Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps
AABSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZEDHORSESHOE MAPS
ABBAS FAKHARI AND MARYAM KHALAJ
Abstract.
In this paper, we study the SRB measures of generalized horseshoe map. We provethat under the conditions of transversality and fatness, the SRB measure is actually absolutelycontinuous with respect to the Lebesgue measure. Introduction
The transversality condition was introduced in the 90s to calculate the Hausdorff dimension ofthe self-similar sets and to find absolutely continuous invariant measures. Roughly speaking, twofamilies F and G of curves in R are transverse if almost all pair ( f, g ) , with f ∈ F and g ∈ G are transversal with uniform slope. Under the transversality condition, Pollicott and Simon [6]computed the Hausdorff dimension of the Cantor setsΛ ( λ ) = { ∞ ∑ k = i k λ k ∶ i k = , } for almost every λ < /
2. Solomyak [11] proved the Lebesgue measure of Λ ( λ ) , for almost every λ > /
2, is positive provided the transversality condition holds. Motivated by this scheme, it isshown that if the transversality condition holds for a certain contracting affine iterated functionsystem in R d , d ≥
2, then the Hausdorff dimension of the attractor is the minimum of d and thesingularity dimension [5].The first motivating classical example in higher dimension is generalized baker map B λ ∶ [− , ] ↺ defined by B λ ( x, y ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩( x − , λy + − λ ) x ≥ ( x + , λy − + λ ) x < . The transversality condition implies that for almost every λ ∈ ( , ] , map B λ admits an absolutelycontinuous ergodic measure [12]. Generalizing Baker maps, Tsujii [13] showed that the SRB measureof any transversal solenoidal attractor T ∶ S × R → S × R defined by T ( x, y ) = ( (cid:96)x, λy + f ( x )) , Mathematics Subject Classification.
Key words and phrases. generalized horseshoe map, fatness, transversality, absolutely continuous invariantmeasure. a r X i v : . [ m a t h . D S ] F e b ABBAS FAKHARI AND MARYAM KHALAJ is absolutley continuous, where f is a C function, (cid:96) ≥ < λ < (cid:96)λ > T ∶ S × R d → S × R d defined by T ( x, y ) = ( f ( x ) , g ( x, y )) , where f is a k to 1 expanding map and g is a contraction.In this paper, we study a class of dynamical systems having the most extended structure called generalized horseshoe maps . The generalized horseshoe map initially defined by Jakobson andNewhouse in [4] to detect the SRB measure in the most general case. The generalized horseshoemap is a piecewise hyperbolic map defined on a countable family of vertical strips. We show thatthe two assumptions of area-expanding and the transversality of the unstable manifolds lead toabsolute continuity of the SRB measure. Theorem A.
For any transversal fat generalized horseshoe map, the SRB measure is absolutlelycontinuous with respect to the Lebesgue measure.
Theorem A has novelties in some ways. The continuity of the map is removed and, unlike theclassical case of skew-product, the boundary of strips have non-zero curvature. These conditionsaccompanied by the infinity of the strips force further calculation for adaptation.The generalized horseshoe map is defined in Section 2. Section 3 is devoted to the proof of theexistence of an SRB measure. The precise definition of the fatness and transversality conditionsare presented in Subsection 4.1. Subsection 4.2 consists of the essential lemmas to control thedistortions. Finally, the absolute continuity of the SRB measure is shown in Subsection 4.3.2.
Generalized Horseshoe Map (GHM)
In this section, we introduce the model we are dealing with in this paper. Suppose that { S , S , ⋯} is a countable collection of closed curvilinear rectangles in S = [ , ] whose interiors are non-overlapping and covering S up to a subset of zero Lebesgue measure. Each S i is full height whoseleft and right boundaries are graphs of smooth functions. Let F i = ( F i , F i ) be a C diffeomorphismon S i and let U i be the image of S i under F i which is full width and bounded by graphs of smoothfunctions from top and bottom. Suppose that for constants 0 < α < K >
1, the map F on S given by F ∣ int S i = F i satisfies the hyperbolicity conditions described in [3], so the following coneconditions hold:(1) DF (C uα ) ⊆ C uα (2) ∣ DF ( v )∣ ≥ K ∣ v ∣ for v ∈ C uα (3) ∣ DF − ( v )∣ ≥ K ∣ v ∣ for v ∈ C sα BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 3 (4) DF − (C sα ) ⊆ C sα where C sα = {( v , v ) ∶ ∣ v ∣ ≤ α ∣ v ∣} and C uα = {( v , v ) ∶ ∣ v ∣ ≤ α ∣ v ∣} , and we use the max normi.e. ∣ v ∣ = ∣( v , v )∣ = max {∣ v ∣ , ∣ v ∣} . The map F with the above conditions is called the generalizedhorseshoe map .Also, for each i , the hyperbolic conditions yield(5) ∣ F i y ( z )∣∣ F i x ( z )∣ ≤ α, (6) ∣ F i x ( z )∣∣ F i x ( z )∣ ≤ α, (7) ∣ F i y ( z )∣∣ F i x ( z )∣ ≤ K + α . Let Σ ∞ N ∶= {( a i ) ∞ i = ∣ a i ∈ N } , [ a ] n ∶= ( a i ) ni = and F [ a ] n ∶= F a n ○ ⋯ ○ F a .The stable and unstable manifolds can be described in two following approaches. ● Analytical Definition.
