TYPICAL POINTS FOR ONE-PARAMETER FAMILIES OF PIECEWISE EXPANDING MAPS OF THE INTERVAL

Transcription

1 TYPICAL POINTS FOR ONE-PARAMETER FAMILIES OF PIECEWISE EXPANDING MAPS OF THE INTERVAL DANIEL SCHNELLMANN Abstract. Let I R be an interval and T a : [0, ] [0, ], a I, a oneparameter family of piecewise expanding maps such that for each a I the map T a admits a unique absolutely continuous invariant probability measure µ a. We establish sufficient conditions on such a one-parameter family such that a given point x [0, ] is typical for µ a for a full Lebesgue measure set of parameters a, i.e. n n i=0 δ T i a (x) weak- µ a, as n, for Lebesgue almost every a I. In particular, we consider C, (L)-versions of β-transformations, skew tent maps, and Markov structure preserving oneparameter families. For the skew tent maps we show that the turning point is almost surely typical.. Introduction Let I R be an interval and T a : [0, ] [0, ], a I, a one-parameter family of maps of the unit interval such that, for every a I, T a is piecewise C 2 and inf x [0,] x T a (x) λ >, where λ is independent on a. Assume that, for all a I, T a admits a unique (hence ergodic) absolutely continuous invariant probability measure (a.c.i.p.) µ a. According to [LaY] and [LiY], for Lebesgue almost every x [0, ], some iteration of x by T a is contained in the support of µ a. From Birkhoff s ergodic theorem we derive that Lebesgue almost every point x [0, ] is typical for µ a, i.e. n n i=0 δ T i a (x) weak- µ a, as n. In this paper we are interested in the question if the same kind of statement holds in the parameter space, i.e. if a chosen point x [0, ] is typical for µ a for Lebesgue a.e. a I, or more general, if, for some given map X : I [0, ], X(a) is typical for µ a for Lebesgue a.e. a in I. In the next section we try to establish some sufficient conditions on a one-parameter family such that the following statement is true. For Lebesgue a.e. a I, X(a) is typical for µ a. The method we use in this paper is a dynamical one. It is essentially inspired by the result of Benedicks and Carleson [BC] where they prove that for the quadratic family f a (x) = ax 2 on (, ) there is a set (, 2) of a-values of positive Lebesgue measure for which f a admits almost surely an a.c.i.p. and for which the The author is supported by grant KAW from the Knut and Alice Wallenberg Foundation.

2 2 DANIEL SCHNELLMANN critical point is typical with respect to this a.c.i.p. The main tool in their work is to switch from the parameter space to the dynamical interval by showing that the a-derivative a fa() j is the same as the x-derivative x fa() j up to some constant. This will also be the essence of the basic condition on our one-parameter family (see condition (I) below), i.e. we require that the a- and the x-derivatives are comparable. Other typicality results related to this paper can be found in [Schm], [B], and [FP]. The one-parameter families T a, a I, considered in these papers have in common that their slopes are constant for a fixed parameter value, i.e. for each a I there is a constant λ > such that T a λ on [0, ]. The advantage of our method and the main novelty of this paper is that we can drop this restriction and, thus, we are able to consider much more general families. This paper consists of two parts. In the first part, which corresponds to Sections 2-4, we establish a general criteria for typicality. In the second part, which corresponds to Sections 5-7, we apply this criteria to several well-studied oneparameter families and derive various typicality results for these families. Even if we formulate a general criteria for typicality the treatment of these families in Sections 5-7 is the main purpose of this paper... β-transformations. The example in Section 5 is a C, (L)-version of the β-transformation. By saying that a function is C, (L), we will mean that it is C and its derivative is Lipschitz continuous with Lipschitz constant L. The usual β-transformation is given by the map x ax mod where x [0, ] and the parameter a is strictly greater than. For a sequence 0 = b 0 < b <... of real numbers such that b k as k and a constant L > 0, let T : [0, ) [0, ] be a right continuous function which is C, (L) on each interval (b k, b k+ ), k 0. Furthermore, we assume that: T(b k ) = 0 for each k 0. For each a >, See Figure. < inf xt(ax) and sup x T(ax) <. x [0,] x [0,] 0 0 Figure. A possible beginning of a graph for T : [0, ) [0, ]. A C, (L)-version of the β-transformation is provided by the map T a : [0, ] [0, ], a >, defined by T a (x) = T(ax), x [0, ]. As we will see in Section 5, for each a >, T a admits a unique a.c.i.p. µ a. In Section 5 we will show the following.

3 TYPICAL POINTS 3 Theorem.. If X : (, ) (0, ] is C and X (a) 0, then X(a) is typical for µ a for Lebesgue a.e. a >. If we choose X(a) = b /a then X (a) < 0 and Ta j (X(a)) = 0 for all j. Hence, if the condition X (a) 0 in Theorem. is not satisfied, we cannot any longer guarantee almost sure typicality for the a.c.i.p. For an illustration of the curves on which we have a.s. typicality see Figure 2 (when a is fixed, we can apply Birkhoff s ergodic theorem and get a.s. typicality on the associated vertical line). Observe that if we choose T : [0, ) [0, ] by T(x) = x mod, then T a (x) = ax mod is the usual β-transformation. Theorem. generalizes considerably a result due to Schmeling [Schm] where it is shown that for the usual β-transformation the point is typical for the associated a.c.i.p. for Lebesgue a.e. a >. x 0 a Figure 2. Lines and curves on which we have a.s. typicality for the C, (L)-version of the β-transformation..2. Skew tent maps. In Section 6 we investigate unimodal maps with slopes constant to the left and to the right of the turning point. Let these slopes be α and β where α, β >. The choice of α and β determines such a map up to an affine conjugacy (and up to a choice of a smaller or larger interval). These maps are called the skew tent maps. By Lemma 3. in [MV], skew tent maps are only meaningful in our context if we assume that α + β (otherwise there does not exist a finite invariant interval and the forward iteration of a.e. point in R tends to ). Fix two points (α 0, β 0 ) and (α, β ) in the set {(α, β) ; α, β > and α +β } such that α α 0, β β 0, and at least one of these two inequalities is sharp. Let α : [0, ] [α 0, α ] and β : [0, ] [β 0, β ] be functions in C ([0, ]) such that (α(0), β(0)) = (α 0, β 0 ), (α(), β()) = (α, β ), and, for all a [0, ], if α 0 α, then α (a) > 0 and if β 0 β, then β (a) > 0. Consider the one-parameter family T a, a [0, ], where T a is a unimodal map on the interval with slope α(a) to the left and β(a) to the right of the turning point. By [LiY], since T a has only two pieces of monotonicity, it follows that there exists a unique a.c.i.p. µ a for T a. In Section 6 we will show that the turning point is a.s. typical for the a.c.i.p. Theorem.2. The turning point for the skew tent map T a is typical for µ a for Lebesgue a.e. a [0, ]. Theorem.2 generalizes a result due to Bruin [B] where almost sure typicality is shown for the turning point of symmetric tent maps (i.e. when α(a) β(a)). It is possible to extend the results in Section 6 to certain C, (L)-versions of skew

