150) \bline(150,0)(0,1)(150) \bline(0,75)(1,0)(150) \bline(75,0)(0,1)(150) \put(135,75){\vector(1,0){6}} \put(15,75){\vector(-1,0){6}} \put(75,20){\vector(0,1){6}} \put(75,130){\vector(0,-1){6}} \thicklines \bline(40,75)(1,1)(35) \bline(40,75)(1,-1)(35) \bline(110,75)(-1,1)(35) \bline(110,75)(-1,-1)(35) \bline(50,75)(1,3)(25) \bline(50,75)(1,-3)(25) \bline(100,75)(-1,3)(25) \bline(100,75)(-1,-3)(25) \put(130,79){$\la_u$}} {One element of the rotated Ulam partition and its preimage in the 2D hyperbolic case. \label{rot-Ulam}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip The following numerical example shows that the untypical instability of the essential spectrum due to the presence of periodic turning points in the one-dimensional case becomes typical for multidimensional maps. Near a periodic point of a multidimensional hyperbolic map stable and unstable foliations are coming arbitrarily close one to another. Therefore an arbitrarily small (random) perturbation can mix them (similarly to the situation near periodic turning points (see \cite{BK95})). We study an example as simple as possible to demonstrate that this type of behavior is generic. \?{Notice that in examples Ulam's conjecture about the convergence of invariant measures of the approximating finite Markov chains to the SBR measure of the original map holds.} \bigskip {\bf Example} 3. Consider the well known ``cat'' map, which is the simplest example of a smooth two dimensional hyperbolic map. This is a map from the unit torus $X=[0,1] \times [0,1]$ into itself defined by $(x,y) \mapsto (x+y \mod1, x+2y \mod1)$. We consider two partitions of $X$ into equal squares. First we simply divide horizontal and vertical axes into $n$ equal intevals, whose products give a partition into $N=n^2$ squares, which we call the standard partition. There is a one to one correspondence of these squares and pairs of integers $(i=nx, j=ny)$, where $(x,y)$ is the pair of coordinates of the lower left corner of a square. Here $i,j \in \{0,1,\dots,n-1\}$. Simple calculation gives the following transition probabilities for the corresponding Markov chain whose elements are numbered as $jn+i+1$ (see also Figure~\ref{Im-Stand-part}): \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline {} & $i+j,i+2j$& $i+j,i+2j+1$& $i+j+1,i+2j+1$& $i+j+1,i+2j+2$ \\ \hline $i,j$ & $1/4$ & $1/4$ & $1/4$ & $1/4$ \\ \hline \end{tabular} \end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Picture for the ``rectangular'' partition. \Bfig(160,110) {\rect(0,0)(160,110) \rect(10,40)(30,30) \rect(90,10)(30,90) \rect(120,10)(30,90) \bline(90,40)(1,0)(60) \bline(90,70)(1,0)(60) \thicklines \put(50,55){\vector(1,0){20}} \bline(90,10)(1,1)(30) \bline(120,40)(1,2)(30) \bline(150,100)(-1,-1)(30) \bline(90,10)(1,2)(30) }{Image of an element of the ``standard'' partition by the ``cat'' map. \label{Im-Stand-part}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Moduli of the ``second'' eigenvalues ($r_2$) of the transition matrices ($n^2 \times n^2$), and their multiplicities (in parenthesis) are shown in the following table: \begin{center} \begin{tabular}{|r|l||r|l||r|l||r|l||r|l|} \hline $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$ \\ \hline 2 &0.0000(3 )& 3 &0.3536(8 )& 4 &0.0000(15)& 5 &0.3299(20)& 6 &0.3536(8 )\\ 7 &0.4886(16)& 8 &0.4454(24)& 9 &0.3847(24)& 10&0.4275(12)& 11&0.5161(20)\\ 12 &0.3783(48)& 13&0.4835(28)& 14&0.4886(16)& 15&0.4045(16)& 16&0.4454(24)\\ 17 &0.4335(24)& 18&0.5957(24)& 19&0.5387(36)& 20&0.4275(12)& 21&0.5357(32)\\ \hline \end{tabular} \end{center} \?