ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaConditionally Invariant Measures forAnosov Maps with Small HolesN. ChernovR. MarkarianS. Troubetzkoy

Vienna, Preprint ESI 412 (1996) December 27, 1996Supported by Federal Ministry of Science and Research, AustriaAvailable via http://www.esi.ac.at

Conditionally invariant measures forAnosov maps with small holesN. Chernov01, R. Markarian02 and S. Troubetzkoy01December 30, 1996AbstractWe study Anosov di�eomorphisms on surfaces in which some small `holes' arecut. The points that are mapped into those holes disappear and never return. Weassume that the holes are arbitrary open domains with piecewise smooth boundary,and their sizes are small enough. The set of points whose trajectories stay awayfrom holes in the past is a Cantor-like union of unstable �bers. We establish theexistence and uniqueness of a conditionally invariant measure on this set, whoseconditional distributions on unstable �bers are smooth. This generalizes previousworks by Pianigiani, Yorke, and others.AMS classi�cation numbers: 58F12, 58F15, 58F11Keywords: conditionally invariant measures, Anosov di�eomorphisms, repellers, scatter-ing theory, chaotic dynamics.1 Introduction1.1. Let T : M ! M be a topologically transitive Anosov di�eomorphism of class C1+�of a compact Riemannian surface M . Let H � M be an open set with a �nite numberof connected components.We denote M = M nH. For any n � 0 we putMn = \ni=0T iM and M�n = \ni=0T�iM; (1.1)and also M+ = \n�1Mn; M� = \n�1M�n; = M+ \M� (1.2)01 Department of Mathematics University of Alabama in Birmingham, Birmingham, AL 35294, USA.02 Instituto de Matem�atica y Estad��stica \Prof. Ing. Rafael Laguardia" Facultad de Ingenier��a,Universidad de la Rep�ublica, C.C. 30, Montevideo, Uruguay.1

All these sets are closed, T�1M+ � M+, TM� � M� and T = T�1 = . The setM+ (resp., M�) consists of points whose trajectories stay away from H in the past (thefuture). The set consists of points whose trajectories never enter H.In this paper, we study the structure of the sets M� and and the dynamics of themap T on these sets. We think of the connected components of H as holes (one can alsothink of H as an absorbing region). The trajectories that fall into H will no longer beconsidered { they disappear. So we may call the set of nonwandering points in M .We denote by T the restriction of T on M , which means that for any set A �M andn � 1 we put T nA = T n(A \M�n) and T�nA = T�n(A \Mn).1.2. The concept of a chaotic dynamical system with holes in its phase space andrelated problems have been formulated by Pianigiani and Yorke in 1979 [11] by way ofthe following pictorial example.Imagine a Sinai billiard table (with dispersing boundary), so that the dynamics ofthe ball are strongly chaotic. Let one or more holes be cut in the table, so that theball can fall through. In particular, one can place those holes at the corners of thetable and make `pockets'. Let the initial position of the ball be chosen at random withsome smooth probability distribution (which may be the equilibrium distribution for theoriginal system, without holes). Denote by p(t) the probability that the ball stays on thetable for at least time t and, if it does, by �(t) its (normalized) distribution in the phasespace at time t. Natural questions are: does p(t) converge to zero at some exponentialrate, as t!1? is there a limit probability distribution �+ = limt!1 �(t); is that limitdistribution independent of the initial distribution �(0)?These questions still remain open. However, since the pioneering work [11], a sub-stantial progress has been made in the study of chaotic dynamical systems with holes.Expanding (noninvertible) maps S with holes have been studied in [11] and later byCollet, Mart��nez and Schmitt [7], where the analogues of the above questions have beenanswered positively. The limit probability distribution �+ was called a conditionallyinvariant measure [11]. The measure �+ was not invariant under S; it could not bebecause of the holes. Instead, its image under S was proportional to itself: �+(S�1A) =�+�+(A) for any Borel set A, with some constant �+ 2 (0; 1). The constant �+ iscalled the eigenvalue of �+ [4, 5]. Another constant, + = � ln�+, is known as anescape rate. The paper [7] also constructed a related S-invariant measure �+ on theset of nonwandering points for S and established an important escape rate formula, + = �+�h(�+), where �+ was the sum of positive Lyapunov exponents on and h(�+)the Kolmogorov-Sinai entropy of �+.In 1981-86 �Cencova [3, 4] studied a class of invertible chaotic transformations withholes, namely smooth Smale's horseshoes. She also answered the analogues of the abovequestions positively. She constructed the invariant measure �+ on the set of nonwanderingpoints by pulling the conditionally invariant measure �+ backward in time.Lopes and Markarian in [9] studied an open billiard system { a particle bouncingo� three convex scatterers placed su�ciently far apart on a plane. Here almost everytrajectory eventually escapes through the openings between the circles. In [9], a condi-2

tionally invariant measure and a related invariant measure were constructed, the latterwas shown to be a Gibbs measure, and the above escape rate formula was also proved.Open billiards and other open Hamiltonian systems have become very popular inphysics in the past ten years. They have been studied numerically and heuristically,see the survey [8] and the references therein. This study is known as chaotic scatteringtheory. Very few results of it, however, are proved mathematically.In a recent manuscript, Ruelle [13] studied Axiom A di�eomorphisms restricted to asmall neighborhood U of a basic set . The leak of mass from U plays the role of escapethrough holes. Ruelle constructed a related invariant Gibbs measure on , proved theescape rate formula, and de�ned the entropy production [13].Independently of Ruelle, two of us (N.Ch. and R.M.) in [5, 6] studied C1+� transitiveAnosov di�eomorphisms T : M ! M with what we called rectangular holes. For anarbitrary �nite Markov partition R1; : : : ; RI 0 of M we de�ned H to be the union of theinteriors of some rectangles, say, H = intRI+1 [ � � � [ intRI 0. Though somewhat special,such Anosov maps with rectangular holes generalize both horseshoes studied in [3, 4] andopen billiards of the paper [9].In [5], we assumed an additional `mixing condition': there is a k0 � 1 such that (inthe notations of 1.1) intRi\ T k0(Rj \M�k0) 6= ; for all i; j � I. We proved the existenceand uniqueness of a conditionally invariant measure and a related invariant measure,established the escape rate formula, and generalized other results of [7, 4, 9, 13]. In[6], we relaxed the mixing condition and extended the results of [5] to nonmixing andnonergodic cases.The subject of this paper is the study of Anosov maps with rather arbitrary openholes, not necessarily rectangles. Our key assumptions are that the holes are smallenough, and dim M = 2. While the latter is assumed only to simplify the arguments,the former is essential { for large holes the conditionally invariant measure is obviouslynot unique. Our main result is the existence and uniqueness of a conditionally invariantmeasure �+ on M+. We plan to study the invariant measure �+ on in a separate paper.1.3. Here we state necessary de�nitions and conventions.For any point x 2 M we denote by W ux and W sx the local unstable and stable �berscontaining x. We denote by Jux and J sx the Jacobians of the map T restricted to W ux andW sx , respectively, at the point x. We put�min = minx2MfJux ; 1=J sxg and �max = maxx2M fJux ; 1=J sxgLet �0 > 0 be the minimum angle between stable and unstable �bers in M . Recall thatfor any two points x; y 2 W s a holonomy map hx;y : W ux ! W uy is de�ned by slidingthe points of W ux along local stable �bers (symmetrically, hx;y : W sx ! W sy is de�ned forx; y 2 W u).A rectangle R � M is a su�ciently small set such that for any x; y 2 R we haveW ux \W sy 2 R. We consider only closed connected rectangles. Those are bounded by two3

