On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors

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1 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors Persson, Tomas Published in: Preprints in Mathematical Sciences 2008 Link to publication Citation for published version (APA): Persson, T. (2008). On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors. Unpublished. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. L UNDUNI VERS I TY PO Box7 2200L und

2 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors Tomas Persson Preprints in Mathematical Sciences 2008:3 Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM

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4 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors Tomas Persson January 2, 2008 With 3 figures. Abstract We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is estimated from below in terms of subshifts of finite type contained in the shift space. Some explicit esimates are done for a specific class of maps. Introduction A general class of piecewise hyperbolic maps was studied by Pesin in [8]. Pesin proved the existence of srb-measures and investigated their ergodic properties. Results from Pesin s article and Sataev s article [9] are described in Section 2. The assumptions in [8] and [9] did not allow overlaps of the images. Schmeling and Troubetzkoy extended in [0] the theory in [8] to allow maps with overlaping images. Using the results of Pesin and techniques from Solomyak s paper [], the author of this paper proved in [6] and [7] that for two classes of piecewise affine hyperbolic maps, there exists, for almost all parameters, an invariant measure that is absolutely continuous with respect to Lebesgue measure, provided that the map expands area. The main difficulty that arises for these classes of maps is that in difference from the fat baker s transformation the symbolic space associated to the systems, changes with the parameters, and also the srb-measure changes in a way that is hard to control. By embedding all symbolic spaces into a larger space it was possible get sufficient control to prove the result. Solomyak s proof in [] uses a transversality property of power series. The proofs in [6] and [7] uses that iterates of points under the maps can be written as power series with such a transversality property. For the possibility of writing iterates as power series, it is important that the directions of contraction is maped onto each other throughout the manifold. The method in [6] and [7] is therefore not good for proving similar results for more general maps. It should also be noted that this method only gives results that holds for almost every map, with respect to some parameter.

5 2 2 Piecewise Hyperbolic Maps Tsujii studied in [2] a class of area-expanding solenoidal attractors and proved that generically these systems has an invariant measure that is absolutely continuous with respect to Lebesgue measure. Tsujii also used a transversality condition, but in a different way. Instead of transversality of power series, Tsujii used transversality of intersections of iterates of curves. This technique makes it possible to show the existence of an absolutely continuous invariant measure for a fixed system, provided that the appropriate transversality condition is satisfied. Tsujii proved that this transversality condition is generically satisfied. In this paper we will use the method from Tsujii s article [2] to estimate the dimension of the attractor from below for some piecewise hyperbolic maps and show how this estimate can be applied to a particular class of systems. In Section 2 we present the general theory of piecewise hyperbolic maps. In Section 3 we introduce a transversality condition. Under the assumtion that this transversality condition holds, a theorem that estimates the dimension from below is stated in Section 4. This theorem is proved in Section 6 and Section 5 contains explicit examples of maps that satisfy the asumptions of this theorem. 2 Piecewise Hyperbolic Maps The first systematical study of piecewise hyperbolic maps was Pesin s article [8]. He studied maps of the following form. Let M be a smooth Riemannian manifold with metic d, let K M be an open, bounded and connected set and let N K be a closed set in K. The set N is called the discontinuity set. Let f : K \ N K. Put K + = { x K : f n (x) N K, n = 0,, 2,... }, D = n N f n (K + ). The attractor of f is the set Ä = D. The maps studied in [8] were assumed to satisfy the following conditions. f : K \ N f (K \ N ) is a C 2 -diffeomorphism. (A) There exists C > 0 and 0 such that (A2) d 2 xf Cd(x, N + ), x K \ N, d 2 x(f ) Cd(x, N ), x f (K \ N ), where N + = N K and N = { y K : z n, z N + : z n z, f (z n ) y }.