4 4 DANIEL SCHNELLMANN tent maps (see Remark 6.5). An important ingredient in the proof of Theorem.2 is the fact that parameters ã [0, ] such that Tã is a Markov map are dense in [0, ] (see Subsection 6.4)..3. One-parameter families of Markov maps. In Section 7 we consider oneparameter families, which preserve a certain Markov structure. A simple example for such a family are the maps T a : [0, ] [0, ] defined by { x if x < a, a T a (x) = x a a otherwise, where the parameter a (0, ). See Figure 3. By [LiY], since this map has only one point of discontinuity, it admits a unique a.c.i.p. µ a (which coincides in this case with the Lebesgue measure on [0, ]). In Example 7.2 in Section 7 we will show the following. Proposition.3. If X : (0, ) (0, ) is a C map such that X (a) 0, then X(a) is typical for µ a for a.e. parameter a (0, ). Observe that if X(a) = p a where p a is the unique point of periodicity 2 in the interval (0, a), then X (a) > 0. Hence, if the condition X (a) 0 is violated in Proposition.3, we cannot any longer guarantee almost sure typicality for the a.c.i.p. The very simple structure of the example in this subsection makes it to a good candidate for serving the reader as a model along the paper. Example 7.2 in Section 7 is formulated slightly more general by composing T a with a C, (L) homeomorphism g : [0, ] [0, ]. 0 0 a Figure 3. A Markov structure preserving one-parameter family T a where a (0, ). 2. Piecewise expanding one-parameter families 2.. Preliminaries. In this subsection we introduce the basic notation and put up a general model for one-parameter families of piecewise expanding maps of the unit interval. A map T : [0, ] R will be called piecewise C, (L) if there exists a partition 0 = b 0 < b <... < b p = of the unit interval such that for each k p the restriction of T to the open interval (b k, b k ) is a C, (L) function. Observe that, by the Lipschitz property, it follows that T restricted to (b k, b k )

5 TYPICAL POINTS 5 can be extended to the closed interval [b k, b k ] as a C, (L) function. Let I R be an interval of finite length and T a : [0, ] [0, ], a I, a one-parameter family of piecewise C, (L) maps where the Lipschitz constant 0 < L < is independent on the choice of the parameter a. We assume that there are real numbers < λ Λ < such that for every a I, () λ inf xt a (x) and sup x T a (x) Λ. x [0,] x [0,] The parameter dependence is assumed to be piecewise C, i.e. for all x [0, ], there exists a partition a 0 < a <... < a p(x) (where a 0 is the left and a p(x) the right boundary point of I) of the parameter interval I such that for each k p(x) the restriction of the map a T a (x) to the open interval (a k, a k ) is a C function, which can be extended to the closed interval [a k, a k ] as a C function. More precise requirements on the parameter dependence will be given shortly (see (i)-(iii) in this subsection). In the sequel, instead of referring to [LaY] and [LiY], we will refer to a paper by S. Wong [Wo] who extended the results in [LaY] and [LiY] on piecewise C 2 maps to a broader class of maps containing also piecewise C, (L) maps. For a fixed a I, by [Wo], there is a finite collection of sets K (a),..., K p0 (a)(a) such that each K k (a), k p 0 (a), is a union of finitely many disjoint intervals and, for Lebesgue a.e. x [0, ], the accumulation points of the forward orbit of x is identical with one of these K k (a) s, i.e. to every x in a full Lebesgue measure set of [0, ], there is a K k (a) such that (2) K k (a) = {Ta n(x)} n=n. N= Furthermore, for each K k (a) there is a unique (hence ergodic) a.c.i.p. µ a,k such that supp(µ a,k ) = K k (a). Since we are always interested in only one K k (a), we can without loss of generality assume that p 0 (a), a I. Henceforth, we write K(a) and µ a instead of K (a) and µ a,, respectively. For a I, let c 0 (a) < c (a) <... < c p (a)(a) be the associated partition for the piecewise C, (L) map T a : K(a) K(a), i.e., if D (a),..., D p2 (a)(a) (p 2 (a) p (a)) denote the (maximal) open intervals in K(a) on which T a is C, (L), then the c k (a) s are the boundary points of these C, (L) domains. We assume that the number of c k (a) s and D k (a) s are constant, i.e. p (a) p and p 2 (a) p 2. Furthermore, we make the following three natural assumptions on our model. (i) For a I, the maps a c k (a), 0 k p, and a D k (a), 0 k p 2, are Lip(L), and there is a constant δ 0 > 0 such that D k (a) δ 0, for all k p 2. (ii) For each x [0, ] and k p 2, if J denotes the set of parameters a I such that x D k (a), then if J is non-empty, it is an interval and the maps a T a (x) and a T a (x) from J to R are Lip(L).

6 6 DANIEL SCHNELLMANN (iii) For each x [0, ] and k p 2, if J denotes the set of parameters a I such that Ta {x} has a pre-image in D k(a), then if J is non-empty, it is an interval and the branch of T (x), which maps J to D k is Lip(L) Partitions. For a fixed parameter value a I, we denote by P j (a), j, the partition on the dynamical interval consisting of the (maximal) open intervals of monotonicity for the map T j a : K(a) K(a). In other words, P j(a) denotes the set of open intervals ω in K(a) such that T j a : ω K(a) is C, (L) and ω is maximal, i.e. for every other open interval ω K(a) with ω ω, T j a : ω K(a) is no longer C, (L). Clearly, P (a) = {D (a),..., D p2 (a)}. For an open set H K(a), we denote by P j (a) H the restriction of P j (a) to the set H. For a set J K(a), which lies completely in one D k (a), k p 2, we denote by symb a (J) the index (or symbol) k. We will define similar partitions on the parameter interval I. Let X : I [0, ] be a C map from I into the dynamical interval [0, ]. The points X(a), a I, will be the candidates of typical points. The forward orbit of a point X(a) under the map T a we denote as x j (a) := T j a(x(a)), j 0. Remark 2.. Since a lot of informations about the dynamics of T a is contained in the forward orbits of the partition points c k (a), 0 k p, it is of interest to know how the forward orbits of these points are distributed. Hence, an evident choice of the map X would be X(a) = lim x c k (a) x D(a) T a (x), where D(a) {D (a),..., D p2 (a)} is an interval adjacent to c k (a). In order that the following definition of partitions on the parameter space makes sense, we assume also that X(a) K(a), for all a I. Let J be an open set in the parameter space I. By P j J, j, we denote the partition consisting of all open intervals ω in J such that for each 0 i < j, there exists a k p 2, such that x i (a) D k (a), for all a ω, and such that ω is maximal, i.e. for every other open interval ω J with ω ω, there exist a ω, 0 i < j, and k p 2 such that x i (a) D k (a). Observe that this partition might be empty. This is, e.g., the case when T a is the usual β-transformation and the map X is chosen to be equivalently equal to 0. However, such trivial situations are excluded by condition (I) formulated in the next subsection. Knowing that condition (I) is satisfied, the partition P j J is the set of the (maximal) open intervals of monotonicity for x j : J [0, ]. We set P 0 J = J, and we will write P j I instead of P j int(i). If for a set J in the parameter space and for some integer j 0 the symbol symb a (x j (a)) exists for all a J, then it is constant and we denote this symbol by symb(x j (J )). Finally, observe that, if a parameter a I is contained in an element of P j I, j, then also the point X(a)(= x 0 (a)) is contained in an element of P j (a) Main statement. In this subsection we will state our main result, Theorem 2.2. Let n be large. To ensure good distortion estimates we will, in the proof of Theorem 2.2, split up the interval I into smaller intervals J I of size /n. The main idea in this paper is to switch from the map x j : J [0, ], j n, to