stable and two unstable �bers (called stable and unstable sides of R). Segments of localunstable and stable �bers insideR that terminate, respectively, on the stable and unstablesides of R are called R-�bers. They are `full-size' local �bers in R stretching completelyacross it. Any subrectangle R0 � R whose stable (unstable) sides are on the stable(unstable) sides of R is called a u-subrectangle (s-subrectangle). Any u-subrectangle(s-subrectangle) in R is a union of unstable (resp. stable) R-�bers.Convention. We say that a measure � on M is smooth if its conditional measures onlocal unstable �bers W u � M are absolutely continuous with respect to the Riemannianlength, and their densities are H�older continuous with H�older exponent �, which is thesame as for the class of smoothness of the map T .Recall that every transitive Anosov di�eomorphism has a unique Sinai-Bowen-Ruelle(SBR) measure [14, 2, 12]. It is an invariant measure, whose conditional distributionson local unstable �bers are smooth in the above sense. Motivated by this, we will callthese conditional distributions u-SBR measures on unstable manifolds. Equivalently, forany local unstable �ber W u its u-SBR measure is a probability measure �u on W u whosedensity �(x) with respect to the Riemannian length satis�es the equation [1]�(x)�(y) = limn!1 JuT�1y � � �JuT�nyJuT�1x � � �JuT�nx (1.3)The u-SBR measures are T -invariant, i.e. the image of �u on W u under T is a u-SBRmeasure on the �ber TW u.We introduce bounds on distorsions as follows. For any r > 0 we denote by D1(r) � 1the supremum of all ratios �(x)=�(y) in (1.3) for all x; y 2 W u on all �bers W u of lengthr (length always means the Riemannian length). Next, D2(r) denotes the supremum ofall the Jacobians of holonomy maps hx;y for points x; y 2 W u;s at distance � r (measuredalong W u;s). We put D(r) = maxfD1(r);D2(r)g. One can think of D(r) as a generalupper bound on distorsions within the distance r in M . Obviously, D(r) ! 1 as r ! 0.For linear Anosov maps D(r) � 1.For any �nite Borel measure � on M we de�ne its norm by jj�jj = �(M). We denoteby T� the adjoint operator on the class of Borel measures on M de�ned by (T��)(A) =�(T�1(A\M1)) for anyA �M . Due to the holes, the operator T� does not preserve norm.We also denote by T+ the (nonlinear) operator on the space of probability measures onM de�ned by T+� = T��=jjT��jj, whenever jjT��jj 6= 0.De�nition. A measure � on M is said to be conditionally invariant under T if thereis a � > 0 such that T�� = ��. The factor � is the eigenvalue of �.Obviously, any conditionally invariant measure � is supported on M+, and we have�(T�1A \M+) = ��(A \M+).1.4. Here we make preliminary assumptions on the holes.The holes (the connected components of H) are domains that satisfy the followingregularity condition. There is a constant B0 > 0 such that for any local unstable �berW u4

and any local stable �ber W s that intersect only one hole H 0 � H the sets W u nH 0 andW s nH 0 consist of not more than B0 smooth components. In particular, if the holes areconvex and the curvature of their boundary is greater than that of unstable and stable�bers, then B0 = 2. Also, if the holes are rectangles bounded by stable and unstable�bers, then again B0 = 2.Let NH be the number of holes. We denote by d0 the minimum distance between theholes, if there is more than one hole, NH > 1. We also assume that d0 is smaller than aquarter of the length of the shortest closed geodesic on M . In the case NH = 1 this willbe the de�nition of d0.We �x D = D(2d0), which will be the only bound on distorsions that we use. Non-linearity of the map T will result in additional factors, all � D, in various estimates.For certain technical reasons, we will assume that �min > 64D2. This is not a restric-tive assumption, because it can be always ful�lled by taking higher iterates of T . (Theconstant D is determined by the stable and unstable �bers in M , so it is the same for alliterates of T ).We denote by h the maximal size of holes de�ned as follows. For any hole H 0 � Hits size is supx2H 0fdiamW ux \H 0; diamW sx \H 0gwhere the diameter is measured along the �bers W u;sx . We will need h to be small enoughcompared to d0, i.e. h < h0(T ; d0; B0). There will be four speci�c upper bounds on hassumed in Sections 2 and 3. They are always clearly stated.1.5. The structure of the paper is as follows. In Section 2 we study the evolutionof an arbitrary unstable �ber W u under T . We estimate the fractions of the images ofthe u-SBR measure on W u that are and are not `eaten' by the holes. In Section 3 weprove the existence of su�ciently long unstable �bers in M+ and study their properties.In Section 4 we de�ne a sequence of approximations of the holes H by unions of rectan-gles of increasingly �ne Markov partitions of M . Accordingly, we obtain a sequence ofconditionally invariant measures, �(k)+ , for those rectangular holes based on the results of[5, 6]. In Sections 5 we prove the existence of the conditionally invariant measures �+ bytaking the weak limit points of the sequence of measures �(k)+ . In Section 6 we establishthe uniqueness of �+.2 Evolution of u-SBR measuresIn this section we study the evolution of unstable �bers and u-SBR measures on themunder T . This requires the following assumption on h.Assumption H1. h < d0=9.Additional cuts. For certain technical reasons, in this section it will be convenient tolimit the length of the unstable �bers in M by 2d0. This will be done by subdividingall maximal unstable �bers W u � M (terminating on @H) whose length is > 2d0 into5

sub�bers of length between d0 and 2d0. This can be accomplished by making a �nitenumber of cuts in M along some local stable �bers. The choice of those stable �bers willnot be important, and they will not be actually removed from M . They just determinethe way we do the bookkeeping of unstable �bers. Obviously, any long unstable �ber inM will be cut into sub�bers that are still longer than d0, and we never cut the �bers oflength � 2d0.We now consider an unstable �berW u �M and a u-SBR measure �u on it. Its imagesunder the iterates of the map T are cut by holes and our additional cuts. Whenever asmooth component of T nW u is cut into subcomponents, the further images of thoseunder the iterates of T are treated separately. So, for every n � 1 the image T nW u =T n(W u \M�n) will consists of many smooth unstable �bers, which carry the image ofthe measure �u under T n� , which we denote by �un. We do not normalize it. Denotethe components of T nW u by W un;i. Let " > 0. For every component W un;i we denote byW un;i(") the "-neighborhood of its endpoints within W un;i.Theorem 2.1 Let W u be long enough, of length > ��1mind0. Then for every n � 0 andany " > 0 we have �un([iW un;i("))�un(T nW u) � C1" (2.1)where C1 = 48D=d0.Proof. We �x an " 2 (0; C�11 ) (the theorem is trivial for " > C�11 ). The componentsW un;i are divided into three groups { long, medium and short. We say that a componentW un;i is long if the length of its complete image TW un;i on the original manifold M is � d0.The other components W un;i are said to be short if their lengths are < (2D)�1", otherwisethey are said to be medium.We put sn;i = �un(W un;i("))�un(W un;i)and s0n;i = Pj �un(T�1W un+1;j("))�un(W un;i)where the sum is taken over j such that W un+1;j � TW un;i.If a component W un;i is long, then its image TW un;i has length d � d0 and can besubdivided by holes and our additional cuts into no more than d=d0 +2 subcomponents.Then we have s0n;i � 2D(d=d0 + 2)d�1" � C 01" (2.2)with C 01 = 6D=d0. 6