6 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 3 One might want to think of N as the image of N + although f is not defined on N +. For > 0 and l =, 2,..., let (A3) D +,l = { x K + : d(f n (x), N + ) l e n, n N }, D,l = { x Ä : d(f n (x), N ) l e n, n N }, D 0 = l (D +,l D,l ). The set D 0 is not empty for sufficiently small > 0. The attractor is called regular if (A3) is satisfied. For a given map, it is usually not apperent whether the condition (A3) is satisfied or not. There exist however conditions that implies (A3) and are such that it easily can be checked if they hold true. These conditions are given in the end of this section. There exists C > 0 and 0 < Ð < such that for every x K \ N + there exists cones C s (x), C u (x) T x M such that the angle between C s (x) and C u (x) is uniformly bounded away from zero, (A4) d x f (C u (x)) C u (f (x)) x K \ N +, d x (f )(C s (x)) C s (f (x)) x f (K \ N + ), and for any n > 0 d x f n (v) CÐ n v, d x f n (v) CÐ n v, x K +, v C u (x), x f n (K + ), v C s (x). The last assumption makes it possible to define stable and unstable manifolds, W s (x) and W u (x) as well as local ones for any x D 0. The condition There exists a point x D 0 and C, t, 0 > 0 such that for any 0 < < 0 and any n 0 (A3 ) Ò u (f n (U (, N + ))) < C t, where Ò u is the measure on the local unstable manifold of x, induced by the Riemannian measure, and U (, N + ) is an open - neigbourhood of N +. implies condition (A3). Pesin proved the following theorem.

7 4 2 Piecewise Hyperbolic Maps Theorem 2. (Pesin [8]). Assume that f satisfies the assumptions (A) (A4) and (A3 ). Then there exists an f-invariant measure Ñ such that Ä can be decomposed Ä = i N Ä i where Ä i Ä j =, if i j, Ñ(Ä 0 ) = 0, Ñ(Ä i ) > 0 if i > 0, f (Ä i ) = Ä i, f Ä i is ergodic, for i > 0 there exists n i > 0 such that (f n i Ä i, Ñ) is isomorphic to a Bernoulli shift. The metric entropy satisfy h Ñ (f ) = Õ i (x) dñ(x), where the sum is over the positive Lyapunov exponents Õ i (x). The measure Ñ in Theorem 2. is called srb-measure (or Gibbs u-measure). For piecewise hyperbolic maps the srb-measures are characterised by the property that their conditional measures on unstable manifolds are absolutely continuous with respect to Lebesgue measure and the set of typical points has positive Lebesgue measure. For a somewhat smaller class of maps Sataev proved in [9] that the ergodic components of the srb-measure (the sets Ä i in Theorem 2.) are finitely many. The maps studied by Pesin and Sataev are all invertible on their images. Schmeling and Troubetzkoy generalised in [0] the results of Pesin to non-invertible maps: If the set K \ N can be decomposed into finitely many sets K i such that f : K i f (K i ) can be extended to a diffeomorphism from K i to f (K i ) (A5) and f satisfies the assumptions (A2) (A4) and (A3 ), then the statement of Theorem 2. is still valid. Note that f (K i ) f (K j ) is allowed to be non-empty so that f : K \ N f (K \ N ) is not a diffeomorphism. Schmeling and Troubetzkoy proved their result by lifting the map and the set K to a higher dimension; Let ˆK = K [0, ], ˆK i = K i [0, ] and ˆ f Ki : (x, t) (f (x), Øt + i/p), i = 0,,..., p, where Ø < and p is the number of sets K i. The map f ˆ is then invertible if Ø is sufficiently small and then f ˆ satisfies the assumptions of Theorem 2., in particular there is an srb-measure ˆÑ on the lifted set ˆK. The projection of this measure to the set K was shown to be an srb-measure of the original map f, in the sence that the set of typical points with respect to the projected measure has positive Lebesgue measure.