7 TYPICAL POINTS 7 the map Ta j : [0, ] [0, ] where a is, say, the right boundary point of J. By this, since the dynamics of the map T a is well-understood, we derive similar dynamical properties for the map x j, which then can be used to prove Theorem 2.2. To be able to switch from x j to Ta j, we put further three conditions on our one-parameter family. These conditions, in particular conditions (II) and (III), are more technical and restrictive. But nevertheless, as we will see in Sections 5-7, important classes of one-parameter families satisfy these conditions. In condition (I) we require that the derivatives of x j and Ta j are comparable. This is the very basic assumption in this paper. Of course, the choice of the map X : I [0, ] plays here an important role. If, e.g., for every parameter a I, X(a) is a periodic point for the map T a, then x j will have bounded derivatives and the dynamics of x j is completely different from the dynamics of T a. Henceforth, we will use the notations T a(x) = x T a (x) and x j(a) = D a x j (a), j. (I) There is a constant C 0 such that for ω P j I, j, we have x j(a) C 0 Ta j (X(a)) C 0, for all a ω. Furthermore, the number of a I, which are not contained in any element ω P j I is finite. For a I, let ϕ a denote the density for µ a. Assume for the moment that ϕ a is bounded from below and from above by a constant C, i.e C ϕ a (x) C for a.e. x K(a). Note that, since the density ϕ a is a fixed point of the Perron- Frobenius operator, we have, for j, ϕ a (y) = for a.e. y K(a). This implies that (3) x K(a) T j a(x)=y x K(a) T j a(x)=y ϕ a (x) T j a (x), T j a (x) C2, for a.e. y K(a). We require a similar estimate for the map x j. (II) There exists a constant C such that the following holds. Let J I be an open interval of length /n. For ω P i J, i, and j n, we have (4) for a.e. y [0, ]. a ω x i+j (a)=y T j a (x i (a)) C Let B [0, ] be a (small) interval. If in addition to condition (I) also condition (II) is satisfied then it follows (see (0) below) that x i ({a ω ; x i+j (a) B}) C 2 0 C B,

8 8 DANIEL SCHNELLMANN which will be the main estimate in the proof of Theorem 2.2. Now, we want to pull back this estimate to the parameter interval I. Assuming that x i ( ω) has a large size, for instance, assume that x i ( ω) = (0, ), then, by a minor distortion estimate for x i, it follows (see () below) that (5) {a ω ; x i+j (a) B} C 2 0C C 3 B ω, where C 3 is some distortion constant. If ω was a very small interval of monotonicity for x i then it might happen that ω is mapped by x i+j entirely into the interval B and not just a B fraction of it, as it is the case in (5). To avoid such cases we want that the total measure of partition elements with a too small image is negligible. Condition (III) requires that we are able to exclude elements ω P j I, j, whose length of x j (ω) is below a fixed constant, say, δ > 0. However, if ω P j I is an element such that x j (ω) < δ, we will not exclude it immediately but, roughly speaking, we will give ω (or at least a part of it) a chance to grow during a certain number of further iterations. The formulation of condition (III) is rather technical. (III) There is a constant δ > 0 such that to every ε > 0 there is an integer n ε growing at most polynomially in /ε such that for n n ε the following holds. Let J I be an open interval of length /n and fix an integer j 2n. Focusing on the j-th until the j+[ n]-th partitions, we define the following exceptional set containing partition intervals in P j+[ n] which originate from small partition intervals in P j which stay small during the subsequent [ n] iterations: E = {ω P j+[ n] J ; ω P j+k J, 0 k [ n], We require that E ε n. Finally, we state the main result of this paper. s.t. ω ω and x j+k ( ω) δ }. Theorem 2.2. Let T a : [0, ] [0, ], a I, be a piecewise expanding oneparameter family as described in Subsection 2., satisfying properties (i)-(iii), and let X : I [0, ] be a C map. If conditions (I)-(III) are fulfilled, then X(a) is typical for µ a for Lebesgue almost every a I. For all the examples considered in this paper we will not verify condition (II) directly. Instead we will verify conditions (IIa) and (IIb) described in Section 4. Knowing that these two conditions are satisfied is then sufficient to deduce that also condition (II) is satisfied (see Proposition 4.4). Furthermore, in the considered examples, we will usually not verify conditions (I)-(III) (or conditions (I), (IIa), (IIb), and (III), respectively) for the whole interval I for which the corresponding family is defined. Instead we will cover I by a countable number of smaller intervals and verify conditions (I)-(III) on these smaller intervals. 3. Proof of Theorem 2.2 The idea of the proof of Theorem 2.2 is inspired by Chapter III in [BC] where Benedicks and Carleson prove the existence of an a.c.i.p. for a.e. parameter in a