If a component W un;i is a medium one of length d0, then its image TW un;i can intersectjust one hole. In this case we have s0n;i � 4D�mind0 "and sn;i � minf1; 2(Dd0)�1"gSince we have a medium component, d0 > (2D)�1", and we haves0n;i � 8D2��1minsn;i (2.3)For any short component W un;i we have sn;i = 1 and s0n;i � 1.The proof of the theorem goes by induction on n. Its validity for the current value ofn means that Pi sn;i�un(W un;i)Pi �un(W un;i) � C1" (2.4)It is then su�cient to show that, under this assumption, we havePi s0n;i�un(W un;i)Pj �un+1(W un+1;j) � C1" (2.5)We split each of the two sums in (2.4) into three parts, corresponding to long, mediumand short components. Those parts we denote by xn; yn; zn for the top sum and pn; qn; rnfor the bottom one, respectively. The two sums in (2.5) are also split into three groupseach, corresponding to the images of long, medium and short components of T nW u.(Note the di�erence in the way we split these sums!) We denote by x0n; y0n; z0n the threeparts of the top sum in (2.5) and p0n; q0n; r0n those for the bottom sum, respectively. Notethat these quantities depend on ", but for brevity we suppress this dependence.It is clear from (2.2) and (2.3) thatx0n + y0n + z0n � C 01"pn + 8D2��1minyn + rnWe have, from (2.4), that yn � C1"(pn + qn + rn)Note that all the short components W un;i lie within [iW un;i("0), where "0 = (4D)�1". Wecertainly can assume that (2.4) is true for all " > 0, in particular for "0. This implies thatrn = rn(") � �un([iW un;i("0)) � C1"0�un([iW un;i)= C1(4D)�1"(pn + qn + rn)Combining the above bounds givesx0n + y0n + z0n � (C 01 + 8D2��1minC1)"pn+ 8D2��1minC1"(qn + rn) + C1(4D)�1"(pn + qn + rn)7

We now recall that �min > 64D2 and C1 = 8C 01. Then we getx0n + y0n + z0n � 12C1"(pn + qn + rn)In order to complete the proof of the theorem, it is enough to show thatp0n + q0n + r0n � 12(pn + qn + rn) (2.6)i.e. the holes cannot eat up more than 50% of the images of curves W un;i under T ,combined. It is clear that if a component W un;i is of length > 4D��1minh, then no morethan 25% of its image under T can be eaten up by holes. Applying the assumption (2.4)with " = 4D��1minh to the other components W un;i shows that their total �un-measure doesnot exceed 4C1D��1minh(pn + qn + rn). Assumption H1 implies that4C1D��1minh = 192D2(d0�min)�1h < 1=3Then (2.6) follows, and the proof of the theorem is completed.Remark. A time-symmetric version of Theorem 2.1 also holds for the backward itera-tions of stable �bers, T�nW s, n � 1. A precise statement of that is obvious. (Of course,additional cuts of stable �bers in M along a �nite number of local unstable �bers can bedone in a completely similar way.)Remark. We have actually proved more than Theorem 2.1 says. Recall that the proofwas done by induction on n: we assumed that the measure �un on T nW u satis�ed (2.1)and deduced the same inequality for �un+1 = T��un. The measure �un was supported on a�nite union of unstable �bers in Mn, on each of which it was proportional to the u-SBRmeasure, but we never used the fact that those �bers were images of one original �berW u under T n. Therefore, we have proved the following:Let � be a �nite measure supported on a �nite union of unstable �bers in M , suchthat its conditional distributions on those �bers are proportional to the u-SBR measures.Let the relative �-measure of the union of "-neigborhoods of the endpoints of those �bersbe � C1" for all " > 0. Then the same property holds for T��.The estimate on the amount of T nW u eaten up by holes under the action of T obtainedin the proof of the above theorem can be greatly improved.Theorem 2.2 Under the conditions of Theorem 2.1, for every n � 0 we have�un((T nW u) \M�1)�un(T nW u) � �h def= 1 �C2h (2.7)where C2 = C1D(�max=�min + 1) = 48D2(�max=�min + 1)=d0Consequently, �u(W u \M�n)�u(W u) � �nh (2.8)8

Proof. We prove (2.7), which is equivalent to�un((T nW u) \ T�1H)�un(T nW u) � C2h (2.9)Every component W un;i of T nW u has length � 2d0 due to our additional cuts. Its imageunder T has length � 2�maxd0. So, it may intersect no more than 2�max + 1 holes. Theintersection with every hole has length � h on TW un;i, so that the total length of thesubset W un;i \ T�1H on the curve W un;i is less than ��1min(2�max + 1)h. The �un measureof that subset does not exceed the �un measure of the (�max=�min + 1)h-neighborhood ofthe endpoints of W un;i times the nonlinearity correction factor D. Now the bound (2.9)follows from Theorem 2.1.>From now on, we remove the additional cuts of unstable �bers introduced in thissection. In the theorems just proved this will amount to gluing together the componentsof T nW u that have been arti�cially cut. Clearly, both theorems will hold after we gluetogether any components of T nW u.3 The structure of the sets M+, M� and On every unstable �ber W u, the set of points whose forward images never fall throughholes has �u-measure zero, because the original Anosov di�eomorphism T is ergodic.However, it follows from Theorem 2.1 that every �ber W u of length > d0=�min containspoints whose forward images never escape (belong in M�), and the set of those points isa Cantor-like set on W u. Furthermore, it follows from Theorem 2.1 that for every n � 1the set Mn contains unstable �bers of length � 2C�11 = (24D)�1d0. Indeed, the theoremimplies that for any " < C�1 the set T nW u n [iW un;i(") is not empty, so that there arecomponents W un;i � T nW u � Mn of length � 2". Moreover, since �min > 64D2, theimages of those components of T nW u will contain components of T n+1W u � Mn+1 oflength � d0. Since M+ = \nMn, the closed set M+ also contains such �bers. Since theforward and backward dynamics are symmetric, we immediately getProposition 3.1 The set M+ and M� are not empty. Moreover, the set M+ containssome unstable �bers of length � d0, and the set M� contains some stable �bers of length� d0.We say that an unstable �ber W u �M is eventually long if for some n � 1 its imageT nW u contains a component of length � d0. Otherwise the �ber W u is said to be forevershort.Lemma 3.2 Every �ber W u �M of length � d1, whered1 = h�min � 3is eventually long. 9

Proof. The curve TW u has length � �mind1. If it intersects more than one hole, ithas a component of length � d0 already. If it intersects one hole, the intersection consistsof curves whose union has diameter � h. Then TW u = (TW u) nH necessarily containsa component of length (�mind1 � h)=2 = 32d1. Thus, the maximal length of componentsof T nW u will grow with n until it necessarily exceeds d0, hence the lemma.Fibers that are forever short may exist, but their in uence on our results will benegligible in view of the following lemma.Lemma 3.3 Let W u be a �ber of length d > 0 that is forever short, and let �u be theu-SBR measure on W u. Then for every n � 1 we have�u(W u \M�n)�u(W u) � Bn0 d1D�nmindProof. For every i � 0 the set T iW u consists of short unstable �bers, whose lengths are< d1. Any such �ber can intersect just one hole. Therefore, the number of components inT iW u can only increase by a factor of B0 at every iteration. The set T nW u then consistsof � Bn0 curves of length < d1. Their preimages under T�n have lengths < d1=�nmin, sotheir total length within W u is less than Bn0 d1=�nmin. Hence the lemma.Assumption H2. h < (1 �B0=�min)=C2, so that �h > B0=�min.Under this assumption, Lemma 3.3 shows that the images of any forever short �berfall through holes at a higher rate than those of eventually long ones, cf. (2.8). Due tothis, only �bers that are eventually long will be essential in our study.We are now going to prove that all su�ciently long unstable and stable �bers con-tained in M+ and M�, respectively, form a sort of connected `net' in M . The followinglemma is a key argument.Lemma 3.4 On any unstable �ber W u � M of length � d2, whered2 = D�mind1sin�0 = D�minh(�min � 3) sin�0there is a closed uncountable Cantor-like subset W u� � W u such that for every x 2 W u�there is a stable �ber W sx � M� containing x whose endpoints are the distance � d0=3away from x.Proof. For any � 2 (0; d0=3] let Hs� be the union of all stable �bers of length �intersecting the set H. In other words, we `stretch' each hole by the distance � in thestable directions. We further enlarge these holes so that if x; y belong in one hole (oneconnected component of Hs� ) and lie on one local unstable �ber, then the segment ofthat �ber between x and y is also included in the hole. The union of these stretched and10