For any X = ( x n ) n ∈ Z in the inverse limit space ←— M = {( x n ) n ∈ Z , f n ( x n ) = x n + } , put E u ( X ) = ⋂ n ≥ DF n ( x − n )(C uα ( x − n )) ,E s ( X ) = ⋂ n ≥ DF − n ( x n )(C sα ( x n )) .E s ( X ) and E u ( X ) are stable and unstable directions at X . By the definition, E s ( X ) onlydepends on the 0th position of X . By Hadamard-Perron Theorem, directions E s and E u are integrable (see [7] for a complete discussion). ● Geometrical Definition.
For any finite word [ a ] n , put S [ a ] n ∶= F − a ( S a n ∩ F [ a ] n − ( S [ a ] n − )) and U [ a ] n ∶= F [ a ] n ( S [ a ] n ) (see Figure 1). For any infinite word a ∈ Σ ∞ N , put W s a ∶= ⋂ n ≥ S [ a ] n , W u a ∶= ⋂ n ≥ U [ a ] n . Actually, the set Λ ∶= ⋃ a ⋂ n ≥ U [ a ] n defines a topological attractor for F . The stable andunstable manifolds W s a and W u a are graphs of C functions defined in any point of Λ.Note that any point of Λ has a unique stable manifold and, probably, non-unique unstablemanifold. ABBAS FAKHARI AND MARYAM KHALAJ S a S [ a ] n U [ a ] n U a n Figure 1.
Stable and Unstable Strips in a Generalized Horseshoe Map Figure 1.
Stable and Unstable Strips in a Generalized Horseshoe Map3.
SRB measure for GHM
There are classical approaches for finding SRB measures for Anosov endomorphisms which arenot applicable in our case because of the existence of discontinuities in the GHM. Jakobson andNewhouse have already proved that the GHM has an SRB measure [3]. Salas has tried in [10] topresent a shorter proof, however, the proof seems not to be complete.In this section, we provide a short proof for the existence of an SRB measure essentially basedon the absolute continuity of the stable manifolds. Roughly speaking, our method is to find aninvariant measure whose disintegration along unstable manifolds is equivalent to the Lebesguemeasure. First, we study the special case of skew-products inspired by Tsujii in [13] and then, wedeal with the general case in which the stable manifolds may have non-zero curvature.3.1.
Skew-product.
Let { I i } ∞ i = be a partition of I = [ , ] into disjoint subintervals, f ∶ I → I bean expanding map, f i ∶= f ∣ I i be one to one and f ( I i ) = [ , ] . Let E i = I i × [ , ] . Consider the map F ∶ I → I given by F ( x, y ) = ( f ( x ) , g ( x, y )) , where g is of class C on each int ( E i ) , also supposethat there exist positive constants α, β < α < ∣ B g / B y ∣ < β . For each x ∈ I let ω ( x ) bethe fixed index such that x ∈ I ω ( x ) which is unique for almost every x ∈ I . Let π y be the projectionmap defined by π y ( x, y ) = y . Let S ( x, a ) ∶= lim n →∞ π y ○ F n ○ F − [ a ] n ( x, ) , where F i = F ∣ E i . Considerthe maps ˆ F ∶ I × Σ ∞ N → I × Σ ∞ N given byˆ F ( x, a ) = ( f ( x ) , ω ( x ) a ) , where ω ( x ) a is the word ( b i ) ∞ i = such that b = ω ( x ) and b i + = a i for i ∈ N . Let h ∶ I × Σ ∞ N → I isgiven by h ( x, a ) = ( x, S ( x, a )) . BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 5
Clearly, F ○ h = h ○ ˆ F . Now, we define an invariant probability measure for ˆ F . Let p i = ∣ I i ∣ , and so ∑ ∞ i = p i =
1, because ⋃ ∞ i = I i = I . For any A ⊆ I i and cylinders C = [ b , . . . , b n ] = { a ∈ Σ ∞ N ∣ a j = b j , j = , . . . , n } , we define ˆ µ ( A × C ) = Leb ( A ) p b . . . p b n . Let S be the semi-ring generated by the sets A × C , where A ⊆ I i for some i and C is a cylinder. By Carath´eodory Theorem, we can extend ˆ µ to the σ -algebra in I × Σ ∞ N generated by S . To see that ˆ µ is invariant under ˆ F , it is enough to show thatˆ µ ( ˆ F − ( A × C )) = ˆ µ ( A × C ) holds for any A ⊆ I i and C = [ b , . . . , b n ] . We haveˆ µ ( ˆ F − ( A × C )) = ˆ µ (( I b ∩ f − ( A )) × [ b , . . . , b n ]) = Leb ( I b ∩ f − ( A )) p b ⋯ p b n = p b Leb ( A ) p b ⋯ p b n = ˆ µ ( A × C ) . So µ = h ∗ ˆ µ is an invariant probability measure for F and infact µ is an SRB measure for F .3.2. General case.
The strategy of the skew-product case is no longer works. There are twoobservations. The first is the stable manifolds are not straight lines in general. The second one isthat map F is not presented explicitly.3.2.1. Absolute Continuity of Stable Manifolds.