9 TYPICAL POINTS 9 certain parameter set (the set ). The argument implies that the critical point is in fact typical for this a.c.i.p. Let B := { (x r, x + r) [0, ] ; x Q, r Q +}. We will show that there is a constant C such that for each B B the function F n (a) = n χ B (x j (a)), n, n fulfills j= (6) lim F n (a) C B, for a.e. a I. n By standard measure theory (see, e.g., [M]), (6) implies that, for a.e. a I, every weak- limit point ν a of (7) n n j= δ xj (a), has a density which is bounded above by C. In particular, ν a is absolutely continuous. Observe that, by the definition of x j (a), the measure ν a is also invariant for T a and, hence, ν a is an a.c.i.p. for T a. By the uniqueness of the a.c.i.p. for T a, we finally derive that, for a.e. a I, the weak- limit of (7) exists and coincides with the a.c.i.p. µ a. This concludes the proof of Theorem 2.2. In order to prove (6), it is sufficient to show that for all (large) integers h there is an integer n h,b, growing for fixed B at most exponentially in h, such that F I n(a) h da const(c B ) h, for all n n h,b (see Lemma A. in [BS]). In the remaining part of this section, we assume that B B is fixed. For h, we have (8) F n (a) h da = χ I n h B (x j (a)) χ B (x jh (a))da. I j,...,j h n For a fixed parameter a, there exists an integer k such that (Ta k, µ a ) is exact and, hence, this system is mixing of all degrees (see [R] and [Wa]). It follows that for sequences of non-negative integers j r,..., jr h, r, with one has ) χ B (T kjr a (x) [0,] lim inf r i l jr i jr l =, ) χ B (T kjr h a (x) dµ a (x) = µ a (T kjr a ) (B)... T kjr h r a (B) µ a (B) h ( ϕ a B ) h. Since the maps T j a and x j are, by conditions (I)-(III), comparable, it is natural to expect similar mixing properties for the maps x j. In fact, in the next subsection, we are going to prove the following statement. Proposition 3.. Under the assumption that conditions (I)-(III) are satisfied, there is a constant C such that the following holds. For all h, there is an integer n h,b growing at most exponentially in h such that, for all n n h,b and for

10 0 DANIEL SCHNELLMANN all integer h-tuples (j,..., j h ) with n j < j 2 <... < j h n and j l j l n, l = 2,..., h, χ B (x j (a)) χ B (x jh (a))da 4 I (C B ) h. I Seen from a more probabilistic point of view, Proposition 3. says that whenever the distances between consecutive j i s are sufficiently large, the behavior of the χ B (x ji (.)) s is comparable to that of independent random variables. Now, the number of h-tuples (j,..., j h ) in the sum in (8), for which min i j i < n or min k l j k j l < n, is bounded by 2h 2 n h /2. Hence, by Proposition 3., F n (a) h da 4 I (C B ) h + 2h2 I 5 I (C B ) h, n whenever I { ( ) } 2h 2 2 n max n h,b,. (C B ) h Since both terms in this lower bound for n grow at most exponentially in h, this concludes the proof of Theorem Proof of Proposition 3.. To be able to make use of conditions (II) and (III), we split up the integral in Proposition 3. and integrate over smaller intervals of length /n. More precisely, under the assumptions of Proposition 3., we are going to show that there exists an integer n h,b growing at most exponentially in h such that, for n n h,b, we have (9) χ B (x j (a)) χ B (x jh (a))da n 3(C B )h, J where J I is an arbitrary interval of length /n. This immediately implies that, for n n h,b, χ B (x j (a)) χ B (x jh (a))da 4 I (C B ) h, I (if n h,b / I ), which concludes the proof of Proposition 3.. Let ω P i J, i, and j n. We have x i ({a ω ; x i+j (a) B}) = B a ω x i+j (a)=y By (I), we obtain x i (a) x i+j (a) C2 0 Ta j (x i (a)), and, thus, we can apply condition (II) to conclude that (0) x i ({a ω ; x i+j (a) B}) C 2 0 C B. x i (a) x i+j (a) dy. In the remaining part of the proof of Proposition 3., we will beside inequality (0) mainly make use of condition (III). We will first give a rough idea of its use. If our one-parameter family satisfies condition (III), we can neglect too small partition elements, and we can without loss of generality assume that the following preferable picture is true: If ω P ji J, i h, and x ji (ω ) B, then we

11 TYPICAL POINTS can write the subinterval of ω which is mapped into B as a disjoint union of intervals ω such that each ω is an element of some partition P ji +k J, k j i+ j i, having a large image, i.e. x ji +k( ω) δ. By Lemma 4.2 a), which is stated and proved in Section 4, we have good distortion estimates on ω, i.e. x j i +k (a) x j i +k (a ) C 3, for a, a ω. Hence, combined with (0), we get () x ji +k({a ω ; x ji+ (a) B}) {a ω ; x ji+ (a) B} C 3 ω x ji +k( ω) C2 0 C C 3 B ω. δ So, for each such large ω only a fraction which is proportional to the length of B can be mapped by x ji+ into B. Observe that the argument for deriving () also applies when ω P k J, k j, satisfying x k ( ω) δ, in which case we obtain (2) {a ω ; x j (a) B} C2 0C C 3 B ω. δ From (2) combined with (), applied h times, we can derive Proposition 3.. In the remaining part of this subsection, we will work this out in more detail. Fix an integer τ such that C0 2λ τ B, and assume that n is so large that n τ, which ensures that there are at least τ iterations between two consecutive j i s. Let Ω 0 = J and Ω i = {ω P ji +τ Ω i ; x ji (ω) B }, for i h. Notice that, by (I) and the assumption on τ, we have x ji (ω) B for all ω P ji +τ J and, thus, Ω i {a Ω i ; x ji (a) 3B}, where 3B denotes the interval being three times as long as B and sharing the same midpoint. In each step we will exclude partition intervals with too short images. To this end we define, for 0 i h, the following exceptional sets (let j 0 = 0): E i = {ω P ji+ Ω i ; ω P ji +k Ω i, τ k j i+ j i, As the ε > 0 in condition (III) we take s.t. ω ω and x ji +k( ω) δ }. ε = (C B )h, h where C is the constant 3C0 2C C 3 /δ. By (III) we derive that there is an integer n ε,τ growing at most polynomially in /ε such that for each 0 i h, E i ε/( n τ) 2, for n n ε,τ. (The need to introduce the integer τ in this subsection is the reason why we require in the formulation of condition (III) that j 2n instead of j n.) If we take n ε,τ (4τ) 2, we get that E i 2ε/n. τ is only dependent on B. By the definition of ε, it follows that n h,b = n ε,τ grows at most exponentially in h. Disregarding finitely many points, Ω i \ E i can