{\begin{center} \begin{tabular}{|r|l||r|l||r|l||r|l||r|l|} \hline $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$ \\ \hline 3 &0.3536(8 )& 4 &0.0017(6 )& 5 &0.3299(20)& 6 &0.3536(8 )& 7 &0.4886(16)\\ 9 &0.3847(24)& 10&0.4275(12)& 11&0.5161(20)& 12&0.3783(48)& 13&0.4835(28)\\ 14&0.4886(16)& 15&0.4045(16)& 17&0.4335(24)& 18&0.5957(24)& 19&0.5387(36)\\ \hline \end{tabular} \end{center}} Compare this with the inverse to the largest eigenvalue of our linear map $1/\Lambda=2/(3+\sqrt5)\approx0.38204$, which (see, for example, \cite{BTV}) is the correct value of the ``second'' eigenvalue in this case. To show that even this is not the worst case we consider also another partition, namely the standard partition shifted by $1/(2n)$ (in both directions). Similarly to the previous case, we associate the square centered at $(x,y)$ with the pair of integers $(i=nx, j=ny)$. Here $i,j \in \{0,1,\dots,n-1\}$. The transition probabilities for the corresponding Markov chain (whose elements are numbered as $jn+i+1$) are shown in the following table: \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline {}&$i+j,i+2j$&$i+j,i+2j+1$&$i+j+1,i+2j+1$&$i+j,i+2j-1$&$i+j-1,i+2j-1$\\ \hline $i,j$& $1/2$ & $1/8$ & $1/8$ & $1/8$ & $1/8$ \\ \hline \end{tabular} \end{center} Moduli of the ``second'' eigenvalues ($r_2$) of the transition matrices ($n^2 \times n^2$), and their multiplicities (in parenthesis) are \begin{center} \begin{tabular}{|r|l||r|l||r|l||r|l||r|l|} \hline $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$ \\ \hline 2 &0.3968(3 )& 3 &0.3953(8) & 4 &0.4543(12)& 5 &0.4029(20)& 6 &0.4443(24)\\ 7 &0.5577(16)& 8 &0.4940(24)& 9 &0.5038(24)& 10&0.4754(12)& 11&0.6203(20)\\ 12&0.4567(48)& 13&0.5371(28)& 14&0.5577(16)& 15&0.5495(16)& 16&0.4940(24)\\ 17&0.5864(36)& 18&0.6733(24)& 19&0.5976(36)& 20&0.4902(24)& 21&0.5838(32)\\ \hline \end{tabular} \end{center} \?{\begin{center} \begin{tabular}{|r|l||r|l||r|l||r|l||r|l|} \hline $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$& $n$& $r_2$ \\ \hline 3 &0.3953(8) & 4 &0.4543(12)& 5 &0.4029(20)& 6 &0.4443(24)& 7 &0.5577(16)\\ 8 &0.4940(24)& 9 &0.5038(24)& 10&0.4754(12)& 12&0.4567(48)& 13&0.5371(28)\\ 14&0.5577(16)& 15&0.5495(16)& 16&0.4940(24)& 17&0.5864(36)& 18&0.6733(24)\\ \hline \end{tabular} \end{center}} In this case all ``second'' eigenvalues are greater in modulus than $1/\Lambda$. Looking at these two tables the structure of limit points of the eigenvalues seems not quite clear for both of the considered families of the partitions, and one might argue that for large enough $n$ the corresponding $r_2$ may converge to $1/\Lambda$. However, we have numerical evidence that for the standard partition for $n=7k$ there is an eigenvalue $0.4886$, while for the shifted standard partition for $n=8k$ there is an eigenvalue $0.4940$. The following general statement justifies this prediction and provides us with the precise description of the structure of the spectrum for the case of a linear automorphism preserving integer points. \begin{theorem} Let $\tilde f: \IR^d \to \IR^d$ be a linear map such that $\tilde f(\IZ^d)=\IZ^d$, and let the map $f:=\tilde f \mod1$ be defined on the $d$-dimensional unit torus. Denote by $P_n$ the matrix corresponding to Ulam's approximation of the map $f$ constructed according to the partition of the unit torus into $n^d$ equal cubes. Then $r \in \Sigma(P_{kn})$ for any positive integer $k$ whenever $r \in \Sigma(P_{n})$. \end{theorem} The proof of this result is based on the fact that due to the selfsimilar structure of Ulam's approximation in this case both the matrix $P_{kn}$ and the eigenvector $e_{kn}$ corresponding to the eigenvalue $r$ consist of repeated blocks of the matrix $P_{n}$ and the eigenvector $e_{n}$ respectively. Therefore if for some $n$ we obtain numerically a ``bad'' value for the ``second'' eigenvalue, it will still be present for large enough multiples of $n$. In recent papers \cite{Fr2,BTV} it was proposed to use Ulam's procedure based on a finite Markov partition to estimate $r_2$. This claim was justified in these papers for 1D smooth expanding maps and 2D Anosov automorphisms. In practice, the usefulness of this approach is limited by the observation that usually such partitions can be found only numerically, and as we shall show a small error here may lead to even worse accuracy of the eigenvalues compared to a non Markov partition. Now we are in a position to answer the question why the spectral gap in the above numerics differs significantly from theoretical predictions. To have a simple model for the analytical study to start with, consider a family of 2D maps from the unit square into itself, defined as follows: $f_\gamma(x,y):=(2x\mod1, \gamma (y-c)+c\mod1)$, $\gamma\ge0, \, 01$ one can prove that the PF-spectrum of this map is just the set of all pairwise products of elements of the spectra of the involved 1D maps. For $\gamma\le1$, however, only the first map contributes to the spectrum. Observe that for $\gamma<1$ the invariant measure is concentrated on the attracting fiber $\Gamma:=\{(x,c): \; 0\le x \le 1\}$, and the spectrum is the PF-spectrum of the piecewise expanding map on $\Gamma$. The dependence of $r_2$ on the parameter $\gamma$ is shown by thick lines in Figure~\ref{r2-direct}. Observe the discontinuous behaviour when the parameter $\gamma$ crosses the value $1$. By dots we indicate the rate of correlations decay with respect to the Lebesgue measure in this system. Observe that these two graphs differ only for $0<\gamma<1$, i.e. when the stable foliation is present. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% $r_2(\gamma)$ for the direct product map. \Bfig(160,110) {\bline(0,0)(1,0)(160) \bline(0,0)(0,1)(110) \put(150,0){\vector(1,0){10}} \put(0,100){\vector(0,1){10}} \bline(50,0)(0,1)(5) \bline(100,0)(0,1)(5) \bline(0,100)(1,0)(5) %\bline(0,50)(1,1)(50) \bezier{20}(0,50)(25,75)(50,100) \thicklines \bline(0,50)(1,0)(50) \bline(50,100)(1,-1)(50) \bline(100,50)(1,0)(50) \put(2,-8){$0$} \put(48,-8){$1$} \put(98,-8){$2$} \put(150,-8){$\gamma$} \put(-8,2){$0$} \put(-8,48){$\frac12$} \put(-8,98){$1$} \put(3,106){$r_2$} }{$r_2(\gamma)$ for the direct product map. \label{r2-direct}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This simple example demonstrates the main difference between expanding and hyperbolic maps, because the ``traditional'' spectrum in the latter case does not take into account the behaviour of the system along the stable foliation. \subsection{Random perturbations of contractive maps.