enlarged holes is denoted by Hs� . Thus, any local unstable �ber intersects any hole in Hs�in at most one interval. The length of that interval does not exceed Dh= sin �0.We set � = d0=3. We now consider iterations T nW u, n � 0, of the given �ber W u,but at the nth iteration we erase all the points of T nW u that fall into the larger holes Hs�rather than H. If, for a point x 2 W u, none of its forward images T nx, n � 0, is erased,then clearly x 2 W u�.On any �berW u of length d2 we erase at most one curve of length < Dh= sin �0. Thenthe remaining part of that �ber necessarily contains two disjoint subcurves of length13 d2 � Dhsin�0! = Dh(�min � 3) sin �0that do not intersect Hs� . Their images under T have lengths > d2, and we can applythe same argument to each of them. Continuing iterating and doubling such subcurvesof length > d2 gives, in the limit, an uncountable subset of points x 2 W u�. Lemma 3.4is proved.Assumption H3. h < (10D)�1d0 sin�0, so that d2 � d0=9.Then Lemma 3.4 says that any unstable �ber of length d2 is crossed by much longerstable �bers in M�. Due to the symmetry, the dual statement is true for any stable �berof length d2. Since the phase space M is compact and connected, we get the following.Theorem 3.5 Denote by M�(d2) the union of all �bers in M� (unstable or stable,respectively) of length > d2. Then the set M+(d2) [ M�(d2) is connected. The setM+(d2) \M�(d2) is a subset of , and it makes a (2d2)-net in the manifold M .So far we have obtained some local properties of the map T on M . The next theoremis the only one in this section requiring a global argument.Theorem 3.6 If h is small enough (i.e, it satis�es assumption H4 below), then thereis a k1 � 1 such that for every two �bers W u;W s � M of length d0 there is a smoothcomponent of T k1W u = T k1(W u \M�k1 ) intersecting W s. Moreover, the endpoints ofthat smooth component are at least a distance d0 away from W s.Proof. First, we enlarge our holes so that if x; y belong in one hole and lie on onelocal stable or unstable �ber, then the segment of that �ber between x and y lies in thehole, too. Thus, any local �ber intersect any hole in at most one interval.Let Pk(W u;W s) be the number of points of intersection between T kW u and W s. Iteasily follows from the general theory of Anosov di�eomorphisms that the sequencePk(d0) = infWu;W s Pk(W u;W s)(in�mumbeing taken over all �bers of length d0) grows exponentially in k. More precisely,Pk(d0) grows asymptotically as ekhtop, where htop > 0 is the topological entropy of T . Wethen put k1 = minfk : Pk(d0) > NHkg11

We now have to show that at least one point in T k1W u \W s is not covered by theset [k1�1i=1 T iH. The image of every hole under T i, 1 � i < k1, covers on the curve T k1W uone or more intervals. The theorem will follow if we show that(i) such an interval is unique for every hole and every i = 1; : : : ; k1 � 1;(ii) every interval has length < dt=3, in particular, it intersects the �ber W s at mostonce;Here dt denotes the minimum length of a stable (unstable) �ber for T in M whoseboth endpoints lie on any one unstable (stable) �ber of length d0. Note that d0 + dt isgreater than the length of the shortest closed geodesic on M , and then dt=3 > d0.Assumption H4. h < 13 dt ��k1max sin�0.This assumption implies (ii) immediately. To prove (i), assume that some hole in Hintersects the curve T iW u for some 1 � i � k1�1 in two intervals. Then there is a stable�ber W s1 of length � 2h= sin �0 whose both endpoints lie on T iW u. Taking its preimageT�iW s1 leads to a contradiction with Assumption H4. The theorem is proved.Remark. Of course, the above theorem has a time-symmetric counterpart: for anytwo �bersW u;W s �M of length d0 there is a smooth component of T�k1W s intersectingW u, and its endpoints are at least a distance d0 away from W u.4 Markov approximationTo further study the dynamics of T on M we will de�ne a sequence of approximations ofthe holes H by unions of `rectangular' holes, H(k), k � 1. To this end we take a sequenceR(k) of increasingly �ne Markov partitions of M , and de�ne H(k) to be the union theinteriors of all rectangles in R(k) that intersect H. Our previous results [5, 6] provide uswith conditionally invariant measures �(k)+ for the map T with holes H(k). As k ! 1,the `rectangular' holes H(k) will be arbitrarily close to the original holes H. This enablesus to construct the conditionally invariant measure �+ as a weak limit of �(k)+ as k !1.Let R = fR1; : : : ; RIg be an arbitrary Markov partition of M . We assume that therectangles Ri 2 R are of su�ciently small diameter so that symbolic dynamics are de�ned[2]. Also, we can assume that every rectangle Ri is closed and connected [10], i.e., everyRi is a curvilinear quadrilateral bounded by two unstable and two stable sides.The union of stable �bers bounding the rectangles R 2 R is invariant under T , whilethe union of unstable �bers is invariant under T�1. Therefore, all these �bers lie on theglobal stable and unstable �bers of a �nite number of periodic points, whose union iscalled the core of the given Markov partition.We call a generic �ber any stable or unstable �ber in M that is not a part of theglobal �ber of a core point of R. There are then only a countable number of nongenericR-�bers in the rectangles R 2 R. 12

For any k � 1 the partition R(k) = k_i=�k T iRis also Markov and consists of connected rectangles.We put R(k) = fR 2 R(k) : R \H = ;gandM (k) = [R2R(k)R, and H(k) = M nM (k). Clearly,H(1) � H(2) � � � � and \kH(k) = H,so that the sets H(k) approximate the holes H `from outside'. In this section we assumethat k is large enough, k � k0(R; d0; h), so that the set H(k) consists of small open holessatisfying the assumptions H1-H4. Therefore, all the results of the previous section applyto the map T restricted to M (k) for k � k0For every k � k0 the set M (k) is a �nite union of Markov rectangles. We put,analogously to (1.1) and (1.2)M (k)n = \ni=0T iM (k); M (k)�n = \ni=0T�iM (k)M (k)+ = \n�0M (k)n ; M (k)� = \n�0M (k)�n ; (k) = M (k)+ \M (k)�The set M (k)+ is a union of some unstable R-�bers in the rectangles R 2 R(k). Likewise,M (k)� is a union of some stable R-�bers in the rectangles R 2 R(k). The set (k) is a closedCantor-like set, which has a direct product structure inside every rectangle R 2 R(k).We denote by T (k) the map T restricted to M (k). A detailed study of Anosov mapswith `rectangular' holes, which are some rectangles of one Markov partition, was per-formed in [5, 6]. In particular, it was shown that the map T (k) always has a conditionallyinvariant measure �(k)+ with an eigenvalue �(k)+ . The conditional distributions of �(k)+on unstable �bers of M (k)+ are u-SBR measures. While the eigenvalue �(k)+ is uniquelydetermined by T (k), the measure �(k)+ may be not unique.The study in [5, 6] was technically simpler than the one we present here. Yet, theproperties of the map T (k) on M (k) greatly depend on the structure of allowed transitionsbetween the rectangles R 2 R(k).To describe the above structure, we invoke the language of symbolic dynamics. TheMarkov partition R(k) of M de�nes a symbolic representation of T in terms of a subshiftof �nite type, which is topologically mixing because so is T . This shift restricted toR(k) is also a subshift of �nite type. That one need not be topologically mixing oreven transitive. It generally consists of several ergodic components, on each of whichthe subshift either is topologically mixing or cyclically permutes several subcomponents,there may be one-way connections between ergodic components and some nonrecurrentrectangles as well, see [6]. The in uence of every ergodic component, E(k)j , on invariantand conditionally invariant measures on (k) is determined by its eigenvalue, �(k)j 2(0; 1) related to the escape rate (k)j = � ln�(k)j . In fact, the eigenvalue �(k)+ of T (k) isthe largest eigenvalue �(k)max = maxj �(k)j of its ergodic components. Furthermore, only13