12 2 DANIEL SCHNELLMANN be seen as a set of disjoint and open intervals ω such that each ω is an element of a partition P ji +k Ω i, τ k j i+ j i, and x ji +k( ω) δ. By () and (2), we obtain {a ω ; x ji+ (a) 3B} C B ω, which in turn implies that, for n n h,b, Ω i+ C B Ω i \ E i + E i C B Ω i + 2ε n. Hence, we obtain Ω h (C B ) h Ω 0 + h 2ε n n 3(C B )h, where in the last inequality we used the definitions of Ω 0 and ε. Since this implies (9). {a J ; x j (a) B,..., x jh (a) B} Ω h, 4. Condition (II) As already mentioned in Subsection 2.3, in the examples considered in this paper, we will not verify condition (II) directly. Instead we will verify two other conditions, conditions (IIa) and (IIb) described below. We will show in this section that conditions (IIa) and (IIb) imply condition (II). 4.. Conditions (IIa) and (IIb). Recall that for inequality (3) we assumed that the density ϕ a is bounded from below and from above. We require that this holds for each density ϕ a, a I, and with a constant independent on a. (IIa) There is a constant C 2 such that for each density ϕ a, a I, we have for a.e. x K(a). C 2 ϕ a (x) C 2, Even if condition (IIa) appears to be a natural requirement on a one-parameter family of piecewise expanding maps, it will take us some effort to verify the lower bound for the examples in Sections 5 and 6. The upper bound follows almost immediately from [Wo] and [LaY] (see the proof of Lemma A.). We turn to condition (IIb). Let a, a 2 be two arbitrary parameters in I such that a < a 2 and fix an integer j. We require that, for a.e. y K(a ) the following holds. To each point x K(a ) satisfying T j a (x) = y there is an associated point x K(a 2 ) satisfying T j a 2 (x ) = y and having the same combinatorics as x, i.e. symb a2 (T i a 2 (x )) = symb a (T i a (x)), 0 i < j. In other words, we require that the combinatorics of T a should be contained in the combinatorics of T a2 and, furthermore, if ω P j (a ) and ω P j (a 2 ) have the same combinatorics, then the image by T j a of ω should be contained in the image by T j a 2 of ω. Condition (IIb) is rather restrictive, see Remark 4.. (IIb) For all a, a 2 I, a a 2, and j there is a mapping U a,a 2,j : P j (a ) P j (a 2 ),

13 such that, for all ω P j (a ), TYPICAL POINTS 3 (3) symb a (T i a (ω)) = symb a2 (T i a 2 (U a,a 2,j(ω))), 0 i < j, and, in particular, (4) T j a (ω) T j a 2 (U a,a 2,j(ω)). Remark 4.. A simple example of a one-parameter family not satisfying (IIb) are the maps T a (x) = βx + α mod, β > and α (0, ), where the parameter a can be chosen to be either β or α. These maps are studied, e.g., in [G] and [FP]. However, using the special property that for every fixed map T a the derivative of T a is constantly equal to β, it is possible to verify condition (II) directly (at least in the case when β 2). The main ingredient in verifying condition (II) for this one-parameter family is that one can show that the number of elements in the partitions P j (a), j, are bounded above by a constant C times β j where the constant C is independent on the parameter value a. This property, that one, roughly speaking, can switch from considering the maps T a to counting the number of partition elements, is also used in [Schn]. To keep this paper in a reasonable size we will not investigate this family T a. For the parameter choice a = β, a.s. typicality in the case when X(a) x [0, ] is proven in [FP] Conditions (I), (IIa), and (IIb) imply condition (II). We prove first a distortion lemma. Let T a, a I, be a one-parameter family as described in Subsection 2., satisfying properties (i)-(iii), and let a X(a) K(a) be a to this family associated C map. Let J I be an interval of length /n. If condition (I) is satisfied, then, for large n, the length of J is huge compared to the length of an element ω P n J, which, by (I), can be estimated from above as ω C 0 /λ n. Nevertheless, as the second part of the following lemma shows, the interval J is small enough to have good distortion estimates. Lemma 4.2. There exists a constant C 3 such that the following holds. a) If the one-parameter family T a, a I, with the associated map X satisfies condition (I), then for ω P j I, j, x j (a) C 3 x j (a ) C 3, for all a, a ω. b) If the one-parameter family T a, a I, with the associated map X satisfies in addition to condition (I) also condition (IIb), then we have the following distortion estimate. Let n and a, a 2 I such that a a 2 and a 2 a /n. For ω P j (a ), j 2n, we have C 3 for all x ω and x U a,a 2,j(ω). Ta j (x) Ta j 2 (x ) C 3, Remark 4.3. If a = a 2 in Lemma 4.2 b), then we get a well-known distortion estimate for piecewise expanding C, (L) maps.

14 4 DANIEL SCHNELLMANN Proof. We proof first part b), which is the more difficult part. Fix τ such that max{4l/τ, Lλ/(λ )τ} δ 0. The constant C 3 in Lemma 4.2 b) will be greater than (Λ/λ) τ. Hence, for 2n τ, the distortion estimate in b) is trivially satisfied and we can assume that τ < j 2n. Observe that, by (IIb), the set T j a (ω) is contained in the set T j a 2 (U a,a 2,j(ω)). Fix a point y in T j a (ω) and let, for i j, r i T j i a (ω), s i T j i a 2 (U a,a 2,j(ω)), be the pre-images of y, i.e. T i a (r i ) = T i a 2 (s i ) = y. Note that, by (IIb), we have symb a (r i ) = symb a2 (s i ). Let k i = symb a (r i ). Claim. The distance between r i and s i, i j, satisfies (5) r i s i Lλ λ n. Proof. In order to show (5), we will show (6) r i s i L n i for i j. Since y has for both parameters a and a 2 a pre-image r = Ta (y) and s = Ta 2 (y), which lies in D k, it follows, by (iii) in Subsection 2., that y has a pre-image in D k (a) for all parameter values a in the interval [a, a 2 ] and, furthermore, the corresponding map a Ta is Lip(L). Hence, we have l=0 λ l, (7) r s = T a (y) T a 2 (y) L(a 2 a ) L n. Assume now that we have shown (6) for some i < j. Since a 2 a /n it follows, by (i), that the length of the intersection of D ki+ (a ) and D ki+ (a 2 ) is at least δ 0 2L/n. If z lies in this intersection, then, by (ii), the map a T a (z) is Lip(L) on the interval [a, a 2 ]. Thus, the length of the intersection of T a (D ki+ (a )) and T a2 (D ki+ (a 2 )) is at least δ 0 4L/n. Since, by the assumption on τ, δ 0 4L/n δ 0 and r i s i Lλ/(λ )n δ 0, we deduce that at least one of the following two situations occurs: The branch of Ta interval [r i, s i ]. The branch of Ta 2 interval [r i, s i ]. which maps r i to D ki+ (a ) is defined on the whole which maps s i to D ki+ (a 2 ) is defined on the whole Assuming the first situation occurs, we obtain, by (), and, as in (7), we derive that T a (r i ) T a (s i ) λ r i s i, T a (s i ) T a 2 (s i ) L n.