8 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 5 It is often hard to check whether (A3 ) holds. It is proved in [0] that if f satisfies (A2), (A4), (A5) and the asumptions (A6) (A8) below, then f satisfies condition (A3 ), and hence also (A3). The sets K and N are unions of finitely many smooth curves such that the angle between these curves and the unstable cones are bounded away from zero. The cone families C u (x) and C s (x) depends continuously on x K i and they can be extend continuously to the boundary. There is a natural number q such that at most L singularity curves of f q meet at any point, and a q > L + where (A6) (A7) (A8) a = inf x K \N inf v C u (x) d x f (v). v 3 A Transversality Condition Let > 0 and 0 < <. We will say that an intersection of two smooth curves and 2 is (, )-transversal if for any balls B and B 2 of radius and centre in and 2 respectively, there exist points x B and x 2 B 2 2 such that the following holds true. If d and d 2 are the induced metrics on and 2 respectively, then the intersection of the open sets B(y, d i (x i, y)), i =, 2, y i B(x i, ) is empty. The symbols B(x, r) denotes the open ball of radius r around x. Note that if and 2 intersect (, )-transversal then the intersection 2 can be empty. Definition 3.. We will say that a piecewise hyperbolic system f : K \N K satisfies condition (T) if there exists numbers, > 0 such that if and 2 are two smooth curves such that every tangent lies in the unstable cone field, and 2 = then the curves f ( ) and f ( 2 ) intersect (, )-transversal. (T) 4 Dimension of the Attractor Consider a map f : K \ N K R 2 that satisfies the conditions (A2), (A4) and (A5) (A8). We denote by Õ s (x) < < Õ u (x) the two Lyapunov exponents at the point x if they exist. If Ä is and ergodic component of the attractor, then the Lyapunov exponents are constant almost everywhere and we write Õ s (x) = Õ s and Õ s (x) = Õ s for almost every x. Let Ä l be an ergodic component of the attractor. We introduce a coding of the system f ˆ : ˆÄl ˆÄ l. If ˆx ˆÄ l then there is a unique sequence ŝ(ˆx) = {i k } k Z such

9 6 5 An Example that ˆ f k (ˆx) K ik for every k Z. We let Ë( ˆÄ l ) be the set of all such sequences, that is Ë( ˆÄ l ) = ŝ( ˆÄ l ). Theorem 4.. Suppose that f : K \ N K R 2 is a piecewise hyperbolic map that satisfies the conditions (T), (A2), (A4) and (A5) (A8). Let Ä be an ergodic component of the attractor. Then the Hausdorff dimension of Ä satisfies dim H Ä + h top(ë finite ) D u D s, where D u = lim sup n D s = lim inf n log sup n log inf n x Ä v = x Ä sup d x (f n )(v), inf d x(f n )(v), v = and Ë finite Ë( ˆÄ ) is a subset of finite type and h top (Ë finite ) denotes the topological entropy of Ë finite. Theorem 4. is proved in Section 6. Note that in [0], it is proved that dim H Ä Õ u /Õ s. Hence, under the assumptions of Theorem 4., dim H Ä satisfies + sup h top(ë finite ) D u D s dim H Ä Õ u Õ s, () where the supremum is over all subshifts of finite type contained in Ë( ˆÄ ). 5 An Example Theorem 4. is not of explicit nature. In this section we give an example of maps satifying the assumptions of Theorem 4., and estimate the supremum in (). Let K = (, ) (, ) be a square. Take < k < and let N = { (x, x 2 ) K : x 2 = kx } be the singularity set. Take Ö 0 and let Ý and Ý 2 be two C 2 functions, such that Ý, Ý 2 < Ö Ý < Ö /2. We take parameters 2 < Ð <, < < 2, a, a 2, b and b 2 such that the map f defined by { (Ðx + a f (x, x 2 ) = + Öx 2 + Ý (x 2 ), x 2 + b ) if x 2 > kx (2) (Ðx + a 2 + Ý 2 (x 2 ), x 2 + b 2 ) if x 2 > kx maps K \ N into K. The case Ö 0, Ý = Ý 2 = 0 and = 2 is threated in [4]. There is a picture of f in Figure. We will use Theorem 4. to prove the following two theorems. Theorem 5.. If a, a 2, b = b 2 = ( ) and (, Ð, k, Ö) { (, Ð, k, Ö) : > 2Ð, Ö 0 }

10 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 7 Figure : A picture of f with Ö = 0., Ý = Ý 2 = 0, =.8, Ð = 0.3, k = 0., a = a 2 = 0 and b = b 2 = 0.8 are numbers such that f : K \ N K, then f : K \ N K defined by (2) has an attractor Ä with dimension + log (, k) log log Ð dim H Ä log log Ð, (3) where (, k) is continuous and (, k) 0 as k 0. Let Ý = Ý 2 = 0, < < 2, 0 < Ð <, a = a 2 = 0 and b = b 2 =. Then if Ö = 0, the attractor is Ä = { (x, x 2 ) : x = 0, x 2 }, and so dim H Ä =. If Ö 0 and > 2Ð then the dimension dim H Ä satisfies the inequalities in (3). The dimension can be made arbitrarily close to 2 by choosing Ð close to. Proof of Theorem 5.. We claim that if > 2Ð and Ö 0 then f satisfies condition (T). Let us prove this claim. It is clear that the cone spanned by the vectors ( ÖÝ Ð, ) and ( Ö + ÖÝ Ð, ) defines an unstable cone family at any point of K \ N. Denote this cone by C u. If K {x 2 > kx } and 2 K {x 2 < kx } are two curves such that if v and v 2 are two tangent vectors of the curves, then v, v 2 C u. The vectors v and v 2 are mapped by d x f to [ Ð Ö + Ý (x u = 2 ) 0 ] [ Ð Ý2 (x v and u 2 = 2 ) 0 ] v 2