The main tool to find the SRB measure is theabsolute continuity of the stable manifolds which is proved in [3].Let ˜ S = ⋃ i int ( S i ) and define inductively ˜ S n = ˜ S ∩ F − ( ˜ S n − ) , for n ≥
1. Let ˜ S = ⋂ n ≥ ˜ S n . Then˜ S is a F -invariant full Lebesgue measure set. Clearly ˜ S is W s -saturated, where W s is the stablelamination. Let D and D be two disks in S transverse to the stable lamination and H D ,D bethe holonomy map along the stable lamination, that is H D ,D ∶ D ∩ ˜ S → D ∩ ˜ SH D ,D ( z ) ∶= W s ( z ) ∩ D , where W s ( z ) is the unique stable leaf of W s through z . Suppose that m D is the Lebesguemeasure induced by the Riemannian metric on the disk D . Jakobson and Newhouse used somegeometric and distortion conditions to prove the absolute continuity of stable lamination, meaningthat the measure ˜ m D = ( H D ,D ) ∗ m D is equivalent to m D . Hence, there is a measurable map J ∶ D → [ , +∞) which is integrable with respect to ˜ m D such that for any Borel set A ⊆ D , wehave m D ( A ) = ∫ A J d ˜ m D . Lifting to an SRB measure.
Here, we use the classical lifting procedure and absolute conti-nuity of stable lamination to generate an SRB measure for F (see also [2]).Suppose that p s ∶ ˜ S → [ , ] is the projection along the stable leaves and let I i = p s ( S i ) . Define g ∶ ⋃ i I i → [ , ] by g ( x ) ∶= p s ○ F ( x, ) . ABBAS FAKHARI AND MARYAM KHALAJ
In this case, g ( I i ) = [ , ] , for each i , and also p s ○ F = g ○ p s on ˜ S . Since the stable lamination isabsolutely continuous and the stable leaves are of codimension one, Jacobian J of the holonomymap is essentially smooth (see [9]), this leads g to be piecewise expanding. According to Folkloretheorem [1, 14], g has an absolutely continuous invariant probability measure, ACIP, say µ g . Proposition 3.1.
The map F has an SRB measure.Proof. For a given continuous function ψ ∶ S → R , let ψ ∶ [ , ] → R defined by ψ ( x ) = ψ ( x, ) .Then we claim that lim n →∞ ∫ ( ψ ○ F n ) dµ g exists. For given (cid:15) >
0, there exists δ > z and z in S satisfying ∣ z − z ∣ < δ , wehave ∣ ψ ( z ) − ψ ( z )∣ < (cid:15) . Since F is contracting along the stable manifolds, there exists n ≥ n + k ≥ n ≥ n we have ∣ F n + k ( x, ) − F n ○ p s ○ F k ( x, )∣ < δ . Therefore ∣∫ ( ψ ○ F n + k ) dµ g − ∫ ( ψ ○ F n ) dµ g ∣ = ∣∫ ( ψ ○ F n + k )( x, ) dµ g − ∫ ( ψ ○ F n ) ○ g k dµ g ∣= ∣∫ ( ψ ○ F n + k )( x, ) dµ g − ∫ ( ψ ○ F n ○ p s ○ F k )( x, ) dµ g ∣ ≤ (cid:15). The first equality holds by the g -invariance of µ g . So we have shown that {∫ ( ψ ○ F n ) dµ g } is aCauchy sequence in R and it converges. Defineˆ µ ( ψ ) = lim n →∞ ∫ ( ψ ○ F n ) dµ g . Obviously ˆ µ is a linear operator on the space of continuous functions ψ ∶ S → R . Also ˆ µ ( ) = µ is non-negative that is ˆ µ ( ψ ) ≥ ψ ≥
0. So, by Riesz representation theorem, there exists aunique measure called µ F such that for any continuous map ψ ˆ µ ( ψ ) = ∫ ψdµ F . By the definition of ˆ µ , µ F is F -invariant and the disintegration of µ F along any unstable manifold W u is the pullback of µ g by the holonomy map H W u ∶ W u → [ , ] . (cid:3) Absolute Continuity of the SRB Measure
Our approach for proving the absolute continuity of µ F is based on two general assumptions oftransversality and fatness which are appeared in a more specific way in [8, 13]. In this context,some more special assumptions are needed for the simplicity of calculations. Recall that F i ( x, y ) =( F i ( x, y ) , F i ( x, y )) . Put ∣ D F i ( z )∣ ∶= max j = , , ( k,l )=( x,x ) , ( x,y ) , ( y,y ) {∣ F ijkl ( z )∣} . We assume that H0 ) there exists a constant C > i ≥ sup z ∈ S i ∣ D F i ( z )∣ < C , BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 7 H1 ) J F i ∶= F i x F i y − F i y F i x and sup i ≥ J F i < ∞ , H2 ) sup i ≥ sup z ∈ S i ,F i ( w )= z ( F i y ( z ) F i x ( w ))/ F i y ( z ) < ∞ , H3 ) ∣ F i y ( z )∣ , ∣ F i x ( z )∣ < / z ∈ ˜ S and i ≥ H0 , there exists a positive constant C such that for anytwo close points z and w lying on an unstable piece in S i ,(8) ∣ F i x ( z )∣∣ F i x ( w )∣ ≤ exp ( C ) , where C = √ ( + α ) C . Theorem A will be proven by using these general and special assumptions.4.1. Fatness and Transversality Conditions.