15 It follows that TYPICAL POINTS 5 r i+ s i+ = T a (r i ) T a 2 (s i ) λ r i s i + L n L n We can do a similar calculation when the second situation occurs, which concludes the proof. By a similar reasoning as in the proof of (5), we note that at least one of the following two situations occurs: [r i, s i ] D ki (a ). [r i, s i ] D ki (a 2 ). If the first situation occurs, we have, by (ii), that the map a T a (s i) is Lip(L) on the interval [a, a 2 ]. Combined with (5) and since x T a (x) is C, (L) on [r i, s i ], we obtain i k=0 λ k. T a (r i ) T a (s i ) + L r i s i T a 2 (s i ) + L(a 2 a ) + L r i s i T a 2 (s i ) + 2Lλ λ n. If the second situation occurs, it follows in a similar way that T a 2 (s i ) T a (r i ) 2Lλ λ n. For τ < i j, let t i = r i and α i = a if the first situation occurs and t i = s i and α i = a 2 otherwise. Altogether, we obtain Ta j ( ) (x) τ Λ j Ta j 2 (x ) T a (Ta j i (x)) λ T i=τ+ a 2 (Ta j i 2 (x )) ( ) τ Λ j T a (r i ) + L Ta j i (ω) λ T a 2 (s i ) L Ta j i 2 (U a,a 2,j(ω)) (8) i=τ+ ( ) τ Λ j λ i=τ+ T α i (t i ) + 2Lλ(λ ) n + Lλ i T α i (t i ) 2Lλ(λ ) n Lλ i. (To ensure that the denominators are positive, we should also assume that τ was chosen so large that 2Lλ(λ ) τ + Lλ τ < λ.) Since j 2n, the product in the last term of inequality (8) is clearly bounded above by a constant independent on n. Hence, this shows the upper bound in the distortion estimate in part b). The lower bound is shown in the same way. This concludes the proof of part b). The proof of part a) is similar but easier than the proof of part b). We will give only a sketch of the proof. Let ω P j I, j, and a, a ω. By condition (I), we have x j (a) j x j (a ) C2 T a (x i(a)) 0 T a (x i (a )). The distance between x i (a) and x i (a ), i j, satisfies, by (I), i=0 x i (a) x i (a ) x i (ω) C 2 0 λ (j i).

16 6 DANIEL SCHNELLMANN This inequality is a counterpart to inequality (5), which is the part in b) where we used condition (IIb). Similarly as in the proof of part b) we derive x j (a) x j (a ) C2 0 ( ) τ j τ Λ T α i (t i ) + 2LC0 2λ (j i) λ T i= α i (t i ) 2LC0 2λ (j i), where either α i = a and t i = x i (a) or α i = a and t i = x i (a ), and τ is chosen so large that 2LC0 2λ τ < λ. The product in this inequality is clearly bounded above by a constant independent on j. This concludes the proof of Lemma 4.2. To prove Lemma 4.2 b), instead of property (4) in condition (IIb), it would be sufficient to assume that dist(t j a (ω), T j a 2 (U a,a 2,j(ω))) /n. However, to establish inequality (4) in condition (II), property (4) is essential as we will see in the proof of the following proposition. Proposition 4.4. If the one-parameter family T a, a I, with the associated map X satisfies conditions (I), (IIa), and (IIb), then it satisfies also condition (II). Proof. Let J I be an open interval of length /n. We assume first that the right endpoint a J of J lies in I. As in condition (II), let ω P i J, i, and j n. The idea of the proof is simple: Using condition (IIb), which provides us with maps between partitions associated to the dynamical interval, we can define in a canonical way a map from the partition P i+j ω associated to the parameter interval to the partition P j (a J ) associated to the dynamical interval. Applying the distortion estimate in Lemma 4.2 b), property (4), and inequality (3), this will then imply condition (II). We define the map U ω,aj,j : P i+j ω P j (a J ) as follows. Let ω P i+j ω and a ω. By the definition of the partitions associated to the parameter interval, x i+l (a) / {c (a),..., c p (a)}, for all 0 l < j. Hence there exists an element ω(x i (a)) in the partition P j (a) containing the point x i (a). We set U ω,aj,j(ω) = U a,aj,j(ω(x i (a))), where U a,aj,j : P j (a) P j (a J ) is the map given by (IIb). Note that ω = U a,aj,j(ω(x i (a))) has the same combinatorics as ω, i.e. symb aj (T l a J (ω )) = symb(x i+l (ω)), 0 l < j. Since there cannot be two elements in P j (a J ) with the same combinatorics the element ω is independent on the choice of a ω. It follows that the map U ω,aj,j is well-defined. Further, since two disjoint elements in P i+j ω cannot have the same combinatorics (which can easily be shown by induction using condition (I)), the map U ω,aj,j : P i+j ω P j (a J ) is one-to-one. Since j n and a J a /n, we derive from the distortion estimate in Lemma 4.2 b) that (9) Ta j (x) C 3 Ta j J (x ),

17 TYPICAL POINTS 7 for x ω(x i (a)) and x U ω,aj,j(ω). By property (4) in condition (IIb), we have T j a (ω(x i(a))) T j a J (U ω,aj,j(ω)), which implies x i+j (ω) T j a J (U ω,aj,j(ω)). Thus, for each ω P i+j ω, whenever there is a parameter value a ω such that x i+j (a) = y, then there is also a point x U ω,aj,j(ω) satisfying T j a J (x) = y. Altogether, we obtain a ω x i+j (a)=y T j a (x i (a)) C 3 x K(a J ) T j a J (x)=y T j a J (x), for all but finitely many y [0, ] (we exclude points y such that x i+j (a) = y for some a ω not contained in any element of P i+j ω). Applying inequality (3) to the right hand side of this inequality, this implies condition (II) with the constant C = C 2 2 C 3. Since C does not depend on a J, we can drop the assumption, made in the beginning of the proof, that the right boundary point of J is contained in I. Thus, this concludes the proof. For the later use, in Subsection 6.4, we notice the following fact derived from the proof of Proposition 4.4. Corollary 4.5. Under the assumption that conditions (I) and (IIb) are satisfied, there is an integer q such that the following holds. Let J I be an interval of length /n such that the right boundary point a J of J is contained in I. If ã I is a parameter such that ã a J and ã a J /n, then, for j 2n, there is an at most q-to-one map U J,ã,j : P j J P j (ã), such that, for ω P j J, we have Ta j (X(a)) C2 3 T j ã (x), for all a ω and x U J,ã,j (ω). Proof. Note that as long as the parameter value ã is greater or equal than a J, the definition of the map U ω,aj,j in the proof of Proposition 4.4 still works if the parameter value a J is replaced by ã. Further, the definition of this map works also for the case i = 0 where we obtain a map U J,ã,j : P j J P j (ã), such that, for ω P j J and ω = U J,ã,j (ω), we have symbã(t l ã(ω )) = symb(x l (ω)) 0 l < j, and x j (ω) T j ã (ω ). In contrast to the case when i, this map is not necessarily one-to-one, since the partition P J might contain more than one element. If q = #{ω P I} then the map U J,ã,j : P j J P j (ã) is at most q-to-one. Let ω P j J, j 2n. If ã a J /n, we can apply Lemma 4.2 b) two times and get, similarly as in (9), that Ta j (X(a)) C2 3 T j ã (x),