} The above examples show that in order to understand how small random perturbations change the behaviour of a hyperbolic system one has to study their influence on a pure contractive map. Let $f_{\gamma,c}(x):=\gamma(x-c)+c$, $0<\gamma,c<1$, be a family of maps from the unit interval $[0,1]$ into itself. If our random perturbation has the transition probability density $q(\cdot,\cdot)$, being a \BV \ function of the first variable, then the corresponding transition operator is well defined as an operator from \BV \ into itself and one can compute its spectrum. Denote by $[x]$ the closest integer to the point $x$, and let $\delta=|[cn]-cn| \in [0,1/2]$ be the distance from the fixed point $c$ to the closest end-point of the Ulam interval to which it belongs, multiplied by $n$. The transition matrix $P_n$ is lower triangular in this case. Therefore the diagonal entries of the matrix are just its eigenvalues. A simple calculation gives the following representation for $\tilde r_2$ as a function of $\delta$: $$ \tilde r_2(\delta) = \function{ 2-\frac1\gamma &\mbox{if } \delta=0 \mbox{ and } \gamma>\frac12 \\ 1-\delta(\frac1\gamma-1) &\mbox{if } 0<\delta\le\frac{\gamma}{1-\gamma}\\ 0 &\mbox{otherwise} .} $$ This result shows the following. First, $\tilde r_2$ sensitively depends on the distance to the closest end-point of the Ulam interval it belongs to: $\tilde r_2(0)=2-\frac 1\gamma$, $\tilde r_2(0_+)=1$, while $\tilde r_2(1/2)=(3-1/\gamma)/2$ (provided $\gamma>1/2$). Second, when the fixed point lies very close to the boundary of one of the Ulam intervals, the estimate is the worst, which yelds a very bad accuracy if one uses an approximation to a Markov partition for Ulam's procedure. Let us show that shift-invariant random perturbations may cure this pathology. Suppose the random perturbation has a shift-invariant transition probability density $q(x,y)=q(y-x)$, $q \in C^2$. We want to advocate that the PF-spectrum of the perturbed operator in zero noise limit in this case is well defined, does not depend on the shape of $q(\cdot)$ and is nontrivial. Let us prove this for $\tilde r_2$. \begin{lemma} Let $q \in C^2$, $q(x)=0$ if $|x|>\ep$, and $\ep\le\max\left\{\gamma c, \gamma(1-c)\right\}$. Then $\tilde r_2=\gamma$. \end{lemma} \proof The random map can be rewritten as $x_{n+1}=\gamma (x_n-c) + c+ \xi_n$, where $(\xi_n)$ is a sequence of iid random variables with probability density $q(\cdot)$. Therefore $$ x_{n+1} = \gamma^n(x_1-c) + c + (\gamma^{n-1}\xi_1 + \gamma^{n-2}\xi_2 + \dots + \xi_{n}) .$$ Let $\xi^{(n)}:=c+\sum_{k=0}^{n-1}\gamma^k\xi_{n-k}$. Since the $\xi_k$ are iid, the sequence $(\xi^{(n)})$ converges in distribution to a random variable $\xi^{(\infty)}$. Then the random variables $x_n$ converge in distribution to $\xi^{(\infty)}$ as $n\to\infty$ and $\tilde r_2$ corresponds to the rate of this convergence in the following sense: $\xi^{(\infty)}$ can be rewritten as $\xi^{(\infty)} {\buildrel \rm d \over =} \gamma^n (\tilde\xi^{(\infty)} -c) + \xi^{(n)}$, where ${\buildrel \rm d \over =}$ means equality in distribution and $\tilde\xi^{(\infty)}$ is a copy of $\xi^{(\infty)}$ which is independent of the $\xi^{(n)}$. Denote by $q_n, q_\infty$, $h_n$, $\phi_n$ and $\psi_n$ the densities of the random variables $\xi^{(n)}, \xi^{(\infty)}$, $x_n$, $\gamma^n(x_1-c)$ and $\gamma^n(\xi^{(\infty)}-c)$ respectively. Then $h_{n+1}=q_n\star\phi_n$ and $q_\infty=q_n\star\psi_n$, and the supports of $\phi_n$ and $\psi_n$ are of order $\gamma^n$. Observing that for any $h \in C^1$ $$ \int|h(x+\delta) - h(x)|\, dx = (|\delta| + o(\delta)) \var(h) , $$ we conclude that for large $n$ $$ \nl{q_n - q_\infty} = \int|q_n(x) - q_n(x-y)|\psi_n(y)\,dy\,dx \le O(\gamma^n)\var(q_n), $$ $$ \var(q_n - q_\infty) = \int|q_n'(x) - q_n'(x-y)\psi_n(y)|\,dy\,dx \le O(\gamma^n)\var(q_n'), $$ and similarly $$ \nl{q_n - h_{n+1}} \le O(\gamma^n)\var(q_n), \qquad \var(q_n - h_{n+1}) \le O(\gamma^n)\var(q_n). $$ As $\xi^{(n)}=\xi_n+\gamma\xi^{(n-1)}$, there is some probability density $h$ such that $q_n=q\star h$. Therefore $\var(q_n)\le\var(q)$ and $\var(q_n')\le\var(q')$ so that $$ \nv{h_n - q_\infty} \le O(\gamma^n) \cdot (\var(q) + \var(q')) . $$ It is easily seen that one can choose the initial density $h_1$ such that this order of convergence is attained (e.g.