Let J be an interval strictly containing I = [ , ] and ˆ S = [ , ] × J . Suppose that ˆ S i and ˆ U i are the neighborhoods of S i and U i in ˆ S respectivelysuch that each F i can be extended to a C hyperbolic diffeomorphism ˆ F i ∶ ˆ S i → ˆ U i which has thesame properties as F i and p s ( ˆ S i ) = I i . For any word a ∈ Σ ∞ N , define ˆ U [ a ] n in a similar way as U [ a ] n . Denote the intersection of U [ a ] n with the stable manifold W s ( x ) through the point ( x, ) by U [ a ] n ( x ) . Define the notation ˆ U [ a ] n ( x ) in a similar way. Let d ([ a ] n ) = max x diam ˆ U [ a ] n ( x ) . Definition 4.1.
The generalized horseshoe F is called fat if there exist K , (cid:15) > such thatdiam I [ a ] n ≤ K ( d ([ a ] n )) + (cid:15) , holds for all [ a ] n ∈ Σ n N , where I [ a ] n = p s ( S [ a ] n ) . To define the transversality condition, we need a bit of notation. For any x ∈ ⋃ I i and a ∈ Σ ∞ N ,put W u a ( x ) ∶= W u a ∩ W s ( x ) . For δ >
0, two words a , b ∈ Σ ∞ N are δ - transversal if d s ( W u a ( x ) , W u b ( x )) > δ or ∣ ddx W u a ( x ) − ddx W u b ( x )∣ > δ holds for almost all x , where d s is the metric induced by the Riemannian metric on the stablemanifold. Two finite words [ a ] n and [ b ] m are δ -transversal if for any u , v ∈ Σ ∞ N , the two infinitewords [ a ] n u and [ b ] m v are δ -transversal. For any r < diam J and a ∈ Σ ∞ N , let n be the biggestnumber satisfying d ([ a ] n ) ≥ r . Let M ( r ) be the set of all such finite words [ a ] n . Put M T rδ ( r ) ∶= {([ a ] n , [ b ] m ) ∈ M ( r ) ; [ a ] n and [ b ] m are δ -transversal } and M NT rδ ( r ) be the complement of M T rδ ( r ) . Definition 4.2.
The map F satisfies the transversality condition if lim δ → lim sup r → r − ∑ ([ a ] n , [ b ] m )∈ M NTrδ ( r ) vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) diam I [ a ] n diam I [ b ] m < ∞ ABBAS FAKHARI AND MARYAM KHALAJ
Control of Distortions.
In the following, we describe the distortion properties of the GHMthrough some propositions and lemmas. The easy case is the control of distortion along the stablemanifolds comes from classical distortion control.
Proposition 4.3 (Bounded Distortion Property Along Stable Manifolds) . There exists K > suchthat for any [ a ] n and any z, w ∈ S belonging to a small stable piece in U [ a ] n , K − < ∣ D sz F − [ a ] i ∣∣ D sw F − [ a ] i ∣ < K for any i = , . . . , n, where D s is the derivative along the stable manifolds. The next proposition is actually Lemma 2.2 in [8]. However, the two generalizations on themodel, the infinity of the strips and the non-zero curvature of stable manifolds force more delicatedetails to prove.
Proposition 4.4.
Suppose that [ a ] n is an arbitrary finite word. Then there exists a constant K such that for almost all x and x in [ , ] we have K − < diam U [ a ] n ( x ) diam U [ a ] n ( x ) < K and similarly for ˆ U [ a ] n . This constant is independent of the choice of [ a ] n . Proposition 4.4 and the following discussion are the key ingredients needed to prove our maingoal. For the proof of Proposition 4.4, we need more subtle details. First, we change the coordinatefrom the standard one to the coordinates induced by splitting E uz ⊕ E sz .To simplify, abuse of notation, denote diffeomorphism F a i ∶ S a i → U a i , for i = , . . . , n by F . For z ∈ ˜ S , and the unstable curve γ through z , consider splitting R = E uz ⊕ E sz , where E uz containsthe unit vector tangent to γ at z and E sz is tangent to the local stable manifold through z . For v ∈ E uz ⊕ E sz , let ∣ v ∣ = ∣ v ∣ z be the max norm which is defined before. Let A z be an affine automorphismof R such that ● A z ( z ) = z ● D z A ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎣ a ( z )⎤⎥⎥⎥⎥⎦ ∈ E uz ● D z A ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎣ b ( z ) ⎤⎥⎥⎥⎥⎦ ∈ E sz . Note that ∣ a ( z )∣ , ∣ b ( z )∣ ≤ α and in this case a ( z ) = α ( z ) F x ( z ) and b ( z ) = β ( z ) F y ( z ) , where ∣ α ( z )∣ , ∣ β ( z )∣ ≤
1. Let ˜ F − be the local representation of F − in this coordinate which means BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 9 ˜ F − = A − F − ( z ) F − A z . Then, the matrix D ˜ F − ( z ) is diagonal. Let D ˜ F − = ⎡⎢⎢⎢⎢⎣ ˜ g x ˜ g y ˜ g x ˜ g y ⎤⎥⎥⎥⎥⎦ = A − F − ( z ) DF − A z . So, for any w ∈ U a i , one gets ● J F J A F − ( z ) ˜ g x ( w ) = F y + b ( F − ( z )) F x − a ( z ) F y − a ( z ) b ( F − ( z )) F x , ● J F J A F − ( z ) ˜ g y ( w ) = b ( z ) F y + b ( z ) b ( F − ( z )) F x − F y − b ( F − ( z )) F x , ● J F J A F − ( z ) ˜ g x ( w ) = − a ( F − ( z )) F y − F x + a ( z ) a ( F − ( z )) F y + a ( z ) F x , ● J F J A F − ( z ) ˜ g y ( w ) = − a ( F − ( z )) b ( z ) F y − b ( z ) F x + a ( F − ( z )) F y + F x ,where J A F − ( z ) = − a ( F − ( z )) b ( F − ( z )) and the partial derivatives of F and F are evaluated at A z ( w ) . Lemma 4.5.