18 8 DANIEL SCHNELLMANN for all a ω and x U J,ã,j (ω). 5. β-transformation We apply Theorem 2.2 to a C, (L)-version of β-transformations. Let the map T : [0, ) [0, ] be piecewise C, (L) and 0 = b 0 < b <... be the associated partition, where b k as k. We assume that: a) T is right continuous and T(b k ) = 0, for each k 0. b) For each a >, < inf x T(ax) and sup x T(ax) <. x [0,] x [0,] See Figure. We define the one-parameter family T a : [0, ] [0, ], a >, by T a (x) = T(ax). There exists a unique a.c.i.p. µ a for each T a as the following lemma asserts. Lemma 5.. For each a > there exists a unique a.c.i.p. µ a for T a. The support K(a) is an interval adjacent to 0 and its length K(a) is piecewise constant in a. Furthermore, the following holds. Let I (, ) be a parameter interval on which K(a) is constant and such that the left endpoint of I does not coincide with or a point of discontinuity for a K(a). There exists an integer t (independent on the parameter value a I), such that the support K(a), a I, is obtained by iterating t times the interval of monotonicity adjacent to 0, i.e. K(a) = closure{t t a ((0, b /a))}. The proof of Lemma 5. is not difficult but tedious. For completeness we add the proof in the end of this section. Henceforth in this section, I (, ) will always denote an interval as described in Lemma 5. and such that, for a I, the number of discontinuities of T a inside K(a) is constant, i.e. the number #{k ; b k /a int(k(a))} is constant on I. For a fixed interval I it is now straightforward to check that the one-parameter family T a, a I, fits into the model described in Subsection 2. fulfilling properties (i)-(iii). Now, we can state the main result of this section. Theorem 5.2. If for a C map a X(a) K(a), a I, condition (I) is satisfied, then X(a) is typical for µ a for a.e. a I. Remark 5.3. As the family T a we could also consider other models as, e.g., x ag(x) mod where g : [0, ] [0, ] is a C, (L) homeomorphism with a strict positive derivative. Even if this model is not included in the families described above, it would be easier to treat since, seen as a map from the circle into itself, it is non-continuous only in the point 0 which, in particular, implies that K(a) = [0, ]. By Theorem 2.2 and Proposition 4.4, in order to proof Theorem 5.2, it is sufficient to check conditions (IIa), (IIb) and (III). As we will show in the following subsection, there is a large class of maps X satisfying condition (I): Corollary 5.4. If X : (, ) (0, ] is C such that X (a) 0, then X(a) is typical for µ a for a.e. a >.

19 Remark 5.5. Observe that the map TYPICAL POINTS 9 X(a) lim T(x), x b k a >, satisfies X(a) > 0 and X (a) 0, and, hence, Corollary 5.4 can be applied to these from a dynamical point of view important values. By disregarding a countable number of points (the discontinuity points of K(a) and the values a such that b k /a K(a) for some k ), we can cover (, ) with a countable number of intervals I as described above. Thus, in order to prove Corollary 5.4, it is sufficient to verify condition (I) for the family T a restricted to a parameter interval I. Henceforth, we will use the notation of Subsection 2. related to the family T a, a I. We assume that the monotonicity domains D k (a), k p 2, are ordered, i.e. D k (a) = (c k (a), c k (a)), for k p 2 (= p ). First we will prove Corollary Proof of Corollary 5.4. Note that if the map X in Corollary 5.4 satisfies X(a) / int(k(a)), then the partition P j I would be empty for all j and, hence, condition (I) is not fulfilled. However, the following calculations in this subsection (see, in particular, (20)) show that, for j, the derivative of x j exists and is strictly positive for all but a finite number of points a I. Combined with property (2), we derive that, disregarding a countable number of points, we can cover I by a countable number of intervals such that for each such interval J there is an integer j 0 such that x j J is C and x j (a) int(k(a)) for each a J. Thus, we can without loss of generality assume that X(a) int(k(a)) for all a I since, otherwise, we can focus on the above described smaller intervals J where we redefine X as x j and I as J. Let j and ω P j I. For a ω we have and, hence, we derive x j(a) = D a T(ax j (a)) = T (ax j (a))(x j (a) + ax j (a)) = T a (x j (a))(x j (a)/a + x j (a)), j x j (a) = i=0 Ta j i (x i (a)) x i(a) + Ta j (X(a))X (a), a (recall that x 0 (a) = X(a)). Furthermore, we obtain x j(a) T j a (X(a)) = j i=0 x i (a) + X (a). Ta i (X(a)) a Let κ = inf a I X(a) and M = sup a I X (a). By the assumptions on I and X, we have κ > 0 and M <. Thus, for a ω, (20) κ a I x j(a) T j a (X(a)) j i=0 λ i + M λ λ + M, where a I denotes the right boundary point of I. This provides us with a lower and an upper bound in (I). It is only left to show that the number of parameters a I not contained in any element of the partition P j I is finite. We show this by induction. Note that