the ergodic components with the largest eigenvalue �(k)max (i.e., with the smallest escaperate) determine the conditionally invariant measure �(k)+ [6]. We call them dominatingergodic components. If there is more than one dominating component in the system, theirin uences on �(k)+ are determined by the structure of one-way connections between them,see [6] for more detail.The next few lemmas show that if h is small enough, then there exists a uniquedominating ergodic component, on which the subshift is topologically mixing.Lemma 4.1 There is an ergodic component E(k)j with eigenvalue �(k)j � �h (the latterwas introduced by (2.7)).The lemma immediately follows from Theorem 2.2.Lemma 4.2 Let a rectangle R 2 R(k) belong to an ergodic component E(k)j with �(k)j ��h. Then its interior contains T (k)-eventually long unstable �bers and T (k)-eventuallylong stable �bers.Proof. If every unstable R-�ber W u � intR were forever short, then according toLemma 3.3 the eigenvalue �(k)j could not exceed B0=�min, which would contradict toAssumption H2.Now let every stable R-�ber W s � intR be forever short. As it was also shown inLemma 3.3, the preimage [T (k)]�nW s consists of not more than Bn0 short �bers. So,[T (k)]�nR\R consists of not more than Bn0 connected subrectangles. Hence, the topolog-ical entropy of the subshift on the ergodic component E(k)j is � lnB0. Then the invariantmeasure �(k)j on the component E(k)j has measure-theoretic entropy h(�(k)j ) � lnB0. Itspositive Lyapunov exponent �(k)j is de�nitely � ln�min. Now, recall [5, 6] that thesequantities satisfy the escape rate formula�(k)j = h(�(k)j )� ln�(k)jThis immediately leads to a contradiction with Assumption H2. The lemma is proved.Lemma 4.3 Let a rectangle R1 2 R(k) contain an eventually long unstable �ber W u �intR1 and a rectangle R2 2 R(k) contain an eventually long stable �ber W s � intR2.Then there is an m0(R1; R2) � 1 such that transitions from R1 to R2 are allowed in anynumber of steps m � m0(R1; R2).Proof. Once the image [T (k)]m1W u contains a component of length � d0, every furtherimage [T (k)]mW u, m � m1, will contain such components. The same is true for all[T (k)]�mW s, m � m2, if only [T (k)]�m2W s contains a long component. Then the lemmafollows from Theorem 3.6, and m0(R1; R2) = m1 +m2 + k1.Corollary 4.4 There is just one ergodic component, E(k)+ with eigenvalue > �h. On thiscomponent the subshift is topologically mixing.14

Let T (k)� be the adjoint operator on Borel measures on M (k), i.e. (T (k)� �)(A) =�([T (k)]�1(A \ M (k)1 )) for any Borel A � M . Let T (k)+ be the operator on probabilitymeasures on M (k) such that T (k)+ � = T (k)� �=jjT (k)� �jj, where the norm of a measure � onM (k) is set to be jj�jj = �(M (k)).The next theorem follows from the results of [5, 6].Theorem 4.5 For all su�ciently large k (k � k0(R; d0; h)) we have the following:(i) the map T (k) on M (k) has a conditionally invariant probability measure �(k)+ supportedon M (k)+ , i.e. T (k)� �(k)+ = �(k)+ �(k)+ for some constant �(k)+ 2 (0; 1);(ii) the measure �(k)+ conditioned on any unstable R-�ber W uR(x) �M (k)+ , R 2 R(k) is theu-SBR measure on that �ber;(iii) for any smooth1 measure � on M (k) the measures [T (k)+ ]n� weakly converge, as n!1, to �(k)+ ;(iv) for any smooth measure � on M (k) there is a constant c� > 0 such that the measuresc�[�(k)+ ]�n[T (k)� ]n� weakly converge to �(k)+ . If � is a probability measure, then c� is boundedaway from 0 and 1.5 Approximation of the conditionally invariant mea-sureHere we de�ne the conditionally invariant measure �+ for the map T as a weak limit ofthe measures �(k)+ .Since M (1) �M (2) � � � �, we have �(1)+ � �(2)+ � � � �, so that there is a limitlimk!1 �(k)+ = �+ 2 (�h; 1) (5.1)The measures �(k)+ are all supported on the compact set M+, hence the sequence ofthese measures has at least one weak limit point. Let �+ be any weak limit point of thissequence, it will be a probability measure on M+. We will �rst investigate the propertiesof any such �+. In the next section we will show that there cannot be two distinct limitpoints, so that the sequence �(k)+ weakly converges to �+. In this section, we denote by�+ an arbitrary weak limit point of the sequence f�(k)+ g.Proposition 5.1 The measure �+ is conditionally invariant under T with eigenvalue�+, i.e. T��+ = �+�+.This follows directly from Theorem 4.5 and (5.1).Proposition 5.2 The measure �+ conditioned on any generic unstable �ber W u �M+is a u-SBR measure on W u.1We remind that our convention on smoothness, cf. Introduction, is still valid.15

Proof. For any k � k0 we putW u;(k)+ = W u\M (k)+ . The diameter of the set T�nW u;(k)+converges exponentially to zero as n!1. Either this set lies in one rectangle R 2 R(k)for every su�ciently large n, or it intersects more than one rectangle for every n. In thelatter case, T�nW u;(k)+ converges, as n!1, to the periodic orbit of a core point of theMarkov partition R, so it is not generic. In the former case, the measure �(k)+ conditionedon T�nW u;(k)+ is the u-SBR measure on it. Since �(k)+ is conditionally invariant, it is alsoa u-SBR measure on W u;(k)+ . Taking the limit as k!1 proves the proposition.It also follows from the above proof that if a �nite number of generic unstable �bersin M+ lie on one global unstable �ber of the original Anosov di�eomorphism T , then themeasure �+ conditioned on their union is also a u-SBR measure on that union.Proposition 5.3 The �+-measure of the union of nongeneric �bers is zero.Proof. Since there are countably many nongeneric �bers, it is enough to show thatthe �+-measure of every one is zero. If a �ber W u � M+ is nongeneric, then accordingto the proof of Proposition 5.2 it can be divided into two subsegments on each of whichthe conditional �+-measure is u-SBR. The �ber W u belongs to a global unstable �ber,�u, of a core periodic point, which is invariant under some iterate of T , say, under T k.Now, if the set �u \M+ has positive �+-measure, the map T k stretches �u by a factor� �kmin decreasing �+ on �u \M+ by at least ��kmin. On the other hand, Proposition 5.1says that the measure �+ under the action of T k decreases by the factor of �k+. Since��1min < �+ due to Assumption H2, we get a contradiction proving Proposition 5.3.Combining the last two propositions shows that the measure �+ conditioned on anyunstable �ber is a u-SBR measure. Also, we can extend the last proposition showingthat any unstable �ber in M+ has zero �+-measure.Our next step is to extend Theorem 2.1 to the measures �(k)+ and �+. For any x 2M (k)+let W u;(k)x;+ be the largest segment of the unstable �ber in M (k)+ containing x. Likewise,W ux;+ is the largest segment of the unstable �ber in M+ containing x 2 M+. For any" > 0 we put U (k)" = fx 2M (k)+ : dist(x; @W u;(k)x;+ ) < "gwhere the distance is measured along the unstable �berW u;(k)x;+ . Removing the superscript(k) in the above formula will de�ne U".Lemma 5.4 For any k � k0 and " > 0 the set M (k)+ n U (k)" is compact (it may be emptyfor large "). The same holds for the set M+ n U".Proof. Let xl 2 M (k)+ n U (k)" be a sequence of points converging to a point x 2 M .Since the map T is smooth and the holes H(k) are open, it is easy to verify that thesegment of the unstable �ber through x of length 2" centered at x belongs in M (k)+ (inother words, its preimages under T�n, n � 0, never cross holes). This proves the lemma.16