Under the notations above, there exist positive constants C , C and C such thatfor any point w close to z lying on the same unstable curve, we have (9) ∣ ˜ g x ( z )∣ ≤ C , (10) ∣ ˜ g y ( w )∣ ≥ C , (11) ∣ ˜ g x ( w )∣∣ ˜ g y ( w )∣ ≤ C . These constants are independent of the choice of z and w . Also, (12) ∣ ˜ g x ( w )∣∣ ˜ g y ( w )∣ ≤ K . Proof.
Due to the above conditions and the explicit formula of ˜ g x , it only suffices to estimate thevalue of H ( z ) ∶= a ( z ) b ( F − ( z )) F x ( z ) J F ( z ) J A F − ( z ) . As mentioned before, ∣ a ( z )∣ ≤ ∣ F x ( z )∣ and ∣ b ( z )∣ ≤ ∣ F y ( z )∣ . So, ∣ H ( z )∣ ≤ ∣( F y ( z ) F x ( F − ( z ))) F x ( z )∣∣( F x ( z ) F y ( z ) − F y ( z ) F x ( z ))( − F y ( F − ( z )) F x ( F − ( z )))∣ ≤ ∣ F y ( z ) F x ( F − ( z ))∣∣ F y ( z )∣ , the last term is bounded due to H2 . This proves (9).For (10) and (11), we claim that ∣ ˜ g y ( w )∣ ≥ C ∣ F x ( z )∣ , for some positive constant C . First, notethat D A z F − ( A z ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ ) belongs to the stable cone C sα and so is a multiple of some vector ⎡⎢⎢⎢⎢⎣ b ⎤⎥⎥⎥⎥⎦ for ∣ b ∣ ≤ α . Thus, D w ˜ F − ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎣ ˜ g y ( w ) ˜ g y ( w )⎤⎥⎥⎥⎥⎦ is a multiple of ⎡⎢⎢⎢⎢⎣ b − b ( F − ( z ))− ba ( F − ( z )) + ⎤⎥⎥⎥⎥⎦ , and hence, RRRRRRRRRRRR⎡⎢⎢⎢⎢⎣ ˜ g y ( w ) ˜ g y ( w )⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR = max {∣ ˜ g y ( w )∣ , ∣ ˜ g y ( w )∣} ≤ ∣ ˜ g y ( w )∣ max { α − α , } . Since A F − ( z ) is uniformly bounded, using (6) and (8), one gets RRRRRRRRRRRR D w ˜ F − ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≥ K RRRRRRRRRRRR D A z ( w ) F − ⎡⎢⎢⎢⎢⎣ b ( z ) ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR≥ K ∣ J F ( z ) (− b ( z ) F x ( A z ( w )) + F x ( A z ( w )))∣≥ K ′ (∣ F x ( A z ( w ))∣ − α ∣ F x ( A z ( w ))∣) ≥ K ′′ ∣ F x ( z )∣ , for some positive constants K , K ′ and K ′′ . This proves (10), because inf z ∈ ˜ S ∣ F x ( z )∣ >
1. On theother hand, by using (5), (6), (7) and (8) in the explicit formula of ˜ g x ( w ) , ∣ ˜ g x ( w )∣ ≤ C ( α )∣ F x ( A z ( w ))∣ ≤ ¯ C ∣ F x ( z )∣ . This proves (11). Using (2) and (3), one gets ∣ ˜ g x ( z )∣∣ ˜ g y ( z )∣ ≤ K , so for w sufficiently close to z , equation (12) holds. (cid:3) Now we are ready to prove Proposition 4.4.
Proof of Proposition 4.4.
Fix [ a ] n . Let z ∈ U [ a ] n ( x ) ∩ ˜ S be an arbitrary point and γ be a C unstable curve through z . Let w = γ ∩ U [ a ] n ( x ) . There exist τ n ∈ U [ a ] n ( x ) and τ n ∈ U [ a ] n ( x ) such that for j = ,
2, the following holds:diam F − [ a ] n ( U [ a ] n ( x j )) = ∣ D s F − [ a ] n ( τ jn )∣ diam U [ a ] n ( x j ) . So, by Proposition 4.3, it suffices to show that(13) ¯ K − < ∣ D sz F − [ a ] n ∣∣ D sw F − [ a ] n ∣ < ¯ K holds for some ¯ K . Let z i = F − [ a ] i ( z ) and w i = F − [ a ] i ( w ) , for i = , . . . , n . As before, we use the affinecoordinates induced by splitting R = E uz i ⊕ E sz i at z i , where E uz i contains the tangent vector to F − [ a ] i ( γ ) at z i and E sz i is tangent to the stable manifold at z i . Let ˜ F − a i be the representation of F − a i BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 11 S a S [ a ] n U [ a ] n U a n Figure 1.