20 20 DANIEL SCHNELLMANN the discontinuity points c k (a), k p, are equal to b k /a (the partition points c 0 (a) 0 and c p (a) are constant) and, thus, strictly decreasing in a. Since X (a) 0 and X(a) int(k(a)), for all a I, it follows that the number of parameters a I such that X(a) = c k (a) for some 0 k p is finite. Hence, the number of parameters a I not contained in any element of P I is finite. Let j and assume that the number of a s not contained in any element of P j I is finite. Let ω P j I. By the first inequality in (20), it follows that x j(a) > 0, a ω. Since the partition points c k (a), 0 k p, of T a are decreasing or constant in a it follows that the number of a ω such that x j (a) = c k (a), 0 k p, is finite. We derive that the number of parameters a I not contained in any element of the partition P j+ I is finite. This concludes the proof of Corollary Condition (IIa). By Lemma A., we can without loss of generality assume that there is a constant C = C(I) such that for each a I the density ϕ a is bounded from above by C and, further, there exists an interval J(a) of length C such that ϕ a restricted to J(a) is bounded from below by C (otherwise, disregarding at most countably many points, by Lemma A., we can cover the interval I by a countable number of smaller intervals on each of which this is true and then proceed with these intervals instead of I). To conclude the verification of condition (IIa) it is left to show that there exists a lower bound for ϕ a on the whole of K(a). To make the definition of the intervals J i (a) below work, we assume that the interval J(a) is closed to the left. Recall that, by (i), c k (a) > c k (a) + δ 0, k p, for some constant δ 0 = δ 0 (I) > 0. Let ε = min{(λ )/2C, λδ 0 } and take l so large that λ l /2C >. We claim that [0, ε) Ta(J(a)). l Let J 0 (a) = J(a) and assume that we have defined the interval J i (a) J(a), i, where J i (a) is a (not necessarily maximal) interval of monotonicity for Ta i. If [0, ε) Ta i(j i (a)), we stop and do not define J i (a). If [0, ε) is not contained in Ta i(j i (a)) then, since J i (a) is a monotonicity interval for Ta i and by the definition of ε (combined with property (i) and property a) of T a ), it follows that there can lie at most one partition point c k (a) in the image Ta i (J i (a)). If there is no partition point in this image then we let J i (a) = J i (a), which is in this case also a monotonicity interval for Ta. i If there is a partition point c k (a) Ta i (J i (a)), then we define J i (a) J i (a) to be the interval of monotonicity for Ta i such that Ta i (J i (a)) = Ta i (J i (a)) [0, c k (a)). Note that Ta i (J i (a)) [c k (a), ] < ε/λ, since otherwise we would have [0, ε) Ta i(j i (a)). Assuming that J l (a) is defined, we obtain T l a (J l(a)) λ( T l a (J l (a)) ε/λ) λ l J 0 (a) ε λl λ λ l (/C /2C) λ l /2C >, where we used the definitions of ε and l. Since J l (a) is a monotonicity interval for Ta l, this is a contradiction and it follows that the maximal integer i 0 such that J i (a) is defined is strictly smaller than l. Hence, Ta l (J(a)) contains [0, ε) as claimed above. This immediately implies that there is an integer l independent on the parameter a I such that [0, c (a)) Ta l (J(a)).

21 TYPICAL POINTS 2 Combined with Lemma 5. we derive that there is an integer j independent on a I such that, K(a) = closure{ta j (J(a))}. Now, by the Perron-Frobenius equality, it follows that, for a I, (2) ϕ a (y) ϕ a (x) Ta j (x) CΛ, j x J(a) T j a(x)=y for a.e. y K(a). This concludes the proof of a lower bound for ϕ a on the whole of K(a) Condition (IIb). We can verify condition (IIb) by induction over j. Let a, a 2 I such that a a 2. If k < p 2 then we have T a (D k (a )) = T a2 (D k (a 2 )). The point c p (a) (b p /a, b p /a) is constant since the length K(a) is constant. It follows that T a (c p (a)) is increasing in a, which implies that T a (D p2 (a )) T a2 (D p2 (a 2 )). Hence, (IIb) holds for j =. Assume that (IIb) holds for j. Let ω P j (a ) and ω = U a,a 2,j( ω) the corresponding element in P j (a 2 ). Note that the image by Ta, i i, of an element in P i (a) is always adjacent to 0. Since Ta j ( ω) Ta j 2 ( ω ) and the c k (a) s are decreasing (or constant in the case k = 0 and k = p ), it follows immediately that for every element ω P j+ (a ) ω there is a unique element ω P j+ (a 2 ) ω fulfilling symb a (Ta i (ω)) = symb a2 (Ta i 2 (ω )), 0 i < j +, and Ta j+ (ω) Ta j+ 2 (ω ). Defining U a,a 2,j+(ω) = ω shows that (IIb) holds also for j Condition (III). Let J I be an open interval of length /n and j 2n. First, we make the following observation. Claim. There is a constant C < such that, for all κ > 0, (22) {a J ; x j (a) κ} Cκ, for all j. Proof. By (I), we have {a J ; x j (a) κ} = [0,κ] a J x j (a)=y x j (a) dy C 0 [0,κ] a J x j (a)=y T j a (X(a)) dy. Since conditions (I), (IIa), and (IIb) are satisfied, by Proposition 4.4, it follows that condition (II) is satisfied. Let q = #{ω P I}. If j = then the sum in the last integral is obviously bounded by q/λ. For j 2 we have, by condition (II), a J x j (a)=y Ta j (X(a)) = ω P J for a.e. y [0, ]. We deduce that a ω x j (a)=y T j a (x (a)) T a (X(a)) qc λ, {a J ; x j (a) κ} qc 0C κ. λ

22 22 DANIEL SCHNELLMANN As the constant δ in condition (III) we take the constant δ 0 from property (i) in Subsection 2.. Observe that if ω P l J, l, is an element not adjacent to a boundary point of J then, by property a) of T and the definition of the partition P l J, its image x l ( ω), is adjacent to 0 (if X : I K(a) is the map in Corollary 5.4, then sign(x j (a)) = + and we just have to assume that ω is not adjacent to the left boundary point of J). This implies: (A) If #{ω P l+ ω} 2 then x l (a) = b /a, for some a ω, and thus, by property (i), we have x l ( ω) δ 0. Let ω L and ω R be the elements in P j J adjacent to the boundary points of J. Assume that ω ω L ω R is an element in P j+[ n] J such that x j+[ n] (ω) < δ 0. If there exists an element ω P j+k J, 0 k < [ n], containing ω but being different from ω then, by (A), x j+k ( ω) δ 0 where k is chosen maximally such that ω P j+k J. If there exists no such an element ω, then ω is also an element of the partition P j J and in particular, by (I), x j (ω) < C 2 0 δ 0λ [ n]. Since for every element ω P j J, ω ω L, ω R, its image x j (ω) is adjacent to 0, the exceptional set E defined in (III) is contained in By (22), {a J ; x j (a) < C 2 0δ 0 λ [ n] } (E (ω L ω R )). {a J ; x j (a) < C0δ 2 0 λ [ n] } CC2 0 δ 0. λ [ n] Regarding the set E ω where ω is either ω L or ω R, we derive from property (A) that #{ω P j+[ n] ω ; ω E} [ n] +. Hence, by (I), we obtain x j (E ω ) C 2 0 and, by Lemma 4.2 a), we finally get Altogether, this implies E ω C 2 0 C 3 E const [ n] λ [ n] [ n] +, λ [ n] [ n] +. λ [ n] ε n, for n greater than some n ε, which can in fact be chosen to grow less than polynomially in /ε Proof of Lemma 5.. For a > let µ a be an a.c.i.p. for T a with support K(a) and let J K(a) be an open interval. Since T a is expanding there exists an integer j such that T j a : J [0, ] is not any longer continuous. It follows that there exists an ε > 0 such that T j a (J) contains [0, ε). If T a had more than one a.c.i.p. then, by [Wo], there would exists two a.c.i.p. s with disjoint supports (disregarding a finite number of points). This shows that the a.c.i.p. µ a is unique. For each a > we define a number y(a) (0, ] and an integer t(a). The number y(a) will be the right boundary point of the support K(a) and t(a) will be