Corollary 5.5 Suppose that a sequence of probability measures f�lg on M (k)+ converges,as l!1, to a probability measure �. Suppose that �l(U (k)" ) � p for some p � 0 and forall l � l0. Then �(U (k)" ) � p. The same holds for U".Theorem 5.6 For any k � k0 and " > 0 we have �(k)+ (U (k)" ) � C1", where C1 = 48D=d0.Proof. Let W u 2 M (k)+ be an unstable �ber whose endpoints are on two stable �bersbounding some rectangles in R(k). Let �u be a u-SBR probability measure supported onW u. It follows from [5] that the measure �un = [T (k)+ ]n�u weakly converges, as n!1, tothe measure �(k)+ .Proposition 3.1 applies to the map T (k) on M (k) for all k � k0. Based on it, we can�nd a �ber W u � M (k)+ longer than d0. Theorem 2.1 then implies that �un(U (k)" ) � C1".Using Corollary 5.5 now completes the proof of Theorem 5.6.Theorem 5.7 For any " > 0 we have �+(U") � C1".Proof. Obviously, U" \M (k)+ � U (k)" for any k. Therefore, �(k)+ (U") � C1". ApplyingCorollary 5.5 to U" and the subsequence of measures �(k)+ that converges to �+ provesthe theorem.Corollary 5.8 For any " > 0 let V" be the union of all maximal unstable �bers in M+of length < ". Then �+(V") = o(") as "! 0.Our last step in this section is to estimate measures of rectangles that are su�cientlylong in the stable direction. Denote by B the set of closed rectangles R � M such that(i) every stable R-�ber has length between d0 and 2d0;(ii) every unstable R-�ber has length � d0.We denote by �SBR the SBR measure on M . For any rectangle R let dumin(R) anddumax(R) be the minimal and maximal length of unstable R-�bers. General properties ofSBR measures imply thatC 0 dumax(R) � �SBR(R) � C 00 dumin(R) (5.2)for some constants C 0; C 00 depending only on T .Lemma 5.9 There is a constant C 03 > 0 such that for every R 2 B, every unstable �berW u � M of length d0 and every k � k0 we have[T (k)� ]k1�u(R) � C 03 dumin(R)Here �u is the u-SBR measure on W u, and the constant k1 appears in Theorem 3.6.17

The lemma readily follows from Theorem 3.6 with (C 03)�1 = d0D�k1max.Combining this lemma with Theorems 5.6 and 5.7 gives the following:Corollary 5.10 There is a constant C3 > 0 such that for every R 2 B and every k � k0we have �(k)+ (R) � C3 �SBR(R) and �+(R) � C3 �SBR(R)Lemma 5.11 There is a constant C4 > 0 such that for every R 2 B and every k � k0we have �(k)+ (R) � C4 �SBR(R) and �+(R) � C4 �SBR(R)Proof. Put " = dumax(R)=2. Let W u be an unstable �ber in M (k)+ (respectively, M+),and �u the u-SBR measure on W u. Then �u(W u \ R) � D�u(W u \ U"). ApplyingTheorems 5.6 and 5.7 proves the lemma with C4 = DC1=(2C 0).6 Uniqueness of the conditionally invariant measureThis section is devoted to the following theorem:Theorem 6.1 There is a unique measure �+ on M+ with the following properties:(i) �+ is a conditionally invariant probability measure with the eigenvalue �+;(ii) the conditional distributions of �+ on unstable �bers in M+ are u-SBR measures;(iii) for any " > 0 we have �+(U") � C1".(iv) for any R 2 B we have C3 �SBR(R) � �+(R) � C4 �SBR(R).Before proving it, we will make some additional constructions.Let �+ be an arbitrary measure satisfying the assumptions of this theorem. LetW s � M be a stable �ber and R a rectangle whose one side is W s. Denote by R+ theunion of all unstable R-�bers that belong in M+. There are two �nite limits then,D1;2(�+;W s) = lim� �+(R+)�SBR(R) (6.1)Here lim� is taken as the rectangle R shrinks in the unstable direction, so that its sta-ble side opposite to W s approaches W s. The two values, D1 and D2 correspond to twopossible locations of R, which may be placed on either side of the curve W s (the sub-scripts, 1 and 2, are assigned arbitrarily). The existence of the limits in (6.1) followsfrom Propositions 5.2 and 5.3 and Corollary 5.8. Note also that Corollary 5.8 impliesthat �+(R+) = �+(R) + o(dumin(R)) = �+(R) + o(�SBR(R)) (6.2)18

Convention. In this section, for any rectangle R we will denote by R+ the union ofunstable R-�bers that belong in M+. Then, for a rectangle denoted by Rn or Rk;l, forexample, we denote by Rn;+ and Rk;l;+, respectively, the union of unstable Rn-�bers orRk;l-�bers belonging in M+.Note that, although R is a rectangle, the preimages of R under T�n, after the re-moval of holes, are no longer rectangles. They are some domains adjacent to the smoothcomponents of T�nW s, where W s is a stable side of the original rectangle R. Since theconnected components of T�nR are not rectangles, it would be di�cult to compute withthem. Instead, we consider R+, which has the following invariance property. For anyn � 1 let Rn;i, i = 1; 2; : : :, be u-subrectangles in T�nR whose stable sides are the smoothcomponents of T�nW s. Then T�nR+ = [iRn;i;+ (6.3)This property is easy to verify by induction on n. Note that, according to (6.2), the lossof measure incurred by the replacement of R by R+ is relatively small as the rectangleR shrinks in the unstable direction.We call a �berW s essential for the measure �+ if D1;2(�+;W s) > 0. We will see later,cf. Corollary 6.5, that this is equivalent to �+(R+) > 0 for at least one (and then forevery su�ciently narrow) rectangle R adjacent to W s. As a result, if a �ber W s containsa piece that is essential, then W s itself is an essential �ber.It follows immediately from Corollary 5.10 and Lemma 5.11 that if a �ber W s haslength � d0, then it is essential and0 < C3 � D1;2(�+;W s) � C4 <1 (6.4)We now begin the proof of Theorem 6.1. Assume that there are two distinct measures,�1 and �2, on M+ satisfying the assumptions of this theorem. For any stable �berW s � M that is essential for both �1 and �2 (see also Corollary 6.5 below) and j = 1; 2there exists a �nite positive limitDj(�1; �2;W s) = lim��1(R+)�2(R+) (6.5)Note that Dj(�1; �2;W s) � Dj(�2; �1;W s) = 1.We then putD(�1; �2) = supfDj(�1; �2;W s) : W s is essential and j = 1; 2gand D�(�1; �2) = supfDj(�1; �2;W s) : W s has length � d0 and j = 1; 2gLemma 6.2 We have 1 � D�(�1; �2) � C4=C319

The lower bound is obvious since �1 and �2 are probability measures. The upperbound immediately follows from (6.4).Lemma 6.3 We have D(�1; �2) = D�(�1; �2)Proof. Let W s be a stable �ber of length < d0, and R a rectangle whose one sideis W s. Since both measures �1 and �2 are conditionally invariant, we have �i(R+) =��n+ T n� �i(R+) = ��n+ �i(T�nR+) for i = 1; 2 and all n � 1. Therefore,�1(R+)�2(R+) = �1(T�nR+)�2(T�nR+)Let W s1;i, i = 1; 2; : : :, be the smooth components of T�1W s. For every i let R1;i be therectangle whose one side is W s1;i and the opposite side belongs to T�1(@R). Note thatT�1R+ = [iR1;i;+ according to (6.3).Some of the curves W s1;i may have length already larger than d0, then the adjacentrectanglesR1;i are not iterated backward any further. If not, we pull them back under T�1and construct rectangles R2;i adjacent to the smooth components of T�2W s in the sameway, etc. At every iteration n � 1, we hold the obtained rectangles Rn;p, p = 1; 2; : : :,which are adjacent to a smooth component of T�nW s of length � d0, and map theother (shorter) rectangles further under T�1. Note that if we map all the eventually held(larger) rectangles Rn;p (for all n; p) back on R under T n, then we get a collection ofdisjoint u-subrectangles in R, which we denote by R0j , j � 1. In this way we obtain a�nite or countable collection of disjoint u-subrectangles R0j � R, such that for every jthere is a nj � 1 for which R0j � Mnj and the length of the stable side of the rectangleR00j = T�njR0j that belongs in T�njW s is � d0.Sublemma 6.4 The union [jR0j essentially covers the set R \M+, i.e.�i(R+) =Xj �i(R0j \R+) = Xj ��nj+ �i(T�nj (R0j \R+))= Xj ��nj+ �i(R00j;+) (6.6)for i = 1; 2. In other words, the measure �i on R+ is supported on the u-subrectanglesR0j that are `eventually long' in the past.Proof. In the iterative process of construction of rectangles R0j , at every step n � 1 wemay have some `unfortunate' rectangles R000n;p � R, p = 1; 2; : : :, whose preimages T�iR000n;pfor all i = 1; : : : ; n belong in shorter rectangles, i.e., those adjacent to smooth componentsof T�iW s of length< d0. Now, by repeating the main argument of the proof of Lemma 3.320