Stable and Unstable Strips in a Generalized Horseshoe Map z i w i v w i v z i F − [ a ] i ( γ ) Figure 2.
Position of z i , w i , v z i and v w i ˆ U [ a ] n ˆ U [ b ] m Figure 3.
Transversality of ˆ U [ a ] n and ˆ U [ b ] m Figure 2.
Position of z i , w i , v z i and v w i in this coordinates and ˜ B i be the small parallelogram centered at z i using the max norm. One hasthat ∣ D sz F − [ a ] n ∣∣ D sw F − [ a ] n ∣ ≤ const. n ∏ i = ∣ D z i ˜ F − a i ( v z i )∣∣ D w i ˜ F − a i ( v w i )∣ , where v z i and v w i are the unit tangent vector to F − [ a ] i ( U [ a ] i ( x )) at z i and F − [ a ] i ( U [ a ] i ( x )) at w i ,respectively (see Figure 2). For proving (13), it is needed to show that n ∑ i = log ∣ D z i ˜ F − a i ( v z i )∣ − log ∣ D w i ˜ F − a i ( v w i )∣ is uniformly bounded. By the H¨older continuity of logarithm, there exists C > C n ∑ i = ∣ D z i ˜ F − a i ( v z i ) − D w i ˜ F − ( v w i )∣ ≤ C n ∑ i = ∣ D z i ˜ F − a i ( v z i − v w i )∣ + C n ∑ i = ∣ D z i ˜ F − a i − D w i ˜ F − a i ∣∣ v w i ∣ . Since ∣ v w i ∣ =
1, using the mean value theorem, the second term on the right is less thanmax z ∈ ˜ S ∣ D ˜ F − a i ∣ n ∑ i = ∣ z i − w i ∣ ≤ max z ∈ ˜ S ∣ D ˜ F − a i ∣ ∞ ∑ i = ∣ z i − w i ∣ , which is bounded, independent of n , since ∣ z i − w i ∣ ≤ const. d u ( z i , w i ) ≤ const. K − i , where d u is themetric induced by the Riemannian metric on the unstable curve. Hence, it is suffices to show that n ∑ i = ∣ D z i ˜ F − a i ( v z i − v w i )∣ is bounded and the bound is independent of n .In these affine coordinates, one has that v z i = ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ . Also, for sufficiently large i , v w i ∈ C sC , where C is from Lemma 4.5. Let v w i = ( u ,i , u ,i ) and D w i − ˜ F − a i − ( v w i − ) = ( ξ i , η i ) . So, ξ i = ˜ g x ( w i − ) u ,i − + ˜ g y ( w i − ) u ,i − η i = ˜ g x ( w i − ) u ,i − + ˜ g y ( w i − ) u ,i − . Since ∣ D w i − ˜ F − ( v w i − )∣ = ∣ η i ∣ , u ,i = ξ i /∣ η i ∣ and u ,i =
1. Then, by (9), one has that ∣ D z i ˜ F − ( v z i − v w i )∣ = ∣ ˜ g x ( z i )∣ ∣ ξ i ∣∣ η i ∣ ≤ C ∣ ξ i ∣∣ η i ∣ . Now, we try to bound the sum of the last fraction. By (11), for sufficiently large i , one has that ∣ η i ∣ = ∣ ˜ g x ( w i − ) u ,i − + ˜ g y ( w i − ) u ,i − ∣ = ∣ ˜ g y ( w i − )∣ ∣ ˜ g x ( w i − ) ˜ g y ( w i − ) u ,i − u ,i − + ∣ ≥ ∣ ˜ g y ( w i − )∣( − C ) . So, ∣ u ,i ∣ = ∣ ξ i ∣∣ η i ∣ ≤ − C ( ∣ ˜ g x ( w i − )∣∣ ˜ g y ( w i − )∣ ∣ u ,i − ∣ + ∣ ˜ g y ( w i − )∣∣ ˜ g y ( w i − )∣ ) . On the other hand, by (11) and equality ˜ g y ( z i − ) =
0, for sufficiently large i there exists τ i − suchthat ∣ ˜ g y ( w i − )∣∣ ˜ g y ( w i − )∣ ≤ ∣ ˜ g yx ( τ i − )∣∣ ˜ g y ( w i − )∣ ∣ w i − − z i − ∣ + ∣ ˜ g yy ( τ i − )∣∣ ˜ g y ( w i − )∣ ∣ w i − − z i − ∣≤ C max τ ∈ ˜ S ∣ D ˜ F − ( τ )∣∣ w i − − z i − ∣ . Then, by (12) and inequality ∣ z i − w i ∣ ≤ const. K − i , one has that ∣ ξ i ∣∣ η i ∣ ≤ ( − C ) K ∣ u ,i − ∣ + ˜ C ( K ) i − . Supposing inductively that ∣ u ,i − ∣ ≤ C ( K ) i − , one gets ∣ u ,i ∣ = ∣ ξ i ∣∣ η i ∣ ≤ C ( − C ) K ( K ) n − + ˜ C ( K ) i − . Assuming that C < and K >
3, one gets ∣ u ,i ∣ ≤ C ( K ) i − . Therefore, n ∑ i = ∣ D z i ˜ F − ( v z i − v w i )∣ ≤ C n ∑ i = ∣ u ,i ∣ < C ˜ C ∞ ∑ i = ( K ) i − . The last sum converges and this finishes the proof of Proposition 4.4. (cid:3)
Remark 4.6.