11 8 5 An Example respectively. One checks that u is contained in the cone spanned by ( Ö Ý Ð ( Ð) + Ö Ö Ý ), and ( (Ö + Ö Ý ) Ð ( Ð) + Ö + Ö Ý ), and u 2 is contained in the cone spanned by ( Ð Ö Ý ( Ð) + Ö ) Ý, and ( Ð (Ö + Ö Ý ) ( Ð) + Ö ) Ý, The intersection of these two cones is trivial if Ð Ö Ý ( Ð) + Ö Ö Ý Ð > (Ö + Ö Ý ) ( Ð) + Ö Ý > 2Ð. This proves the claim. By Theorem 4. it now follows that + sup h top(ë finite ) log log Ð dim H Ä log log Ð. It remains to estimate the supremum of h top (Ë finite ) where Ë finite is a subshift of finite type contained in the shift Ë generated by the map. Fix all parameters except for k. The map defined by (2) with parameter k will be denoted f k. For each k we let Ë k denote the shift generated by the map f k. Let k 0 > 0 be fixed. For any k such that k < k 0 the maps f k and f k0 coincide on the set K \ { (x, x 2 ) : x 2 > k 0 }. Let k0 be the set of points in Ä such that the orbit has empty intersection with the set { (x, x 2 ) : x 2 k 0 }. We will describe the subshifts of finite type that lie inside k0. For this purpose we can consider the map f k0 instead of f k since they coinside on k0. We note that x { (x, x 2 ) : x 2 k 0 } if and only if Hence k0 = f k0 (x) K k0 = [, ] [ + k 0, k 0 ]. n= f n k 0 ( { (x, x 2 ) Ä : f m k 0 (x, x 2 ) K k0, m 0 } ). Since the dynamics of (f k0, k0 ) is determined by the second coordinate, we are led to study the map g : I I where I = [ ( ), ] and g : x { x ( ), if x > 0, x + ( ), if x 0. Hence g is the restriction of f 0 to the second coordinate, and k0 corresponds to the set k0 = { x I : ( ) + k 0 g n (x) ( ) k 0 }.

12 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 9 We let I = I { x > 0 } and I = I { x < 0 }. For x I we let s(x) denote the sequence {s n } n=0, where gn (x) I sn for all n 0. It is easy to see that S( ) := s(i) = { {s n } n=0 : s( ) < {s n} n=0 s( ) }, where the inequalities are in the sence of the lexicographic order of {, } 0 with larger than. Moreover S k0 ( ) := s( k0 ) = { {s n } n=0 : s( k 0 ) < {s n } n=0 s( k 0 ) }. We note that the natural extension of S( ) to a two-sided infinity shift is the shift Ë 0, and the extension of S k0 ( ) is contained in... The shift S( ) is of finite type if and only if s( ) is periodic. Moreover h top (S( )) = log for any >. We now use that if s ( k 0 ) = s ( 0 0 ) for some 0, then S k0 ( ) = S( 0 ). The fact that the function s is continuous in the product topology of {, } 0, now provides us with the existence of a funcion with the properties in the theorem. This finishes the proof. Let us end this section with an explicit estimate of the attractor of the map in Figure. We will use the notations from the proof of Theorem 5.. For this map, we have If 0 is such that i 0, i,... : = s ( k) =,,,,,,,... j 0, j,... := s 0 ( 0 ) =, (, ). then 0 is the unique positive root of the equation j n = n=0 n. (4) Moreover, s 0 ( 0 ) < s ( k), and this implies that S( 0 ) S k ( ). Hence log (, k) log 0 > log.44. The dimension of the attractor satisfies.93 < dim H Ä <.489. There is a picture of the attractor Ä in Figure 2. Similarly, if we had k = 0 then (, k) = 0 and we get the stronger estimate.328 < dim H Ä <.489. Since (, k) does not depend on Ð, we can estimate the dimension of Ä when =.8, Ð = 0.5 and k = 0., by.270 < dim H Ä <.848.