one can see that for every n � 1 the number of those short rectangles T�nR000n;p does notexceed Bn0 . The �i-measure of every short rectangle T�nR000n;p is smaller than const���nmin,because its width in the unstable direction is less than const���nmin. Therefore,�i([pR000n;p) � const � B0�+�min!n (6.7)which exponentially approaches zero as n ! 1 due to our Assumption H2. The sub-lemma is proved.Now, since all the rectangles T�njR0j are long, the decomposition (6.6) completes theproof of Lemma 6.3. Here we made use of a simple fact that(c1 + c2 + � � �)=(d1 + d2 + � � �) � maxi fci=dig (6.8)for positive ci; di.The proof of Sublemma 6.4 justi�es the following corollary:Corollary 6.5 A stable �ber W s is essential to �+ if and only if it is eventually longin the past, i.e. for some n � 1 a smooth component of T�nW s is of length � d0. Italso follows from (6.7) that for every nonessential �ber W s we have �i(R+) = 0 for alladjacent rectangles R and i = 1; 2.This corollary is an analogue of Lemma 3.3. Since the property of being eventuallylong in the past does not refer to any measure, a �ber W s in our de�nition (6.5) isessential either for both �1 and �2 or for neither.Lemma 6.6 We have D(�1; �2) = D�(�1; �2) = 1Proof. Assume that D(�1; �2) = D�(�1; �2) > 1. Then for any � > 0 we can �nd astable �ber W s of length between d0 and 2d0 such thatDj(�1; �2;W s) > (1� �)D�(�1; �2)for j = 1 or j = 2.Next, we introduce additional cuts of stable �bers in the same way as we cut unstable�bers in Sect. 2. That is, we �x a �nite number of local unstable �bers so that everymaximal stable �ber in M (terminating on @H) with length > 2d0 is cut into segmentsof length between d0 and 2d0. None of the maximal stable �bers of length � 2d0 shouldbe cut.Now, for any n � 1 the set T�nW s consists of a �nite number of stable �bers, say,W sn;q, q = 1; 2; : : :. (These are de�ned just like the components W un;i in Sect. 2). Denote21

by Qn the number of these �bers. It follows from our construction and the de�nition ofB0 that Qn � 4B0�nmax (6.9)Let R be a su�ciently narrow rectangle whose one side is W s. For every q let Rn;q bea narrow rectangle such that (i) one side of it is W sn;q and (ii) its opposite side belongs inT�n(@R). Then T�nR+ = [qRn;q;+ (here Rn;q;+ is the union of the unstable Rn;q-�bersthat belong in M+). Since the measures �1 and �2 are conditionally invariant with thesame eigenvalue, we have�1(R+)�2(R+) = Pq �1(Rn;q;+)Pq �2(Rn;q;+) > (1� 2�)D�(�1; �2) (6.10)The last inequality is ensured if the rectangle R is thin enough in the unstable directionfor the given n, which will be our standing assumption in the proof of this lemma. Thenwe also have�1(Rn;q;+)�2(Rn;q;+) � (1 + �)Dj(�1; �2;W sn;q) � (1 + �)D(�1; �2) = (1 + �)D�(�1; �2) (6.11)for every q such that the �ber W sn;q is essential. For nonessential �bers, both measuresin (6.11) are zero, cf. Corollary 6.5.Note that, according to (6.2), �i(Rn;q;+) � �i(Rn;q)=2 for i = 1; 2. If for some q0 thecurve W sn;q0 has length � d0, then the assumption (iv) of Theorem 6.1 and (5.2) implythat �i(Rn;q0;+) � 12�i(Rn;q0) � C 0C32 dumax(Rn;q0)� C 0C32 ��nmax dumin(R) � C 0C32D ��nmax dumax(R)� C 0C32D ��min�max�n maxq fdumax(Rn;q)g� C 0C32DC 00C4 ��min�max�n maxq f�i(Rn;q)g� C 0C38DC 00C4B0 �min�2max!n Xq 6=q0 �i(Rn;q;+) (6.12)for i = 1; 2, where in the last line we have used the bound (6.9). DenoteGn = C 0C38DC 00C4B0 �min�2max!nIt is a simple calculation to combine the above estimates (6.10) and (6.11) that gives�1(Rn;q0;+) � (1� 2�)D�(�1; �2)�2(Rn;q0;+)� 3�D�(�1; �2) Xq 6=q0 �2(Rn;q;+)22

Here we used again the simple fact (6.8). Now, applying the bound (6.12) with i = 2gives �1(Rn;q0;+)�2(Rn;q0;+) � (1� 2� � 3G�1n �)D�(�1; �2)Taking a limit as R shrinks in the unstable direction givesDj(�1; �2;W sn;q0) � (1� 2� � 3G�1n �)D�(�1; �2) (6.13)We now �x an unstable �ber, ~W u � M of length � d0. According to Theorem 3.6(see also the remark after it), there is a curveW sk1 ;q0 intersecting ~W u whose endpoints areat least a distance d0 away from ~W u. By choosing � su�ciently small we can make theright hand side of (6.13) with n = k1 arbitrary close to D�(�1; �2). We can also switch �1and �2 and repeat the above construction. As a result, we get the following sublemma.Sublemma 6.7 Let ~W u � M be an unstable �ber of length d0. For any � > 0 there aretwo stable �bers, W s1 and W s2 crossing ~W u, whose endpoints are at least a distance d0away from ~W u, such that(1� �)D�(�1; �2) � Dj1 (�1; �2;W s1 ) � D�(�1; �2) (6.14)and (1� �)D�(�2; �1) � Dj2 (�2; �1;W s2 ) � D�(�2; �1) (6.15)for some j1; j2 2 f1; 2g.Recall that Dj2(�1; �2;W s2 ) = 1=Dj2 (�2; �1;W s2 ). Thus, by choosing � su�cientlysmall we can make the di�erenceDj1(�1; �2;W s1 )�Dj2(�1; �2;W s2 )arbitrary close to the �xed positive value� = D�(�1; �2)� 1=D�(�2; �1) > 0We now have two stable �bers, W s1 and W s2 , which are long (longer than 2d0), and bysliding one of them along unstable �bers (applying a holonomy map) a distance less thanDd0 we can cover at least 2D�1d0 of the other �ber. We now iterate both �bers backward,and erase the parts of their preimages that fall through holes. Also, whenever a part ofthe preimage of one �ber is erased, its image under the holonomy map on the preimageof the other �ber is erased, too. It is also convenient to make additional cuts, like inthe proof of Lemma 6.6, to ensure that the preimages consist of smooth components oflength � 2d0. Then, for any n � 1 on the curves T�nW s1 and T�nW s2 we obtain a �nitenumber of subsegments, W s1;n;l and W s2;n;l (l = 1; 2; : : :), respectively, such that W s1;n;l isthe image of W s2;n;l under the holonomy map for every l.23