\ take $h_1$ close to a $\delta$-function). This yields the claim of the lemma. \qed \bigskip \n{\em Acknowledgements}. M.B. gratefully acknowledges support by the Volkswagen-Stiftung and by INTAS-RFBR 95-0723 and RFFI grants. G.K. was partially supported by the DFG under grant Ke 514/3-2. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %\newpage \begin{thebibliography}{99} \label{bibl} \bibitem{Ba} V. Baladi, {\em Periodic orbits and dynamical spectra}, Preprint, 1997. %{\em Dynamical zeta functions. Real and %complex dynamical systems}, NATO Adv. Sci. Inst. Ser. C Math. Phys. %Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 1--26. \bibitem{BY} V. Baladi, L.-S. Young, {\em On the spectra of randomly perturbed expanding maps}, Comm. Math. Phys. {\bf 156}:2 (1993), 355--385; {\bf 166}:1 (1994), 219--220. \bibitem{Bl5} M.L. Blank, {\em Small perturbations of chaotic dynamical systems}, Uspekhi Matem. Nauk. {\bf 44}:6 (1989), 3--28. (English transl. Russ. Math. Surveys {\bf 44}:6 (1989), 1--33.) \bibitem{Bl17} M.L. Blank, {\em Chaotic maps and stochastic Markov chains}, Abstracts of Congress IAMP--91, 1992, 6p. \bibitem{Bl20} M.L. Blank, {\em Discreteness and continuity in problems of chaotic dynamics}, Monograph, Amer. Math. Soc., 1997. \bibitem{BK95} M.L. Blank, G. Keller, {\em Stochastic stability versus localization in chaotic dynamical systems}, Nonlinearity {\bf 10}:1 (1997), 81-107. \bibitem{BTV} F. Brini, G. Turchetti, S. Vaienti, {\em Decay of correlations for the automorphism of the torus $T^2$}, Preprint Luminy, 1997. \bibitem{dellnitz} M. Dellnitz, O. Junge, {\em On the approximation of complicated dynamical behavior}, to appear in SIAM Journal on Numerical Analysis, 1998. \bibitem{Fr2} G. Froyland, {\em Computer-assisted bounds for the rate of decay of correlations}, Comm. Math. Phys. {\bf 189}:1 (1997), 237-257. %\bibitem{Hu} F. Hunt, {\em A Monte Carlo approach to the approximation of %invariant measures}, Random \& Computational Dynamics %{\bf 2}:1 (1994), 111--133. \bibitem{Ke1} G. Keller, {\em Stochastic stability in some chaotic dynamical systems}, Mh. Math. {\bf 94} (1982), 313--333. \bibitem{Ke2} G. Keller, {\em On the rate of convergence to equilibrium in one-dimensional systems}, Comm. Math. Phys. {\bf 96}:2 (1984), 181--193. \bibitem{Ki3} Yu. Kifer, {\em Random perturbations of dynamical systems}, Boston: Birkhauser, 1988. \bibitem{Ki4} Yu. Kifer, {\em Computations in dynamical systems via random perturbations}, Discrete Contin. Dynam. Systems {\bf 3}:4 (1997), 457--476. %\bibitem{LY} A. Lasota, J.A. Yorke, {\em On the existence of invariant %measures for piecewise monotone transformations}, Trans. Amer. Math. Soc. %{\bf 186} (1973), 481--488. \bibitem{Li} T.Y. Li, {\em Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture}, J. Approx. Th. {\bf 17} (1976), 177--186. \bibitem{Ul} S. Ulam, {\em Problems in modern mathematics}, Interscience Publishers, New York, 1960. \bibitem{Yo} L.-S. Young, {\em Statistical properties of dynamical systems with some hyperbolicity}, Preprint UCLA, 1996 \end{thebibliography} \end{document}