13 0 6 Proof of Theorem 4. Figure 2: The attractor Ä of the map in Figure. Figure 3: The attractor Ä of the map in Figure, with Ð = 0.5. A picture of this attractor is in Figure 3. We can strengthen these estimates as follows. Since f (K \ N ) ( Ð Ö, Ð + Ö ) (, ), we may consider the seqence s ( (Ð + Ö)k) instead of s ( k) in the estimates above. For the attractor in Figure 2, we get i 0, i,... : = s ( (Ð + Ö)k) =,,,,,,,,.... Hence, if 0 is such that j 0, j,... := s 0 ( 0 ) =,, (,, ). then 0 is the unique positive root of the equation (4) and s 0 ( 0 ) < s ( (Ð+Ö)k). As above we can use this to estimate log (, k) log 0 log.68. This implies that the attractor in Figure 2 satisfies.269 < dim H Ä < Proof of Theorem 4. Assume that f satisfies condition (T) with ( 0, )-intersections. Let ˆ f be the lift of f as described in Section 2. We let ˆÄ denote the attractor of ˆ f. Let Ë = { a {, 2,..., p} Z : ˆx ˆK such that ˆ f k (ˆx) ˆK ak, k Z },

14 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors as defined in Section 4. Then there is an one-to-one correspondance Ö: Ë ˆK, defined in the natural way. Let Ô: ˆK K be the projection Ô(x, y) = x. A cylinder is a set of the form k[a] l := { b Ë : b i = a i, i = k, k +,..., l }. We assume that ˆ f is ergodic with respect to the srb-measure. (If not, we can just take an ergodic component.) We let Ë 0 be a subshift of finite type contained in Ë( ˆÄ). Let a Ë 0. Then there is a unique point ˆx ˆÄ such that ˆx = Ö(a). Let x = Ô(ˆx). Then for any sequence b Ë 0, there is a point ŷ = Ö(c) ˆÄ), such that a =...,a 2, a,a 0, a, a 2,..., c =...,a 2, a,a 0, a, a 2,..., a M, b 0, b,.... That is, ŷ is contained in the unstable manifold of ˆx. Since Ë 0 is of finite type, this can be done for any point x Ä 0 := Ä Ô(Ö(Ë 0 )), with uniform bounds on M. The uniform bound on M provides a bound from below of the length of the unstable manifold around a point x Ä. This proves that there exists a number l 0 > 0 such that there exists a local unstable manifold Wl u 0 (x) of length l 0 around x, for any x Ä. We take l 0 so small that l 0 < 0. Take > 0 and a constant C such that for any n and any x Ä holds. sup x inf x sup d x (f n )(v) Ce (Du+ )n, v = inf d x(f n )(v) C e (Ds )n. v = We estimate the dimension of Ä 0. For this purpose we define measures Ñ n with support in Ä 0. For a cylinder n [a] 0 Ë 0 take a point x( n [a] 0 ) Ô(Ö( n [a] 0 Ë 0 )). Then W u l 0 (x( n [a] 0 )) exists. We let W n be the collection of such unstable manifolds. Let Ñ n = #W n Ò W u l0, Wl u W n 0 where Ò W u l0 is the normalised Lebesgue measure on the set W u l 0. By taking a subsequence we can achieve that Ñ n converges weakly to a measure Ñ with support in Ä 0. We will use the following method, originating from Frostman [5], to estimate the dimension of Ä 0. If dñ(x)dñ(y) x y s <, then dim H Ä dim H Ä 0 s. For a proof of this, see Falconer s book [3].