Fix two finite words [ a ] n and [ b ] m , ● there exists a constant K such that the components of ˆ U [ a ] n ( x ) ∖ U [ a ] n ( x ) have length notsmaller than K ⋅ diam ˆ U [ a ] n ( x ) , ● there exists a constant K such that K − ≤ d ([ a ] n [ b ] m ) d ([ a ] n ) d ([ b ] m ) ≤ K . The first part of the remark above is a simple application of the classical bounded distortionproperty. The second part follows from Proposition 4.4 and the fact that F [ a ] n [ b ] m = F [ b ] m ○ F [ a ] n .Let B sr ( z ) = { w ∈ W s ( z ) ∣ d s ( z, w ) < r } for r ∈ ( , +∞) . Combining Proposition 4.4 and Remark4.6, one gets the following corollary: Corollary 4.7. If r < K K − d ([ a ] n ) and B sr ( z ) intersects U [ a ] n then z ∈ ˆ U [ a ] n . BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 13
Proof of Theorem A.
The analytical approach in [13] needs the explicit formula of F andit is not applicable here, although the sufficient condition for the SRB measure µ F to be absolutelycontinuous prepared in [13] is still the key tool to prove Theorem A.For the SRB measure µ F , there exist probability measures µ x along the stable manifolds W s ( x ) which we may write µ F = ∫ µ x dµ g ( x ) . Tsujii proved the following remarkable proposition in [13].
Proposition 4.8.
For a positive real number r , let ∥ µ x ∥ r = ∫ R ( µ x ( B sr ( z ))) dz and I ( r ) = r − ∫ ∥ µ x ∥ r dx. If lim inf r → I ( r ) < ∞ then µ F is absolutely continuous with respect to the Lebesgue measure andits density function is square integrable. We use the geometric approach of Rams in [8] in order to prove that the SRB measure is anACIP. Denote the inverse branches of g by g i for i ∈ N and let g [ a ] n ∶= g a ○ ⋯ ○ g a n . Since µ g isequivalent to the Lebesgue measure on [ , ] , there exist positive constants l and L such that l < dµ g dx < L, so clearly the following inequalities hold(14) lL diam I [ a ] n ≤ ddx g [ a ] n ≤ Ll diam I [ a ] n and(15) diam I [ a ] n [ b ] m ≤ Ll diam I [ a ] n diam I [ b ] m . Remark 4.9. ● For any two δ -transversal finite words [ a ] n and [ b ] m
1. any two subword [ a ] ij = ( a i , . . . , a j ) and [ b ] kl = ( b k , . . . , b l ) are δ -transversal,2. if a ≠ b , then vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) ≤ δ − d ([ a ] n ) d ([ b ] m ) . See Figure 3. ● For any < c < c , ∑ [ a ] n ; c < d ([ a ] n )< c diam I [ a ] n ≤ + log c − log c log m , where m is the infimum of g ′ on ⋃ i ∈ N I i . The inequality holds since for an interval of length c , there are at most ( log c − log c )/ log m interval of length c . Now, we are prepared to prove Theorem A. S a S [ a ] n U [ a ] n U a n Figure 1.
Stable and Unstable Strips in a Generalized Horseshoe Map z i w i v w i v z i F − [ a ] i ( γ ) Figure 2.
Position of z i , w i , v z i and v w i ˆ U [ a ] n ˆ U [ b ] m Figure 3.
Transversality of ˆ U [ a ] n and ˆ U [ b ] m Figure 3.
Transversality of ˆ U [ a ] n and ˆ U [ b ] m Proof of Theorem A.