It follows from Theorem 2.1 (see the �rst remark after it) that the larger n the moreof the segments W s1;n;l and W s2;n;l are long, i.e. have length � d0. More precisely, there isa sequence Nn (independent of the choice of the �bers W s1 and W s2 ) such that Nn !1as n ! 1 and the number of long pairs of unstable �bers W s1;n;l, W s2;n;l is larger thanNn. We denote by L�n the set of indices flg for which both �bers W s1;n;l and W s2;n;l arelong. Then cardL�n � Nn.The distance between each W s1;n;l and its counterpart W s2;n;l in the unstable directionis less than d0D��nmin. For any given n we can argue like in the proof of Sublemma 6.7and choose � small enough to make the value of Dj1 (�1; �2;W s1;n;l) arbitrarily close toD�(�1; �2) and the value of Dj2(�1; �2;W s2;n;l) arbitrarily close to 1=D�(�2; �1) for alll 2 L�n. Therefore, for every n there is a �n > 0 such that for any � � �n we haveDj1(�1; �2;W s1;n;l)�Dj2(�1; �2;W s2;n;l) � �=2for all l 2 L�n. Equivalently,Dj1(�1;W s1;n;l)Dj1(�2;W s1;n;l) � Dj2(�1;W s2;n;l)Dj2(�2;W s2;n;l) � �2 (6.16)for all l 2 L�n.We represent the left hand side of (6.16) by a=b � c=d. Then, according to (6.4) wehave ad� bcbd = (a� c)d + (d� b)cbd � ja� cjC4 + jd� bjC4C23This gives the following alternative: For every l 2 L�n eitherjDj1(�1;W s1;n;l)�Dj2 (�1;W s2;n;l)j � 14�C23=C4 (6.17)or jDj1(�2;W s1;n;l)�Dj2 (�2;W s2;n;l)j � 14�C23=C4 (6.18)We put �1 = 14�C23=C4 > 0. Denote by L(1)n and L(2)n the sets of indices in L�n for which(6.17) and (6.18) hold, respectively. Note that L(1)n [L(2)n = L�n. Without loss of generality,we assume that for some in�nite sequence fnmg of indices we have cardL(1)nm � Nnm=2.Let l 2 L(1)n . Denote the rectangle whose two stable sides are W s1;n;l and W s2;n;lby Rn;l. Its width in the unstable direction is smaller than d0D��nmin. Let Rn;l be arectangle containing Rn;l as an s-subrectangle, such that its unstable sides are at distancedumax(Rn;l)=100 from those of Rn;l. Now letR1 and R2 be two very thin rectangles adjacentto W s1;n;l and W s2;n;l as prescribed by the indices j1 and j2, respectively. In particular,R1; R2 � Rn;l. For i = 1; 2 we put Ri;++ = Ri \Rn;l;+ andD+ji (�1;W si;n;l) = lim��1(Ri;++)�SBR(Ri)24

similarly to (6.1). Note that the sets R1;++ and R2;++ consist of pieces unstable �bersthat lie on the sameRn;l-�bers. It is then a direct calculation with the help of the uniformcontinuity of the u-SBR density (1.3) and the jacobian of the holonomy map thatjD+j1(�1;W s1;n;l)�D+j2(�1;W s2;n;l)j � g(dumax(Rn;l)) � g(2d0D��nmin) (6.19)where the function g(x)! 0 as x! 0 depends on T only. Let n = g(2d0D��nmin).For large n = nm such that n < �1 and for any l 2 L(1)nm , the inequality (6.17) canonly hold due to the contribution from the sets Ri;+nRi;++, i = 1; 2, which lie on unstable�bers in M+ terminating inside Rnm;l.Sublemma 6.8 Let nm be so large that nm < �1=2. Then for any l 2 L(1)nm and small" > 0 (i.e., " < "0(nm; l)) we have�1(Rnm;l \ U") � 13C 0D�1�1" (6.20)where C 0 > 0 appears in (5.2).Proof. Let � > 0. If the rectangles R1 and R2 are thin enough (in the unstabledirection), it follows from (6.17) and (6.19) that������1(R1;+ nR1;++)�SBR(R1) � �1(R2;+ nR2;++)�SBR(R2) ����� � �1(1� �)=2Now, without loss of generality we can assume that�1(R1;+ nR1;++)�SBR(R1) � �1(1� �)=2Due to (5.2), we have �1(R1;+ nR1;++) � C 0dumax(R1)�1(1� �)=2Let " = dumax(R1) (this is not a restrictive choice, since R1 is an arbitrary su�cientlynarrow rectangle adjacent to W s1;n;l). The set R1;+ n R1;++ is a union of unstable R1-�bers that belong in M+. They all have length � dumin(R1) � D�1". Moreover, thecontinuations of these �bers inM+ terminate within our rectangle Rnm;l (otherwise those�bers would be in R1;++). We now slide every �ber in this set along the �ber in M+that contains it, down to the endpoint of the latter lying in Rnm;l. We then get another�ber, of the same length, which is between " and D�1", so that the new �ber lies in U".By sliding the �ber we may change its measure, but at most by a factor of D (actually,very little since the rectangle Rnm;l is also thin in the unstable direction). Therefore, theunion of the new �bers has measure � "D�1C 0�1(1� �)=2. Sublemma 6.8 is proved.Now note that the number of rectangles Rnm;l satisfying (6.20) is � Nnm=2.25

Claim. The number of disjoint rectangles satisfying (6.20) can be made arbitrarilylarge by increasing nm.Proof. Assume that the number of disjoint rectangles satisfying (6.20) is uniformlybounded, say � K0. For large nm we have cardL(1)nm � Nn=2� K0, so that the rectanglesRnm;l, l 2 L(1)nm , have to form� K0 clusters, and they necessarily converge to a �nite unionof some stable curves �W sk �M , 1 � k � K0. For any r � 1 denote by �Nr the number ofsmooth components of the set T�r([k �W sk ). It follows from the time-symmetric versionof Theorem 2.1, cf. a remark after it, that �Nr !1 as r !1. Under our assumptionson the sequence fnmg there is an in�nite sequence frlg such that at least �Nrl=2 smoothcomponents of T�rl([k �W sk ) must be also curves to which some clusters of rectanglesRnm;l, l 2 L(1)nm , converge as nm !1. This proves the claim.The existence of arbitrarily many disjoint rectangles Rnm;l satisfying (6.20) clearlycontradicts Theorem 5.7. The proof of Lemma 6.6 is completed. Theorem 6.1 nowreadily follows.Corollary 6.9 The sequence of measures �(k)+ converge weakly, as k ! 1, to a uniquemeasure �+ satisfying all the assumptions of Theorem 6.1.Lastly, note that the conditionally invariant measure �+ for the map T is also suchfor any iteration T k, k � 2, of the map T , with eigenvalue �k+. Moreover, it is fairlystraighforward that all our arguments apply to any iteration of T . Therefore, the measure�+ is the only conditionally invariant measure for the map T k for every k � 2.Acknowledgements. This work was started when R.M. visited the University ofAlabama at Birmingham, for which he is the most indebted. N.Ch. acknowledges thesupport of NSF grant DMS-9401417, R.M. the support of CSIC (Univ. de la Rep�ublica)and CONICYT (Uruguay).References[1] D.V. Anosov and Ya.G. Sinai, Some smooth ergodic systems, Russ. Math. Surveys22 (1967), 103{167.[2] R. Bowen, Equilibrium states and the ergodic theory of Anosov di�eomorphisms,Lect. Notes Math. 470, Springer-Verlag, Berlin, 1975.[3] N.N. �Cencova, A natural invariant measure on Smale's horseshoe, Soviet Math.Dokl. 23 (1981), 87{91.[4] N.N. �Cencova, Statistical properties of smooth Smale horseshoes, in: MathematicalProblems of Statistical Mechanics and Dynamics, R.L. Dobrushin, Editor, pp. 199{256, Reidel, Dordrecht, 1986. 26

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