15 2 6 Proof of Theorem 4. Let M be a number. Then { } min M, x y s dñ n (x)dñ n (y) { max M, } x y s dñ(x)dñ(y), as n, and { } min M, x y s dñ(x)dñ(y) dñ(x)dñ(y), as M. x y s We will therefore estimate E s (n, M) = { } min M, x y s dñ n (x)dñ n (y). It is clear that E s (n, M) M. By the definition of the measure Ñ n we immediately get that E s (n, M) = W W n (#W n ) 2 V W n { } min M, x y s dò V (x)dò W (y). (5) with We rewrite (5) as J = J 2 = (#W n ) 2 W W n W W n V W n V W (#W n ) 2 E s (n) = J + J 2, { min M, } x y s dò W (x)dò W (y), { } min M, x y s dò V (x)dò W (y). To estimate J we note that { } min M, x y s dò W (x)dò W (y) M. Hence and so J 0 as n. J W W n M (#W n ) 2 = M, #W n

16 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 3 We will now estimate J 2. First we estimate that if m = m(w, V ) is the largest number such that n m [a] n = n m [b] n then x y s dò V (x)dò W (y) C 0 e (Du Ds+2 )(s )m(w,v ), (6) where C 0 does not depend on V and W. Indeed, if m = m(w, V ), then V and W intersect ( 0, C 2 e (Du Ds+2 )m )-transversal and we can estimate x y s dò V (x)dò W (y) C 0 x y s dxdy, where and 2 are the curves = { (x, x 2 ) : x = 0, x 2 < l 0 }, 2 = { (x, x 2 ) : x 2 < l 0, x 2 = C 2 e (Du Ds+2 )m x }, and C 0 is a constant, that depend only on the second derivative of the map. To prove (6), one easily checks that there exists a constant C such that C 0 x y s dxdy C e (Du Ds+2 )(s )m. 2 Since Ë 0 is of finite type there is a constant C 2 such that for W W n This yield J 2 = n #{ V W n : m(w, V ) = k } C 2 e htop(ë 0)(n k) k=0 W W n V W n m(w,v )=k n k=0 W W n V W n m(w,v )=k k=0 (#W n ) 2 2 { } min M, x y s dò V (x)dò W (y) C e (Du Ds+2 )(s )m(w,v ) (#W n ) 2 n C C2 2 0)n ehtop(ë e htop(ë0)(n k) e (Du Ds+2 )(s )k e htop(ë 0)2n. This sum is bounded, uniformly over n, if h top (Ë 0 ) > (D u D s + 2 )(s ), or equivalently s < + h top(ë 0 ) D u D s + 2. (7) We conclude that, if s satisfies (7) then the integral { } min M, x y s dñ(x)dñ(y)

17 4 6 Proof of Theorem 4. is uniformly bounded and hence converges as M. This proves that x y s dñ(x)dñ(y) < if (7) holds true. Hence Let 0. dim H Ä dim H Ä 0 + h top(ë 0 ) D u D s + 2.

18 On the Hausdorff Dimension of Fat Generalised Hyperbolic Attractors 5 References [] J. C. Alexander, J. A. Yorke, Fat baker s transformations, Ergodic Theory and Dynamical Systems, 4 (984), 23 [2] P. Erdös, On a family of symmetric Bernoulli convolutions, American Journal of Mathematics, 6 (939), [3] K. Falconer, Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, Chichester, 990, isbn [4] K. Falconer, The Hausdorff Dimension of Some Fractals and Attractors of Overlapping Construction, Journal of Statistical Physics 47 (987), [5] O. Frostman, Potentiel d équilibre et capacité des ensambles avec quelques applications a la théorie des fonctions, Meddelanden från Lunds universitets matematiska seminarium, band 3 [6] T. Persson, A piecewise hyperbolic map with absolutely continuous invariant measure, Dynamical Systems: An International Journal, 2:3 (2006), [7] T. Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Ergodic Theory and Dynamical Systems, [8] Ya. Pesin, Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergodic Theory and Dynamical Systems, 2 (992), 23 5 [9] E. Sataev, Invariant measures for hyperbolic maps with singularities, Russian Mathematical Surveys, 47 (992), 9 25 [0] J. Schmeling, S. Troubetzkoy, Dimension and invertibility of hyperbolic endomorphisms with singularities, Ergodic Theory and Dynamical Systems, 8 (998), [] B. Solomyak, On the random series ±Ð i (an Erdös problem), Annals of Mathematics, 42 (995), [2] M. Tsujii, Fat solenoidal attractors, Nonlinearity 4 (200), Tomas Persson, Centre for Mathematical Sciences, Box 8, se-2200 Lund, Sweden tomasp@maths.lth.se, tomasp

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20 Preprints in Mathematical Sciences 2008:3 ISSN LUTFMA Mathematics Centre for Mathematical Sciences Lund University Box 8, SE Lund, Sweden