For any r < diam J , let N ( r ) ∈ N be the biggest number such that for any i ∈ { , . . . , N ( r )} , inf z ∈ S i D s F ( z ) > r . For any a ∈ Σ ∞ N ( r ) = {( a i ) ∞ i = ∣ a i ∈ { , . . . , N ( r )}} , let cylinder Z [ a ] n be the set of all words in Σ ∞ N ( r ) that begin with [ a ] n . Then, the cylinders { Z [ a ] n ; [ a ] n ∈ M ( r )} form a finite disjoint cover of Σ ∞ N ( r ) , and(16) lim r → ∑ [ a ] n ∈ M ( r ) diam I [ a ] n = . Since µ F is invariant, for any x ∈ [ , ] , z ∈ W s ( x ) and r > µ x ( B sr ( z )) = lim n →∞ ∑ [ a ] n ; B sr ( z )∩ U [ a ] n ( x )≠∅ y ∈ I [ a ] n ,g n ( y )= x µ y ( F − [ a ] n ( B sr ( z )))( g n ) ′ ( y ) ≤ lim n →∞ ∑ [ a ] n ; B sr ( z )∩ U [ a ] n ( x )≠∅ diam I [ a ] n . Fix a small positive r and let R = K − K r . According to Corollary 4.7, (14) and (17), one has that µ x ( B sr ( z )) ≤ L ∑ [ a ] n ∈ M ( R ) ; z ∈ ˆ U [ a ] n ( x ) diam I [ a ] n . Then ∥ µ x ∥ r ≤ L ∑ [ a ] n ∈ M ( R ) ∑ [ b ] m ∈ M ( R ) diam ( ˆ U [ a ] n ( x ) ∩ ˆ U [ b ] m ( x )) diam I [ a ] n diam I [ b ] m and(18) I ( r ) ≤ r − L ∑ [ a ] n ∈ M ( R ) ∑ [ b ] m ∈ M ( R ) vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) diam I [ a ] n diam I [ b ] m . Now, for δ > I T rδ ( r ) = r − L ∑ ([ a ] n , [ b ] m )∈ M Trδ ( R ) vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) diam I [ a ] n diam I [ b ] m and I NT rδ ( r ) = r − L ∑ ([ a ] n , [ b ] m )∈ M NTrδ ( R ) vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) diam I [ a ] n diam I [ b ] m . By (18), I ( r ) ≤ I T rδ ( r ) + I NT rδ ( r ) . In view of the transversality condition, there is δ such that I NT rδ ( r ) < ∞ . Hence, to bound I ( r ) , it is sufficient to bound I T rδ ( r ) . Following Rams [8], for i ≥ BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 15 put(19) I i ( r ) = r − L ∑ [ a ] n ∑ [ b ] m ∑ [ c ] i vol ( ˆ U [ c ] i [ a ] n ∩ ˆ U [ c ] i [ b ] m ) diam I [ c ] i [ a ] n diam I [ c ] i [ b ] m , where the sum is taken over such words that ([ c ] i [ a ] n , [ c ] i [ b ] m ) ∈ M T rδ ( R ) and a ≠ b . So, I T rδ ( r ) = I ( r ) + ⋯ + I i ( r ) + ⋯ . According to Remark 4.9, (16) and H1 , one gives ∑ [ a ] n ∈ M ( R ) ∑ [ b ] m ∈ M ( R ) diam I [ a ] n diam I [ b ] m < , and so I ( r ) ≤ K for some constant K .Now, fix i >
0. By Remark 4.6 and the definition of M ( R ) , one has that(20) RK d ([ c ] i ) ≤ d ([ a ] n ) ≤ K Rd ([ c ] i ) . The above inequality also holds for d ([ b ] m ) . Hence(21) K − ≤ d ([ a ] n ) d ([ b ] m ) ≤ K From (15), (19) and (20), we have I i ( r ) ≤ ∑ [ a ] n ∑ [ b ] m ∑ [ c ] i L K K − K l − diam I [ a ] n diam I [ b ] m ( diam I [ c ] i ) d ([ a ] n ) d ([ b ] m )( d ([ c ] i )) vol ( ˆ U [ c ] i [ a ] n ∩ ˆ U [ c ] i [ b ] m ) . For fixed [ a ] n and [ b ] m , one has that ⋃ [ c ] i ( ˆ U [ c ] i [ a ] n ∩ ˆ U [ c ] i [ b ] m ) = ⋃ [ c ] i F [ c ] i ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) , since the contraction of each map F [ c ] i along the vertical direction is at least K d ([ c ] i )/ diam J times and its expansion along the horizontal direction is at most Ll − / diam I [ c ] p . Therefore(22) ∑ [ c ] i diam I [ c ] i d ([ c ] i ) vol ( ˆ U [ c ] i [ a ] n ∩ ˆ U [ c ] i [ b ] m ) ≤ K Ll diam J vol ( ˆ U [ a ] n ∩ ˆ U [ b ] m ) . So (22) and Lemma 4.9 imply that I i ( r ) ≤ ∑ [ a ] n ∑ [ b ] m L K K δ l K diam J diam I [ a ] n diam I [ b ] m sup diam I [ c ] i d ([ c ] i ) . Let η i = inf z ∈⋃ N ( R ) j = S j D s F [ c ] i ( z ) and η i = sup z ∈⋃ N ( R ) j = S j D s F [ c ] i ( z ) . By bounded distortion property, Proposition 4.3, we have K − ≤ η i / η i ≤ K (where K is independentof the choice of [ c ] i ). So by (20) and the fact that η i diam J ≤ d ([ c ] i ) ≤ η i diam J , and Remark 4.9,we get ∑ [ c ] i [ a ] n ∈ M ( R ) for some [ c ] i diam ( I [ a ] n ) = + K + log η i − log η i log m ≤ + K + log K log m , and the same holds for I [ b ] m . Therefore I i ( r ) ≤ L K K δ l K diam J ( + K + log K log m ) sup diam I [ c ] i d ([ c ] i ) . The fatness condition implies thatdiam I [ c ] i d ([ c ] i ) ≤ K ( d ([ c ] i )) (cid:15) ≤ K ( diam J ) (cid:15) M i(cid:15) , where M = sup z ∈ Λ D s F ( z ) <
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BSOLUTELY CONTINUOUS INVARIANT MEASURE FOR GENERALIZED HORSESHOE MAPS 17
Department of Mathematics, Shahid Beheshti university, Tehran 19839, Iran
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